Intuition For The Law Of Cosines

The Law of Cosines is presented as a geometric result that relates the parts of a triangle:

\displaystyle{ c^2 = a^2 + b^2 - 2ab\cos(C) }

While true, there’s a deeper principle at work.

The Law of Interactions: The whole is based on the parts and the interaction between them.

The wording “Law of Cosines” gets you thinking about the mechanics of the formula, not what it means. Part of my learning strategy is rewording ideas into ones that make sense.

  • The Law of Cosines, after cranking through geometric steps we’re prone to forget, looks like c2 = a2 + b2 – 2abcos(C).

  • This is suspiciously like the expansion that if c = (a + b), then c2 = a2 + b2 + 2ab

  • The difference is that 2ab has an extra factor, cos(C), which measures the “actual overlap percentage” (2ab assumes we fully overlap, i.e. where cos(C) = 1).

  • So, the Law of Cosines is really a generalization of how c2 = (a + b)2 expands when components aren’t fully lined up. We’re treating geometric lines as terms in an algebraic expansion.

Analogy: The Assistant Chef

Imagine a restaurant with a single chef, Alice. She’s overworked, so Bob is hired as her assistant (sous chef).

Based on Alice’s current performance, and Bob’s performance in his interview, what happens when they work together?

Surely the new result must be their combined effort:

\displaystyle{ \text{Total Contribution} = \text{Alice's Work} + \text{Bob's Work}}

Hah! Office workers everywhere are rolling their eyes. You can’t just assume people contribute identically when they’re put together: there are interactions to account for.

Beyond their individual contributions, the two might slow each other down (Where’d you put the whisk again?), or find ways to work together (I’m peeling carrots anyway, use some of mine.).

In a system with several parts, start with the individual contributions and then ask if their interaction will:

  • Help each other
  • Hurt each other
  • Ignore each other

The original idea that “Total = Alice + Bob” is more generally expressed as:

\displaystyle{ \text{Total Contribution} = \text{Alice's Work} + \text{Bob's Work} + \text{Interaction effects} }

Exploring The Scenario

We need to separate the list of participants (Alice, Bob) from the result of their interaction.

Take the numbers 5 and 3. We can write them like so:

  • Parts = (5, 3)

and we’re pretty sure they combine to make 8. But is there another way to get that conclusion?

Yes: we multiply. Beyond repeated counting, multiplication shows what happens when the parts of a system interact:

\displaystyle{(5 + 3)(5 + 3) = 5^2 + (5)(3) + (3)(5) + 3^2 = 25 + 15 + 15 + 9 = 64 }

We’ve gone from “parts view”, (5, 3), to “interaction view”, (5 + 3)2. The result of interaction mode says the system would result in 64 if it did interact with itself.

One caveat: when going to interaction view, we wrote down (5 + 3)(5 + 3), but we can’t simplify (5 + 3) = 8 on the outset. We’re using addition for bookkeeping until multiplication can combine the parts.

Oh, another caveat: why can we just add the interactions, but not the parts? Great question. The individual parts might be pointing in different dimensions, and don’t line up nicely on the same scale. The interacting parts turn into area, which can be combined to the same result no matter the orientation.

(I’ll investigate this concept more in a follow-up. It’s a neat idea that area is a generic, easily combinable quantity but individual paths are not.)

Generalizing the Principle

Simple setups like (5, 3) are easy to think through, like eyeballing 2x + 3 = 7 and guessing x = 2. But a more complex scenario like x2 + 3x = 15 requires a systematic approach.

The Law of Cosines is a systematic approach to working through the parts:

  1. List the parts
  2. Get every interaction as area
  3. Add to find the total contribution
  4. Convert into the equivalent “single part”

The last step is often implied. Once we’ve merged the jumble of interactions, we want the single part that could represent the entire system. Is there a single person (Charlie) whose efforts are identical to that of Alice and Bob working together?

The Law of Cosines gives us a way to find Charlie.

What’s the Deal with Cosine?

When two parts interact, they can help, hurt, or ignore each other:

  • Perfect alignment means they help 100% (5 and 3)
  • Perfect mis-alignment means they hurt 100% (5 and -3)
  • Partial alignment or mis-alignment means they help or hurt by a percentage
  • No alignment means they ignore each other

How do we measure alignment? With cosine.

Using our trig analogy, cosine is the percentage an angle moves along the ground.

A 0-degree angle follows the ground perfectly (100%), and moving vertically doesn’t follow it at all (0%). Other angles are a fraction in-between.

If the parts in our system can be written as paths, and we know the angle between them is theta (theta), then we can measure the overlap with cosine. One path acts as the ground, and the other is the path we’re following:

\displaystyle{\text{Overlap percentage} = \cos(\text{angle between them}) }

\displaystyle{\text{Actual interaction} = \text{Max Interaction} \cdot \text{Overlap percentage} = ab\cos(\theta)) }

When paths are perfectly aligned, their full strength is used (ab and ba). The interaction factor cos(theta) modifies that strength to show much they actually work together.

So, our jumble of interactions becomes:

\displaystyle{\text{Overall behavior} = a^2 + b^2 + ab\cos(\theta) + ba\cos(\theta) }

\displaystyle{\text{Overall behavior} = a^2 + b^2 + 2ab\cos(\theta)}

\displaystyle{\text{Single part} = \sqrt{ a^2 + b^2 + 2ab\cos(\theta)}}

Phew! And that’s the Law of Cosines: collect every interaction, account for the alignment, and simplify it to a single part. (The formula is usually written without the square root, but usually you want c, not c2.)

Now, why is the Law of Cosines often written with a negative sign? Well, the assumption is that in a typical triangle, a small internal angle C means the sides are negatively aligned, while theta (theta) is an external look at their alignment:

Similarly, a large internal angle means the sides are positively aligned, and will help each other. Typically, a small angle means you’re moving in the same direction, but this internal/external difference means we reverse the sign.

Personally, I don’t memorize whether there’s a positive or negative sign: I think about whether the parts will help or hurt each other in the scenario, and make the interaction positive or negative. Don’t be a slave to the formula.

Quick Practice Problem

Let’s say my triangle has side a = 10 and side b = 20. What is side c when the angle between a and b is:

45 degrees in alignment

Here, we need the Law of Cosines. a and b are pointing partially in the same direction. We switch to interaction mode to get to a common, combinable unit (area):

  • a2 = 100
  • b2 = 400
  • 2ab = 2 · 10 · 20 = 400, but we need to adjust by the interaction factor. That is cos(45) = .707, so the real interaction factor is 400 · .707 = 282.8

The overall interactions are:

\displaystyle{100 + 400 + 282.8 = 782.8}

and the equivalent single side (c) is:

\displaystyle{ \sqrt{782.8} = 27.97 \text{cm} }

70 degrees in mis-alignment

Again, we need the Law of Cosines. We can see that the angles fight each other, so the interaction will be negative:

\displaystyle{\text{total interaction} = a^2 + b^2 - 2AB\cos(\theta) = 100 + 400 - (2)(10)(20)\cos(70) = 363.19} \displaystyle{ c = \sqrt{636.8} = 19.05 \text{cm} }

Our intuition says this arrangement should be smaller than the previous one (since the sides aren’t working together), and it is.

Full alignment or mis-alignment

When our “triangle” has an angle of 0 degrees (or 180), all the parts are lying flat. Here, the parts are in the same dimension, and can be treated as regular numbers:

  • Fully aligned: 10 + 20 = 30
  • Fully mis-aligned: 10 – 20 = -10 (pointing in direction of B).

The Law of Cosines still works, of course:

  • Full alignment: a2 + b2 + 2abcos(theta) = 100 + 400 + 400cos(0) = 900 and c = √(900) = 30
  • Full mis-alignment: a2 + b2 – 2abcos(theta) = 100 + 400 + 400cos(180) = 100 which means c = √(100) = 10 (pointing backwards).

Again, we shouldn’t robotically follow the formula: have a rough idea what the result should be, and think through the calculations. (“The overall interaction is this, so the individual side would that…”).

Thinking of interactions is one interpretation: next time, we’ll see it as the Law of Projections.

Happy math.

Appendix: Pythagorean Theorem

The Law of Cosines resembles the Pythagorean Theorem, no?

Now you might suspect why. The Pythagorean Theorem is the special case of zero interaction, which happens when the sides are at right angles. After all, 90 degree angle is vertical, and has 0% overlap with the ground.

The Law of Cosines becomes:

\displaystyle{ c^2 = a^2 + b^2 + interaction }

\displaystyle{ c^2 = a^2 + b^2 + 0}

If we know the parts won’t interact, we can ignore interaction effects. However, the self-interactions are still there and must be combined: a2 and b2 are fine, but the crossover terms ab and ba disappear.

Here’s another version of the Pythagorean Theorem. We can’t combine a and b directly, so combine their interactions and reduce them to a single part:

\displaystyle{c = \sqrt{a^2 + b^2} }

Appendix: The Geometric Proof

You might be hankering for a geometric proof. Here’s one from quora, based on a paper by Knuth:

The insight is that we take our original a-b-c triangle and scale it by a (giving the a2-ab-ac triangle) and b (giving the ab-b2-bc triangle). These two triangles build a larger, similar triangle ac-bc-c2, and with some trig, the bottom portion can be shown to equal a2 + b2 – 2abcos(theta).

While interesting, I don’t like these types of proofs up front. The Law of Cosines is about interactions, not re-arranging triangles. Does this explanation get you thinking about what cosine represents? About when it should be positive, negative, or zero?

ADEPT Summary

Concept Law of Cosines
Analogy Imagine an assistant chef whose interactions may (or may not) be helpful.
Example Suppose a = 10 and b = 20 in a triangle. If they are aligned 45-degrees, their interaction is a2 + b2 + 2abcos(45) = 782.8 and the remaining side is √(782.8) = 27.97 units long.
Plain-English The Law of Interactions: The whole is based on the parts and the interaction between them.
Technical Triangle with internal angle C: c2 = a2 + b2 – 2abcos(C)
General interaction: c2 = a2 + b2 + 2abcos(theta)

Intuition For The Law Of Sines

The Law Of Sines is something I memorized in a class once, but didn’t internalize:

\displaystyle{\frac{\sin(A)}{a} = \frac{\sin(B)}{b} = \frac{\sin(C)}{c} }

Ok, that’s a neat connection, and maybe we can prove it by drawing some right triangles (of course) and re-arranging terms.

But what does it mean?

Rather than the Law of Sines, think of the Law of Equal Perspectives:

Each angle & side can independently find the circle that wraps up the whole triangle. This connection lets us start with one angle and work out facts about the others.

Analogy: Kids Describing A Monster

I occasionally frighten the neighborhood children by unchaining the mutant gorilla in my front yard.

The kids run screaming, telling different stories of what they’ve seen:

“Alice claims the monster was 20 feet tall, but we all know she exaggerates by doubling. And Billy’s a bit of a crybaby, and said it was 30 feet tall. Charlie’s fairly no-nonsense and said the beast was exactly 10 feet high.”

If we know a kid’s “exaggeration factor” and the size they claim, we can deduce the true size of the monster. (Furious George has a name, you know.)

Even better, we can predict what other kids might have said: If Alice claimed it was 40 feet, what would Charlie have said?

Triangles And The Monster Circle

What do kids running from monsters have to do with triangles? Well, every triangle is trapped inside its own Monster Circle:

Whatever triangle we draw, there’s some circle trying to gobble it up (technically, “circumscribe it”). Try this page to explore an example on your own.

Now here’s the magic: just knowing a single angle and its corresponding side, we can figure out the Monster Circle.

Here’s how. Let’s say we have a triangle like this:

We don’t know anything except the angle A (call it 30 degrees) and the length of side a (call it an inch).

First off: is this the correct drawing of the triangle? Probably not! We don’t know the other sides, so this is equally valid:

It still has the same angle (A = 30 degrees) and the size of the base hasn’t changed (still one inch).

What if we start drawing more possibilities?

Whoa. From A‘s point of view all the possible triangles that have “A=30 degrees, a=1 inch” are on this circle. Whatever B and C end up being, they need to pick an option from this circle.

Similarly, we can argue this from the other perspectives:

  • We can lock down angle B and side b, and trace out a circle of possibilities
  • We can lock down angle C and side c, and trace out a circle of possibilities

This is the meaning of the Law of Sines: each angle unknowingly generates the same circle as the others.

(How do we prove, not just see that the possibilities lie on a circle? That’s the Inscribed Angle Theorem, for another day.)

Calculating The Actual Size

We’ve figured out that there is a Monster Circle, now let’s see how big it is. Um… how?

Remember, we can slide around the circle and keep A (30 degrees) and a (1 inch) the same. So let’s slide until we make a right triangle:

Ah! Now we can use sine. Remember that sine is the percentage height compared to the max possible. The max possible height is the full diameter (d) of the Monster Circle.

(Why is a 90-degree angle across from the full diameter? Draw a square inside the circle, touching the sides. It must be symmetric, the diagonals pass through the center along the diameter, and are opposite a 90-degree angle.)

With a little re-arranging, we get:

\displaystyle{ \frac{a}{\sin(A)} = d }

Using the same logic for the other sides, we get:

\displaystyle{ \frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)} = d }

In a way, sin(A) is the “exaggeration factor” that converts the size the angle measured (a) to the full diameter (d). Each angle is a different kid, and some really misjudge the size of the full circle based on what they see. (90-degrees is right on target.)

Practice Problem

In our example above, A is 30 degrees and a is 1 inch.

We can calculate the diameter pretty fast. First, we get the sine:

\displaystyle{\sin(30) = 0.5}

That means our length a is 50% of the max height, so the full diameter must be 2 inches.

This isn’t enough to figure out the triangle by itself. Let’s say angle B comes along and says it is 45 degrees. How long is b?


\displaystyle{sin(45) = .707}

which means that b is .707 of the max diameter. Therefore,

\displaystyle{ b = .707 \cdot \text{2 inches} =  \text{1.414 inches}}

Previously, I would plug numbers into the Law of Sines formula and chug away algebraically. Now I can think in terms of the Monster Circle: “Ok, I have the max diameter. I take the sine, and get the fraction of the max diameter for that side.”

Most books write the formula with sin(A) in the numerator. It might read better “Sine A over A” but it distorts the conclusion that frac(a)(sin(A)) is the size of the circle.

Put the concept in your own words. The “Law of Sines” is a generic description of what’s in the formula, but the “Law of Equal Perspectives” explains what it means:

  • All parts of the triangle have a perspective on the whole
  • Sine is the “exaggeration factor” that scales up an individual side to the full diameter. (Sine is the percentage of the max possible, and we divide by it.)

Happy math.

Appendix: Obtuse Angles

Technically, because B is over 90 degrees, we can’t ever spin it and have either A or C be a right angle (if we could, the triangle would have over 180 degrees).

What to do? Realize the 180-degree complement of B (call it ) acts like a stand-in on the other side:

has the same sine as B, which should make sense: they both point upwards along the same trajectory. To help us sleep better at night, we start with in the right-angle setup:

\displaystyle{\sin(B*) = \sin(B) = \frac{b}{d} }

and get to the same conclusion as before. Phew.

However, the fact that B and can be swapped can lead to problems.

If I have a triangle where I know A (30 degrees) and a (1 inch), and then say b is 1.5 inches, what can you deduce?

The max diameter is 2 inches as before, so

\displaystyle{\sin(B) = \frac{1.5}{2} = .75}

Unfortunately, there are two angles with that sine value: a calculator says sin-1(.75) = 48 degrees, but 180 – 48 = 132 degrees would work too (more details).

Also, the triangle may not be possible given a hypothetical scenario. If I say b is 3 inches, you know something’s amiss. The max diameter was already calculated to be 2. Even a 90-degree angle, the best possible, could only have a side of 2 inches.

ADEPT Summary

ADEPT Topic Law of Sines
Analogy Imagine kids describing the same monster with varying degrees of exaggeration.
Example Suppose A=30 and a=1 inch. Since sin(A) = 0.5, the Monster Circle is 1 / 0.5 = 2 inches wide. Given another angle, I can figure out the length of its side. If B = 45 degrees, then side b takes up sin(45) = .707 of the diameter, and is 1.414 inches.
Plain-English Any angle + side can deduce the size of the wrapping circle.
Technical \displaystyle{\frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)} = \text{diameter of circle} }

Learn Difficult Concepts with the ADEPT Method

After a decade of writing explanations, I’ve simplified the strategy I use to get new concepts to click.

Make explanations ADEPT: Use an Analogy, Diagram, Example, Plain-English description, and then a Technical description.

ADEPT method of learning

Here’s how to teach yourself a difficult idea, or explain one to others.

Analogy: What Else Is It Like?

Most new concepts are variations, extensions, or combinations of what we already know. So start there!

In our decades of life, we’ve encountered thousands of objects and experiences. Surely one of them is vaguely similar to this new topic and can be the starting point.

Here’s an example: Imaginary numbers. Most lessons introduce them in a void, simply saying “negative numbers can have square roots too.”

Argh. How about this:

  • Negative numbers were distrusted until the 1700s: How could you have less than nothing?
  • We overcame this by realizing numbers could exist on a number line, allowing us to move forward or backward from zero.
  • Imaginary numbers express the idea that we can move upwards and downwards, or rotate around the number line.

Instead of just going East/West, we can go North/South too – or even spin around in a circle. Neat!

Analogies are fuzzy, not 100% accurate, and yet astoundingly useful. They’re a raft to get across the river, and leave behind once you’ve crossed.

Diagram: Engage That Half Of Your Brain

We often think diagrams are a crutch if you aren’t macho enough to directly interpret the symbols. Guess what? Academic progress on imaginary numbers took off only after the diagrams were made!

Favor the easiest-to-absorb explanation, whether that comes from text, diagram, or interpretative dance. From there, we can work to untangle the symbols.

So, here’s a visualization:

imaginary numbers

Imaginary numbers let us rotate around the number line, not just move side-to-side.

Starting to get a visceral sense for what they can do, right?

Half our brain is dedicated to vision processing, so let’s use it. (And hey, maybe for this topic, twirling around in an interpretative dance would help.)

Example: Let Me Experience The Idea

Oh, now’s our chance to hit the student with the fancy terminology, right?

Nope. Don’t tell someone the way things are: let them experience it. (How fun is hearing about the great dinner I had last night? The movie you didn’t get to see?)

But that’s what we do for math. “Someone smarter than you thought this through, found out all the cool connections, and labeled the pieces. Memorize what they discovered.”

That’s no fun: let people make progress themselves. Using the rotation analogy, what happens after 4 turns?

How about 2 turns? 4 turns clockwise?

Plain-English Description: Use Your Own Words

If you genuinely experienced an idea, you should be excited to describe it:

  • Imaginary numbers seem to point North, and we can get to them with a single clockwise turn.
  • Oh! I guess they can point South too, by turning the other way.
  • 4 turns gets us pointing in the positive direction again
  • It seems like two turns points us backwards

These are all correct conclusions, just not yet written in the language of math. But you can still reason in plain English!

Technical Description: Learn The Formalities

The final step is to convert our personal understanding to the formal notation. It’s like sharing a song you’ve made up: you can hum it to yourself, but need sheet music for other people to use.

Math is the sheet music we’ve agreed upon to share ideas. So, here’s the technical terminology:

  • We say i (lowercase) is 1.0 in the imaginary dimension
  • Multiplying by i is a 90-degree counter-clockwise turn, to face “up” (here’s why). Multiplying by -i points us South
  • It’s true that starting at 1.0 and taking 4 turns puts us at our starting point:

\displaystyle{1 * i * i * i * i = 1 }

And two turns points us negative:

\displaystyle{1 * i * i = -1 }

which simplifies to:

\displaystyle{i^2 = -1}


\displaystyle{i = \sqrt{-1}}

In other words, i is “halfway” to -1. (Square roots find the halfway point when using multiplication.)

Starting to get a feel for it? Just spitting out “i is the square root of -1″ isn’t helpful. It’s not explaining, it’s telling. Nothing was experienced, nothing was internalized.

Give people the chance to make an idea their own.

The Mental Checklist

I used to be satisfied with a technical description and practice problem. Not anymore.

ADEPT is a checklist of what I need to feel comfortable with an idea. I don’t think I’ve actually learned a topic unless I have a metaphor that ties everything together. Here’s a few places to look:

Unfortunately, there aren’t many resources focused on analogies, especially for math, so you have to make your own. (This site exists to share mine.)

Modifying the Learning Order

It seems logical to assume we can present facts in order, like transmitting data to a computer. But who actually learns like that?

I prefer the blurry-to-sharp approach to teaching:

Start with a rough analogy and sharpen it until you’re covering the technical details.

Sometimes, you need to untangle a technical description on your own, so must work backwards to the analogy.

Starting with the technical details:

  • Can you explain them in your own words?
  • Can you solve an example problem, describing the steps in your own words?
  • Can you create a diagram that represents how the concept fits together for you?
  • Can you relate the concept to what you already know?

With this initial analogy, layer in new details and examples, and see if it holds up. (It doesn’t need to be perfect, but iterate.)

If we’re honest, we’ll admit that we forget 95% of what we learn in a class. What sticks? A scattered analogy or diagram. So, make them for yourself, to bootstrap the rest of the understanding as needed.

In a year, you probably won’t remember much about imaginary numbers. But the quick analogy of “rotation” or “spinning” might trigger a flurry of recognition.

The Goal: Explanations That Actually Work

I’m wary of making a contrived acronym, but ADEPT does capture what I need to internalize a new concept. Let’s stop being shy about thinking out loud: does a fact-only presentation really work for you? What other components do you need? I have a soft, squishy brain that needs the connecting glue, not just data.

Scott Young uses the Feynman Technique to explain concepts in everyday words and work backwards to an analogy and diagram. (Richard Feynman was a world-class expositor and physicist, and one of my teaching heroes.)

Prof. Barb Oakley runs an excellent, free course on Learning How To Learn. I was honored to do an interview with her for the class:

Click to watch the interview — I recommend the full course. The first session had over 180,000 students and was a great success.

Beyond any technique, raise your standards to find (or create) explanations that truly work for you. It’s the only way to have concepts stick.

Happy math.


“BE” is a nice prefix for the style to use when teaching:

  • Brevity is beautiful.

  • Empathy makes us human. Use your natural style, relate to common experience, and anticipate questions in your explanation.

I’ve yet to complain that a lesson respected my time too much, or related too well to how I thought.

Appendix: ADEPT Summaries

ADEPT is like a nutrition label for an explanation: what are the key ingredients?

Concept Euler’s Formula
Analogy Imaginary numbers spin exponential growth into a circle.
Example Let’s figure out the value of 3^i. (It’s on the unit circle.)
Plain-English Raising an exponent to an imaginary power spins you on the unit circle. The same destination can be written with polar (distance and angle) or rectangular coordinates (real part and imaginary part).
Technical \displaystyle{e^{ix} = \cos(x) + i\sin(x)}

Concept Fourier Transform
Analogy Like filtering a smoothie into ingredients, the Fourier Transform extracts the circular paths within a pattern.
Diagram Smoothie being filtered:
Example Split the sequence (4 0 0 0) into circular components:
Plain-English / Technical

Concept Distributed Version Control
Analogy Distributed Version Control is like sharing changes to a group shopping list with your friends.
Diagram / Example
Plain-English We check out, check in, branch, and share differences (“diffs”).
Technical git checkout -b branchname
git diff branchname

Combine ingredients with your own style. Steps might merge, but shouldn’t be skipped without a good reason (“Zombies coming, no time for biochem, use this serum for the cure.”). The site cheatsheet has a large collection of analogies.

Learning math? Think like a cartoonist.


What’s the essential skill of a cartoonist? Drawing ability? Humor? A deep well of childhood trauma?

I’d say it’s an eye for simplification, capturing the essence of an idea.

For example, let’s say we want to understand Ed O’Neill:

A literal-minded artist might portray him like this:

While the technical skill is impressive, does it really capture the essence of the man? Look at his eyes in particular.

A cartoonist might draw this:

Wow! The cartoonist recognizes:

  • The unique shape of his head. Technically, his head is an oval, like yours. But somehow, making his jaw wider than the rest of his head is perfect.

  • The wide-eyed bewilderment. The whites of his eyes, the raised brows, the pursed lips – the cartoonist saw and amplified the emotion inside.

So, who really “gets it”? It seems the technical artist worries more about the shading of his eyes than the message they contain.

Numbers Began With Cartoons

Think about the first numbers, the tally system:


Those are… drawings! Cartoons! Caricatures of an idea!

They capture the essence of “existing” or “having something” without the specifics of what it represents.

Og the Cavemen Accountant might have tried drawing individual stick figures, buffalos, trees, and so on. Eventually he might realize a shortcut: draw one buffalo symbol to show the type, then a line for each item. This captures the essence of “something is there” and our imaginations do the rest.

Math is an ongoing process of simplifying ideas to their cartoon essence. Even the beloved equals sign (=) started as a drawing of two identical lines, and now we can write “3 + 5 = 8″ instead of “three plus five is equal to eight”. Much better, right?

So let’s be cartoonists, seeing an idea — really capturing it — without getting trapped in technical mimicry. Perfect reproductions come in after we’ve seen the essence.

Technically Correct: The Worst Kind Of Correct

We agree that multiplication makes things bigger, right?

Ok. Pick your favorite number. Now, multiply it by a random number. What happens?

  • If that random number is negative, your number goes negative
  • If that random number is between 0 and 1, your number is destroyed or gets smaller
  • If that random number is greater than 1, your number will get larger

Hrm. It seems multiplication is more likely to reduce a number. Maybe we should teach kids “Multiplication generally reduces the original number.” It’ll save them from making mistakes later.

No! It’s a technically correct and real-life-ily horrible way to teach, and will confuse them more. If the technically correct behavior of multiplication is misleading, can you imagine what happens when we study the formal definitions of more advanced math?

There’s a fear that without every detail up front, people get the wrong impression. I’d argue people get the wrong impression because you provide every detail up front.

As George Box wrote, “All models are wrong, but some are useful.”

A knowingly-limited understanding (“Multiplication makes things bigger”) is the foothold to reach a more nuanced understanding. (“People generally multiply positive numbers greater than 1, so multiplication makes things larger. Let’s practice. Later, we’ll explore what happens if numbers are negative, or less than one.”)


I wrap my head around math concepts by reducing them to their simplified essence:

  • Imaginary numbers let us rotate numbers. Don’t start by defining i as the square root of -1. Show how if negative numbers represent a 180-degree rotation, imaginary numbers represent a 90-degree one.

  • The number e is a little machine that grows as fast as it can. Don’t start with some arcane technical definition based on limits. Show what happens when we compound interest with increasing frequency.

  • The Pythagorean Theorem explains how all shapes behave (not just triangles). Don’t whip out a geometric proof specific to triangles. See what circles, squares, and triangles have in common, and show that the idea works for any shape.

  • Euler’s Formula makes a circular path. Don’t start by analyzing sine and cosine. See how exponents and imaginary numbers create “continuous rotation”, i.e. a circle.

Avoid the trap of the guilty expert, pushed to describe every detail with photorealism. Be the cartoonist who seeks the exaggerated, oversimplified, and yet accurate truth of the idea.

Happy math.

PS. Here’s my cheatsheet full of “cartoonified” descriptions of math ideas.

How To Think With Exponents And Logarithms

Here’s a trick for thinking through problems involving exponents and logs. Just ask two questions:

Are we talking about inputs (cause of the change) or outputs (the actual change that happened?)

  • Logarithms reveal the inputs that caused the growth
  • Exponents find the final result of growth

Are we talking about the grower’s perspective, or an observer’s?

  • e and the natural log are from the grower’s instant-by-instant perspective
  • Base 10, Base 2, etc. are measurements convenient for a human observer

In my head, I put the options in a table:

exponent points of view

I have thoughts like “I need the cause, from the grower’s perspective… that’s the natural log.”. (Natural log is abbreviated with lowercase LN, from the high-falutin’ logarithmus naturalis.)

I was frustrated with classes that described the inner part of the table, the raw functions, without the captions that explained when to use them!

That won’t fly, let’s get direct practice thinking with logs and exponents.

Scenario: Describing GDP Growth

Here’s a typical example of growth:

  • From 2000 to 2010, the US GDP changed from 9.9 trillion to 14.4 trillion

Ok, sure, those numbers show change happened. But we probably want insight into the cause: What average annual growth rate would account for this change?

Immediately, my brain thinks “logarithms” because we’re working backwards from the growth to the rate that caused it. I start with a thought like this:

\displaystyle{\text{logarithm of change} \rightarrow \text{cause of growth} }

A good start, but let’s sharpen it up.

First, which logarithm should we use?

By default, I pick the natural logarithm. Most events end up being in terms of the grower (not observer), and I like “riding along” with the growing element to visualize what’s happening. (Radians are similar: they measure angles in terms of the mover.)

Next question: what change do we apply the logarithm to?

We’re really just interested in the ratio between start and finish: 9.9 trillion to 14.4 trillion in 10 years. This is the same growth rate as going from $9.90 to $14.40 in the same period.

We can sharpen our thought:

\displaystyle{\text{natural logarithm of growth ratio} \rightarrow \text{cause of growth} }

\displaystyle{\ln(\frac{14.4}{9.9}) = .374}

Ok, the cause was a rate of .374 or 37.4%. Are we done?

Not yet. Logarithms don’t know about how long a change took (we didn’t plug in 10 years, right?). They give us a rate as if all the change happened in a single time period.

The change could indeed be a single year of 37.4% continuous growth, or 2 years of 18.7% growth, or some other combination.

From the scenario, we know the change took 10 years, so the rate must have been:

\displaystyle{ \text{rate} = \frac{.374}{10} = .0374 = 3.74\%}

From the viewpoint of instant, continuous growth, the US economy grew by 3.74% per year.

Are we done now? Not quite!

This continuous rate is from the grower’s perspective, as if we’re “riding along” with the economy as it changes. A banker probably cares about the human-friendly, year-over-year difference. We can figure this out by letting the continuous growth run for a year:

\displaystyle{\text{exponent with rate \& time} \rightarrow \text{effect of growth} }

\displaystyle{e^{\text{rate} \cdot \text{time}} = \text{growth}}

\displaystyle{e^{.0374 \cdot 1} = 1.0381}

The year-over-year gain is 3.8%, slightly higher than the 3.74% instantaneous rate due to compounding. Here’s another way to put it:

  • From an instant-by-instant basis, a given part of the economy is growing by 3.74%, modeled by e.0374 · years
  • On a year-by-year basis, with compounding effects worked out, the economy grows by 3.81%, modeled by 1.0381years

In finance, we may want the year-over-year change which can be compared nicely with other trends. In science and engineering, we prefer modeling behavior on an instantaneous basis.

Scenario: Describing Natural Growth

I detest contrived examples like “Assume bacteria doubles every 24 hours, find its growth formula.”. Do bacteria colonies replicate on clean human intervals, and do we wait around for an exact doubling?

A better scenario: “Hey, I found some bacteria, waited an hour, and the lump grew from 2.3 grams to 2.32 grams. I’m going to lunch now. Figure out how much we’ll have when I’m back in 3 hours.”

Let’s model this. We’ll need a logarithm to find the growth rate, and then an exponent to project that growth forward. Like before, let’s keep everything in terms of the natural log to start.

The growth factor is:

\displaystyle{\text{logarithm of change} \rightarrow \text{cause of growth} }

\displaystyle{\ln(\text{growth}) = \ln(2.32/2.3) = .0086 = .86\%}

That’s the rate for one hour, and the general model to project forward will be

\displaystyle{\text{exponent with rate \& time} \rightarrow \text{effect of growth} }

\displaystyle{e^{.0086 \cdot \text{hours}} \rightarrow \text{effect of growth} }

If we start with 2.32 and grow for 3 hours we’ll have:

\displaystyle{2.32 \cdot e^{.0086 \cdot 3} = 2.38}

Just for fun, how long until the bacteria doubles? Imagine waiting for 1 to turn to 2:

\displaystyle{1 \cdot e^{.0086 \cdot \text{hours}} = 2}

We can mechanically take the natural log of both sides to “undo the exponent”, but let’s think intuitively.

If 2 is the final result, then ln(2) is the growth input that got us there (some rate × time). We know the rate was .0086, so the time to get to 2 would be:

\displaystyle{ \text{hours} = \frac{\ln(2)}{\text{rate}} = \frac{.693}{.0086} = 80.58}

The colony will double after ~80 hours. (Glad you didn’t stick around?)

What Does The Perspective Change Really Mean?

Figuring out whether you want the input (cause of growth) or output (result of growth) is pretty straightforward. But how do you visualize the grower’s perspective?

Imagine we have little workers who are building the final growth pattern (see the article on exponents):

compound interest

If our growth rate is 100%, we’re telling our initial worker (Mr. Blue) to work steadily and create a 100% copy of himself by the end of the year. If we follow him day-by-day, we see he does finish a 100% copy of himself (Mr. Green) at the end of the year.

But… that worker he was building (Mr. Green) starts working as well. If Mr. Green first appears at the 6-month mark, he has a half-year to work (same annual rate as Mr. Blue) and he builds Mr. Red. Of course, Mr. Red ends up being half done, since Mr. Green only has 6 months.

What if Mr. Green showed up after 4 months? A month? A day? A second? If workers begin growing immediately, we get the instant-by-instant curve defined by ex:

continuous growth

The natural log gives a growth rate in terms of an individual worker’s perspective. We plug that rate into ex to find the final result, with all compounding included.

Using Other Bases

Switching to another type of logarithm (base 10, base 2, etc.) means we’re looking for some pattern in the overall growth, not what the individual worker is doing.

Each logarithm asks a question when seeing a change:

  • Log base e: What was the instantaneous rate followed by each worker?
  • Log base 2: How many doublings were required?
  • Log base 10: How many 10x-ings were required?

Here’s a scenario to analyze:

  • Over 30 years, the transistor counts on typical chips went from 1000 to 1 billion

How would you analyze this?

  • Microchips aren’t a single entity that grow smoothly over time. They’re separate editions, from competing companies, and indicate a general tech trend.
  • Since we’re not “riding along” with an expanding microchip, let’s use a scale made for human convenience. Doubling is easier to think about than 10x-ing.

With these assumptions we get:

\displaystyle{\text{logarithm of change} \rightarrow \text{cause of growth} }

\displaystyle{\log_2(\frac{\text{1 billion}}{1000}) = \log_2(\text{1 million}) \sim \text{20 doublings}}

The “cause of growth” was 20 doublings, which we know occurred over 30 years. This averages 2/3 doublings per year, or 1.5 years per doubling — a nice rule of thumb.

From the grower’s perspective, we’d compute ln(text(1 billion)/1000) / text(30 years) = 46% continuous growth (a bit harder to relate to in this scenario).

We can summarize our analysis in a table:


Learning is about finding the hidden captions behind a concept. When is it used? What point view does it bring to the problem?

My current interpretation is that exponents ask about cause vs. effect and grower vs. observer. But we’re never done; part of the fun is seeing how we can recaption old concepts.

Happy math.

Appendix: The Change Of Base Formula

Here’s how to think about switching bases. Assuming a 100% continuous growth rate,

  • ln(x) is the time to grow to x
  • ln(2) is the time to grow to 2

Since we have the time to double, we can see how many would “fit” in the total time to grow to x:

\displaystyle{\text{number of doublings from 1 to x} = \frac{\ln(x)}{\ln(2)} = \log_2(x)}

For example, how many doublings occur from 1 to 64?

Well, ln(64) = 4.158. And ln(2) = .693. The number of doublings that fit is:

\displaystyle{\frac{\ln(64)}{\ln(2)} = \frac{4.158}{.693} = 6}

In the real world, calculators may lose precision, so use a direct log base 2 function if possible. And of course, we can have a fractional number: Getting from 1 to the square root of 2 is “half” a doubling, or log2(1.414) = 0.5.

Changing to log base 10 means we’re counting the number of 10x-ings that fit:

\displaystyle{\text{number of 10x-ings from 1 to x} = \frac{\ln(x)}{\ln(10)} = \log_{10}(x) }

Neat, right? Read Using Logarithms in the Real World for more examples.

Understand Ratios with “Oomph” and “Often”

Ratios summarize a scenario with a number, such as “income per day”. Unfortunately, this hides the explanation for how the result came about.

For example, look at two businesses:

  • Annie’s Art Gallery sells a single, $1000 piece every day
  • Frank’s Fish Emporium sells 250 trout at $4/each every day

By the numbers, they’re identical $1000/day operations, right? Hah.

Here’s how each business actually behaves:

\displaystyle{\mathit{\frac{Dollars}{Day} = \frac{Dollars}{Transaction} \cdot \frac{Transactions}{Day} }}

Transactions are the workhorse that drive income, but they’re lost in the dollars/day description. When studying an idea, separate the results into Oomph and Often:

\displaystyle{\mathit{ Result = Oomph \cdot Often = \frac{Dollars}{Transaction} \cdot \frac{Transactions}{Day} }}

With Oomph and Often, I visualize two distinct levers to increase. A ratio like dollars/day makes me stumble through thoughts like: “For better results, I need 1/day to improve… which means the day gets shorter… How’s that possible? Oh, that must be the portion of the day used for each transaction…”.

Why make it difficult? Rewrite the ratio to include the root case: What’s the Oomph, and how Often does it happen?

Horsepower, Torque, RPM

In physics, we define everyday concepts like “power” with a formal ratio:

\displaystyle{\mathit{ Power = \frac{Work}{Time} }}

Ok. Power can be explained by a ratio, but we’re already in inverted-thinking mode. Just another hassle when exploring an already-tricky concept.

How about this:

\displaystyle{\mathit{ Power = Oomph \cdot Often }}

Easier, I think. What could Oomph and Often mean?

Well, Oomph is probably the work we do (such as moving a weight) and Often is how frequently we do it (how many reps did you put in?).

In the same minute, suppose Frank lifted 100lbs ten times, while Annie lifted 1000lbs once. From the equation, they have the same power (though to be honest, I’m more frightened by Annie.)

An engine mechanic might internalize power like this:

\displaystyle{\mathit{ Power = \frac{Work}{Revolution} \cdot \frac{Revolutions}{Time} }}

\displaystyle{\mathit{ Horsepower = Torque \cdot RPM }}

What does that mean?

  • Torque is the Oomph, or how much weight (and how far) can be moved by a turn of the engine (i.e., moving 500lbs by 1 foot)

  • RPM (revolutions per minute) is how frequently the engine turns

A motorcycle engine is designed for reps, i.e. spinning the wheels quickly. It doesn’t need much torque — just enough to pull itself and a few passengers — but it needs to send that to the wheels again and again.

A bulldozer is designed for “Oomph”, such as knocking over a wall. We don’t need to tap into that work very frequently, as one destroyed wall per minute is great, thanks.

I’m not a physicist or car guy, but I can at least conceptualize the tradeoffs with the Oomph/Often metaphor.

Gears can change the tradeoff between Oomph and Often in a given engine. If you’re going uphill, fighting gravity, what do you want more of? If you’re cruising on a highway? Trying to start from a standstill? Driving over slippery snow? Lost the brakes and need to slow down the car?

Oomph/Often gets me thinking intuitively, Work/Time does not.

Variation: Electric Power

Electric power has the same ratio as mechanical power:

\displaystyle{\mathit{ Electric \ Power = \frac{Work}{Time} }}

Yikes. It’s not clear what this means. How about:

\displaystyle{\mathit{ Electric \ Power = Oomph \cdot Often }}

It’s hard to have ideas out of the blue, but we might imagine something (a mini-engine?) is moving the Oomph around inside the wire. If we call it a “charge” then we have:

\displaystyle{\mathit{ Electric \ Power = \frac{Work}{Charge} \cdot \frac{Charges}{Time} }}

And we can give those subparts formal names:

  • Voltage (Oomph): How much work each charge contributes

  • Current (Often): How quickly charges are moving through the wire

Now we get the familiar:

\displaystyle{\mathit{ Electric \ Power = Voltage \cdot Current }}

Boomshakala! I don’t have a good intuition for electricity, at least my goal is clear: find analogies where voltage means Oomph, and current means Often.

And still, we can take a crack at intuitive thinking: when you get zapped by a doorknob in winter, was that Oomph or Often? What attribute should batteries maximize? What’s better for moving energy through stubborn power lines? (Vive la résistance!)

The ratios think every type of power reduces to a generic Work/Time calculation. The Oomph/Often metaphor gets us thinking about Torque/RPM in one scenario and Voltage/Current in another.

What’s Really Going On? Parameters, Baby.

The Oomph/Often viewpoint lets us think about the true cause of the ratio. Instead of dollars and days, we wonder how the actual transactions affect the outcome:

  • Can we increase the size of each transaction?

  • Can we increase the number each day?

In formal terms, we’ve introduced a new parameter to explain the interaction. To change a ratio from a/b to one parameterized by x, we can do:

\displaystyle{\frac{a}{b} = \frac{(a/x)}{(b/x)} = (a/x) \cdot \frac{1}{(b/x)} = \frac{a}{x} \cdot \frac{x}{b} }

We change our viewpoint to see x as the key component. In math, we often switch viewpoints to simplify problems:

  • Instead of asking what happens to the observer, can we change parameters and ask what the mover sees? (Degrees vs. radians.)

  • Can we see a giant function as being parameterized by smaller ones? (See the chain rule.)

  • Can we express probabilities as odds, instead of percentages? (It makes Bayes Theorem easier.)

Adjusting parameters is a way to morph an idea that doesn’t click into one that does. Since I don’t naturally think with inverted units, I’ve made it easier on myself: deal with two multiplications, instead of a division.

Happy math.

How To Learn Trigonometry Intuitively

Trig mnemonics like SOH-CAH-TOA focus on computations, not concepts:

body proportions

TOA explains the tangent about as well as x2 + y2 = r2 describes a circle. Sure, if you’re a math robot, an equation is enough. The rest of us, with organic brains half-dedicated to vision processing, seem to enjoy imagery. And “TOA” evokes the stunning beauty of an abstract ratio.

I think you deserve better, and here’s what made trig click for me.

  • Visualize a dome, a wall, and a ceiling
  • Trig functions are percentages to the three shapes

Motivation: Trig Is Anatomy

Imagine Bob The Alien visits Earth to study our species.

Without new words, humans are hard to describe: “There’s a sphere at the top, which gets scratched occasionally” or “Two elongated cylinders appear to provide locomotion”.

After creating specific terms for anatomy, Bob might jot down typical body proportions:

  • The armspan (fingertip to fingertip) is approximately the height
  • A head is 5 eye-widths wide
  • Adults are 8 head-heights tall

body proportions

How is this helpful?

Well, when Bob finds a jacket, he can pick it up, stretch out the arms, and estimate the owner’s height. And head size. And eye width. One fact is linked to a variety of conclusions.

Even better, human biology explains human thinking. Tables have legs, organizations have heads, crime bosses have muscle. Our biology offers ready-made analogies that appear in man-made creations.

Now the plot twist: you are Bob the alien, studying creatures in math-land!

Generic words like “triangle” aren’t overly useful. But labeling sine, cosine, and hypotenuse helps us notice deeper connections. And scholars might study haversine, exsecant and gamsin, like biologists who find a link between your fibia and clavicle.

And because triangles show up in circles…

body proportions

…and circles appear in cycles, our triangle terminology helps describe repeating patterns!

Trig is the anatomy book for “math-made” objects. If we can find a metaphorical triangle, we’ll get an armada of conclusions for free.

Sine/Cosine: The Dome

Instead of staring at triangles by themselves, like a caveman frozen in ice, imagine them in a scenario, hunting that mammoth.

Pretend you’re in the middle of your dome, about to hang up a movie screen. You point to some angle “x”, and that’s where the screen will hang.

Trig dome

The angle you point at determines:

  • sine(x) = sin(x) = height of the screen, hanging like a sign
  • cosine(x) = cos(x) = distance to the screen along the ground [“cos” ~ how “close”]
  • the hypotenuse, the distance to the top of the screen, is always the same

Want the biggest screen possible? Point straight up. It’s at the center, on top of your head, but it’s big dagnabbit.

Want the screen the furthest away? Sure. Point straight across, 0 degrees. The screen has “0 height” at this position, and it’s far away, like you asked.

The height and distance move in opposite directions: bring the screen closer, and it gets taller.

Tip: Trig Values Are Percentages

Nobody ever told me in my years of schooling: sine and cosine are percentages. They vary from +100% to 0 to -100%, or max positive to nothing to max negative.

Let’s say I paid $14 in tax. You have no idea if that’s expensive. But if I say I paid 95% in tax, you know I’m getting ripped off.

An absolute height isn’t helpful, but if your sine value is .95, I know you’re almost at the top of your dome. Pretty soon you’ll hit the max, then start coming down again.

How do we compute the percentage? Simple: divide the current value by the maximum possible (the radius of the dome, aka the hypotenuse).

That’s why we’re told “Sine = Opposite / Hypotenuse”. It’s to get a percentage! A better wording is “Sine is your height, as a percentage of the hypotenuse”. (Sine becomes negative if your angle points “underground”. Cosine becomes negative when your angle points backwards.)

Let’s simplify the calculation by assuming we’re on the unit circle (radius 1). Now we can skip the division and just say sine = height.

Every circle is really the unit circle, scaled up or down to a different size. So work out the connections on the unit circle and apply the results to your particular scenario.

Try it out: plug in an angle and see what percent of the height and width it reaches:

The growth pattern of sine isn’t an even line. The first 45 degrees cover 70% of the height, and the final 10 degrees (from 80 to 90) only cover 2%.

This should make sense: at 0 degrees, you’re moving nearly vertical, but as you get to the top of the dome, your height changes level off.

Tangent/Secant: The Wall

One day your neighbor puts up a wall right next to your dome. Ack, your view! Your resale value!

But can we make the best of a bad situation?

Trig dome

Sure. What if we hang our movie screen on the wall? You point at an angle (x) and figure out:

  • tangent(x) = tan(x) = height of screen on the wall
  • distance to screen: 1 (the screen is always the same distance along the ground, right?)
  • secant(x) = sec(x) = the “ladder distance” to the screen

We have some fancy new vocab terms. Imagine seeing the Vitruvian “TAN GENTleman” projected on the wall. You climb the ladder, making sure you can “SEE, CAN’T you?”. (Yeah, he’s naked… won’t forget the analogy now, will you?)

Let’s notice a few things about tangent, the height of the screen.

  • It starts at 0, and goes infinitely high. You can keep pointing higher and higher on the wall, to get an infinitely large screen! (That’ll cost ya.)

  • Tangent is just a bigger version of sine! It’s never smaller, and while sine “tops off” as the dome curves in, tangent keeps growing.

How about secant, the ladder distance?

  • Secant starts at 1 (ladder on the floor to the wall) and grows from there
  • Secant is always longer than tangent. The leaning ladder used to put up the screen must be longer than the screen itself, right? (At enormous sizes, when the ladder is nearly vertical, they’re close. But secant is always a smidge longer.)

Remember, the values are percentages. If you’re pointing at a 50-degree angle, tan(50) = 1.19. Your screen is 19% larger than the distance to the wall (the radius of the dome).

(Plug in x=0 and check your intuition that tan(0) = 0, and sec(0) = 1.)

Cotangent/Cosecant: The Ceiling

Amazingly enough, your neighbor now decides to build a ceiling on top of your dome, far into the horizon. (What’s with this guy? Oh, the naked-man-on-my-wall incident…)

Well, time to build a ramp to the ceiling, and have a little chit chat. You pick an angle to build and work out:

Trig dome

  • cotangent(x) = cot(x) = how far the ceiling extends before we connect
  • cosecant(x) = csc(x) = how long we walk on the ramp
  • the vertical distance traversed is always 1

Tangent/secant describe the wall, and COtangent and COsecant describe the ceiling.

Our intuitive facts are similar:

  • If you pick an angle of 0, your ramp is flat (infinite) and never reachers the ceiling. Bummer.
  • The shortest “ramp” is when you point 90-degrees straight up. The cotangent is 0 (we didn’t move along the ceiling) and the cosecant is 1 (the “ramp length” is at the minimum).

Visualize The Connections

A short time ago I had zero “intuitive conclusions” about the cosecant. But with the dome/wall/ceiling metaphor, here’s what we see:

Trig overall

Whoa, it’s the same triangle, just scaled to reach the wall and ceiling. We have vertical parts (sine, tangent), horizontal parts (cosine, cotangent), and “hypotenuses” (secant, cosecant). (Note: the labels show where each item “goes up to”. Cosecant is the full distance from you to the ceiling.)

Now the magic. The triangles have similar facts:

Trig identities

From the Pythagorean Theorem (a2 + b2 = c2) we see how the sides of each triangle are linked.

And from similarity, ratios like “height to width” must be the same for these triangles. (Intuition: step away from a big triangle. Now it looks smaller in your field of view, but the internal ratios couldn’t have changed.)

This is how we find out “sine/cosine = tangent/1″.

I’d always tried to memorize these facts, when they just jump out at us when visualized. SOH-CAH-TOA is a nice shortcut, but get a real understanding first!

Gotcha: Remember Other Angles

Psst… don’t over-focus on a single diagram, thinking tangent is always smaller than 1. If we increase the angle, we reach the ceiling before the wall:

Trig alternative

The Pythagorean/similarity connections are always true, but the relative sizes can vary.

(But, you might notice that sine and cosine are always smallest, or tied, since they’re trapped inside the dome. Nice!)

Summary: What Should We Remember?

For most of us, I’d say this is enough:

  • Trig explains the anatomy of “math-made” objects, such as circles and repeating cycles
  • The dome/wall/ceiling analogy shows the connections between the trig functions
  • Trig functions return percentages, that we apply to our specific scenario

You don’t need to memorize 12 + cot2 = csc2, except for silly tests that mistake trivia for understanding. In that case, take a minute to draw the dome/wall/ceiling diagram, fill in the labels (a tan gentleman you can see, can’t you?), and create a cheatsheet for yourself.

In a follow-up, we’ll learn about graphing, complements, and using Euler’s Formula to find even more connections.

Appendix: The Original Definition Of Tangent

You may see tangent defined as the length of the tangent line from the circle to the x-axis (geometry buffs can work this out).


As expected, at the top of the circle (x=90) the tangent line can never reach the x-axis and is infinitely long.

I like this intuition because it helps us remember the name “tangent”, and here’s a nice interactive trig guide to explore:

Trig interactive

Still, it’s critical to put the tangent vertical and recognize it’s just sine projected on the back wall (along with the other triangle connections).

Appendix: Inverse Functions

Trig functions take an angle and return a percentage. sin(30) = .5 means a 30-degree angle is 50% of the max height.

The inverse trig functions let us work backwards, and are written sin-1 or arcsin (“arcsine”), and often written asin in various programming languages.

If our height is 25% of the dome, what’s our angle?

Now what about something exotic, like inverse secant? Often times it’s not available as a calculator function (even the one I built, sigh).

Looking at our trig cheatsheet, we find an easy ratio where we can compare secant to 1. For example, secant to 1 (hypotenuse to horizontal) is the same as 1 to cosine:

\displaystyle{\frac{sec}{1} = \frac{1}{cos}}

Suppose our secant is 3.5, i.e. 350% of the radius of the unit circle. What’s the angle to the wall?

\frac{\sec}{1} &= \frac{1}{\cos} = 3.5 \\
\cos &= \frac{1}{3.5} \\
\arccos(\frac{1}{3.5}) &= 73.4

Appendix: A Few Examples

Example: Find the sine of angle x.

Sine Example

Ack, what a boring question. Instead of “find the sine” think, “What’s the height as a percentage of the max (the hypotenuse)?”.

First, notice the triangle is “backwards”. That’s ok. It still has a height, in green.

What’s the max height? By the Pythagorean theorem, we know

3^2 + 4^2 &= \text{hypotenuse}^2 \\
25 &= \text{hypotenuse}^2 \\
5 &= \text{hypotenuse}

Ok! The sine is the height as a percentage of the max, which is 3/5 or .60.

Follow-up: Find the angle.

Of course. We have a few ways. Now that we know sine = .60, we can just do:

\displaystyle{\asin(.60) = 36.9}

Here’s another approach. Instead of using sine, notice the triangle is “up against the wall”, so tangent is an option. The height is 3, the distance to the wall is 4, so the tangent height is 3/4 or 75%. We can use arctangent to turn the percentage back into an angle:

\displaystyle{\tan = \frac{3}{4} = .75 }

\displaystyle{\atan(.75) = 36.9}

Example: Can you make it to shore?

Boat Example

You’re on a boat with enough fuel to sail 2 miles. You’re currently .25 miles from shore. What’s the largest angle you could use and still reach land? Also, the only reference available is Hubert’s Compendium of Arccosines, 3rd Ed. (Truly, a hellish voyage.)

Ok. Here, we can visualize the beach as the “wall” and the “ladder distance” to the wall is the secant.

First, we need to normalize everything in terms of percentages. We have 2 / .25 = 8 “hypotenuse units” worth of fuel. So, the largest secant we could allow is 8 times the distance to the wall.

We’d like to ask “What angle has a secant of 8?”. But we can’t, since we only have a book of arccosines.

We use our cheatsheet diagram to relate secant to cosine: Ah, I see that “sec/1 = 1/cos”, so

\sec &= \frac{1}{\cos} = 8 \\
\cos &= \frac{1}{8} \\
\arccos(\frac{1}{8}) &= 82.8

A secant of 8 implies a cosine of 1/8. The angle with a cosine of 1/8 is arccos(1/8) = 82.8 degrees, the largest we can afford.

Not too bad, right? Before the dome/wall/ceiling analogy, I’d be drowning in a mess of computations. Visualizing the scenario makes it simple, even fun, to see which trig buddy can help us out.

In your problem, think: am I interested in the dome (sin/cos), the wall (tan/sec), or the ceiling (cot/csc)?

Happy math.

Update: The owner of Grey Matters put together interactive diagrams for the analogies (drag the slider on the left to change the angle):



Site Update: New Design + Intuition Cheatsheet

After months of work with the help of Neil, a great designer, and my Excel-blogging friend Andrew, I’m happy to launch a brand-new design.

My goals were to be friendly, readable, and easy-to-navigate. Here’s a quick before-and-after:

New Logo

Neil did a fantastic job here — I’d been looking for a way to convey a welcoming, conversational tone.

New Homepage

A site about explanations should describe what it does simply, right?

Better Readability

The fonts are bumped up, there’s more breathing room, and pages are optimized for iPads/iPhones. Instead of a text-dense cram session, I want an unhurried walkthrough of insights.

Intuition Cheatsheet

My favorite feature is a site summary that reduces insights to a few words. Previously, I had trouble navigating the various articles, and I bet you did too :). Readers of the newsletter got a sneak peek, and I have a PDF version I’ll be sending out to subscribers as well.

Overall, BetterExplained is an excited friend who shares what really helps ideas click, not an authority trying to be the grand poombah of math. Let’s have a good time on this journey of learning.

Hope you enjoy the new site, feedback is welcome!


A Quick Intuition For Parametric Equations

Algebra is really about relationships. How are things connected? Do they move together, or apart, or maybe they’re completely independent?

Normal equations assume an “input to output” connection. That is, we take an input (x=3), plug it into the relationship (y=x2), and observe the result (y=9).

But is that the only way to see a scenario? The setup y=x2 implies that y only moves because of x. But it could be that y just coincidentally equals x2, and some hidden factor is changing them both (the factor changes x to 3 while also changing y to 9).

As a real world example: For every degree above 70, our convenience store sells x bottles of sunscreen and x2 pints of ice cream.

We could write the algebra relationship like this:

\displaystyle{ice \ cream = (sunscreen)^2}

And it’s correct… but misleading!

The equation implies sunscreen directly changes the demand for ice cream, when it’s the hidden variable (temperature) that changed them both!

It’s much better to write two separate equations

\displaystyle{sunscreen = temperature - 70}

\displaystyle{ice \ cream = (temperature - 70)^2}

that directly point out the causality. The ideas “temperature impacts ice cream” and “temperature impacts sunscreen” clarify the situation, and we lose information by trying to factor away the common “temperature” portion. Parametric equations get us closer to the real-world relationship.

parametric steps

Don’t Think About Time. Just Look for Root Causes.

A reader pointed out that nearly every parametric equation tutorial uses time as its example parameter. We get so hammered with “parametric equations involve time” that we forget the key insight: parameters point to the cause. Why did we change? (Maybe it was time, or temperature, or perhaps sunscreen really does make you hungry for ice cream.)

Most algebraic equations lay out a connection like y = x2. Parametric equations remind us to look deeper (lost on me until recently; I’d been stuck in the “time/physics” mindset).

Sure, not every setup has a hidden parameter, but isn’t it worth a look?

BetterExplained Calculus Course Now Available (Public Beta)

Hi all! I’m happy to announce a public availability of the BetterExplained Guide To Calculus. You can read it online:

And here’s a peek at the first lesson:


(Like the new look? I’ve been working with a great designer and will be refreshing the main site too.)

The goal is an intuition-first look at a notoriously gnarly subject. This isn’t a replacement for a stodgy textbook — it’s the friendly introduction I wish I’d had. A few hours of reading that would have saved me years of frustration.

The course text is free online, with a complete edition available, which includes:

  • Course Text
  • Video walkthroughs for each lesson
  • Per-lesson class discussions
  • PowerPoint files for all diagrams
  • Quizzes to check understanding (In development)
  • Print-friendly PDF ebook (In development)
  • Invitations to class webinars (In development)

Buy The Beta Course $99 149

The price of the course will increase when the final version is released, so hop onto the beta to snag the lower price.

Building A Course: Lessons Learned

A few insights jumped out while making the course. This may be helpful if you’re considering teaching a course one day (I hope you are).

Incentives Matter

I struggled with what to make free vs. paid. I love sharing insights with people… and I also love knowing I can do so until I’m an old man, complaining that newfangled brain-chip implants aren’t “real learning”.

Incentives always exist. I want to make education projects sustainable, designed to satisfy readers, not a 3rd party.

Similar to the fantastic Rails Tutorial Book, the course text is free, with extra resources available. Having the core material free with paid variations & guidance helps align my need to create, share, and be sustainable.

Being Focused Matters

Historically, I’m lucky to write an article a month. But this summer, I wrote 16 lessons in 6 weeks. What was the difference?

Well, pressure from friends, for one: I’d promised to do a calculus course this summer. But mostly, it was the focus of having a single topic, brainstorming on numerous analogies/examples, and carving a rough path through on a schedule (2-3 articles/week).

I hope this doesn’t sound disciplined, because I’m not. A combination of fear (I told people I’d do this) and frustration (Argh, I remember being a student and not having things click) pushed me. When I finished, I took a break from writing and vegged out for a few weeks. But I think it was a worthwhile trade — in my mind, a year’s worth of material was ready.

Fundamentals Matter

There’s many options for making a course. Modules. Quizzes. Interactive displays. Tribal dance routines. Hundreds of tools to convey your message.

And… what is that message, anyway? Are we transmitting facts, or building insight?

Until the fundamentals are working, the fancy dance routine seems useless. I’d rather read genuine insights from a pizza box than have an interactive hologram that that recites a boring lecture.

When lessons are lightweight and easy to update, you’re excited about feedback (Oh yeah! A chance to make it better!).

The more static the medium, the more you fear feedback (Oh no, I have to redo it?). A fixed medium has its place, ideally after a solid foundation has been mapped out.

I’ll be polishing the course in the coming weeks, feedback is welcome!


It’s Time For An Intuition-First Calculus Course

Summary: I’m building a calculus course from the ground-up focused on permanent intuition, not the cram-test-forget cycle we’ve come to expect.

Update: The course is now live at

The Problem: We Never Internalized Calculus

First off: what’s wrong with how calculus is taught today? (Ha!)

Just look at the results. The vast majority of survivors, the STEM folks who used calculus in several classes, have no lasting intuition. We memorized procedures, applied them to pre-packaged problems (“Say, fellow, what is the derivative of x2?”), and internalized nothing.

Want proof? No problem. Take a string and wrap it tight around a quarter. Take another string and wrap it tight around the Earth.

Ok. Now, lengthen both strings, adding more to the ends, so there’s a 1-inch gap all the way around around the quarter, and a 1-inch gap all the way around the Earth (sort of like having a ring floating around Saturn). Got it?

Quiz time: Which scenario uses more extra string? Does it take more additional string to put a 1-inch gap around the quarter, or to put a 1-inch gap around the Earth?

Think about it. Ponder it over. Ready? It’s the… same. The same! Adding a 1-inch gap around the Earth, and a quarter, uses the same 6.28 inches.

And to be blunt: if you “learned” calculus but didn’t have the answer within 3 seconds, you don’t truly know it. At least not deep down.

Now don’t feel bad, I didn’t know it either. Only one engineer in the dozens I’ve asked came up with the answer instantly, without second-guesses (my karate teacher, Mr. Rose).

This question has a few levels of understanding:

  • Algebra Robot: Calculating change in circumference: 2*pi*(r + 1) - 2*pi*r = 2*pi. They are the same. Calculation complete.

  • Calculus Disciple: Oh! We know circumference = 2*pi*r. The derivative is 2*pi, a constant, which means the current radius has no impact on a changing circumference.

  • Calculus Zen master: I see the true nature of things. We’re changing a 1-dimensional radius and watching a 1-dimensional perimeter. A dimension in, a dimension out, it’s like making a fence 1-foot longer: the initial size doesn’t change the work needed. The gap could be made around a circle, square, rectangle, or Richard Nixon mask, and it’s the same effort for similar shapes. (And, silly me, I’d forgotten the equation for circumference anyway!)

We can be calculus warrior-monks, cutting through problems with our intuition. Notice how the most advanced approach didn’t need specific equations — it was just thinking about the problem! Equations are nice tools, but are they your only source of understanding?

See, according to standardized tests and final exams, I “knew” calculus — but clearly only to the beginning level. I didn’t immediately recognize how calculus could help with a question about making a string longer. If you asked someone for the amount of cash in a wallet with six $20 bills, and they didn’t think to use multiplication, would you say they’ve internalized arithmetic? (“Oh geez, you didn’t tell me this would be a multiplication question! Could you set up the problem for me?”)

I want you to have the intuition-first calculus class I never did. The goal is lasting intuition, shared by an excited friend, and built with the test of “If you haven’t internalized the idea, the material must change.”

How Can We Make Learning Intuitive And Interesting?

With Progressive Refinement. You may have seen these two methods to download and display an image:

  • Baseline Rendering: Download it start-to-finish in full detail
  • Progressive Rendering: Download a blurry version, and gradually refine it

baseline vs progressive

Teaching a subject is similar:

  • Baseline Teaching: Cover individual concepts in full-depth, one after another
  • Progressive Teaching: See the big picture, how the whole fits together, then sharpen the detail

The “start-to-finish” approach seems official. Orderly. Rigorous. And it doesn’t work.

What, exactly, do you know when you’ve seen the first 20% of a portrait in full resolution? A forehead? Do you even know the gender? The age? The teacher has forgotten that you’ve never seen the full picture and likely can’t appreciate that you’re even seeing a forehead!

Progressive rendering (blurry-to-sharp) gives a full overview, a rough approximation of what the expert sees, and gets you curious about more. After the overview, we start filling in the details. And because you have an idea of where you’re going, you’re excited to learn. What’s better: “Let’s download the next 10% of the forehead”, or “Let’s sharpen the picture”?

Let’s admit it: we forget the details of most classes. If we’ll have a hazy memory anyway, shouldn’t it be of the entire picture? That has the best shot of enticing us to sharpen the details later on.

How Do We Know If A Lesson Is Any Good?

With the Pizza Box Test. Imagine you pass a dumpster while walking home. You see a message scrawled on a discarded pizza box. Is the note so insightful and compelling that you’d take the pizza box home to finish reading it?

Ignore the sparkle of a lesson being digital, mobile-friendly, gamified, interactive, or a gesture-based hologram. Would you take this lesson home if it were written on a pizza box?

If yes, great! Clean it up and add in the glitz. But if the core lesson is not compelling without the trimmings, it must be redone.

Everyone’s “pizza box” standard varies; just have one. Here’s a few things I wish were written on the boxes outside my high school:

  • Psst! Think of e as a universal component in all growth rates, just like pi is a universal factor in all circles…

  • Hey buddy! Degrees are from the observer’s perspective. Radians are from the mover’s. That’s why radians are more natural. Let me show you…

  • Yo! Imaginary numbers are another dimension, and multiplication by i is a 90-degree rotation into that dimension! Two rotations and you’re facing backwards, aka -1.

How Do We Know What’s Best For The Student?

By focusing on what future-you would teach current-you.

Teachers, like all of us, face external incentives which may interfere with their goals (publish or perish, mandated curriculum, need to impress others with jargon, etc.). The test of “What would future-me teach present-me?” helps me focus on the essentials:

  • Use the shortest lessons possible. There’s no word count to meet. The same insight in fewer words is preferred.

  • Use the simplest language possible. It’s future-me talking to current-me. There’s nobody to impress here.

  • Use any analogy that’s memorable. I’m not embarrassed by “childish” analogies. If a metaphor excites me, and helps, I’m going to use it. Nyah.

  • Be a friend, not lecturer. I want a buddy, a guide who happened to experience the material before I did, not a pompous schoolmarm I can’t question.

  • Point out the naked emperor. Most calculus classes cover “limits, derivatives, integrals” in that order because… why? Limits are the most nuanced concept, invented in the mid-1800s. Were mathematicians like Newton, Leibniz, Euler, Gauss, Taylor, Fourier and Bernoulli inadequate because they didn’t use them? (Conversely: are you better than them because you do?). Most courses are too timid (or oblivious) to question the strategy of covering the most elusive, low-level topic first.

  • Learn for the long haul. The elephant in the room is that most math courses are a stepping-stone to some credential. Future-me doesn’t play that game: he only benefits when current-me permanently understands something.

Sign Up To Learn More

Let’s learn calculus intuition-first. The goal is a lasting upgrade to your intuition and storehouse of analogies. If that doesn’t happen, the course isn’t working, and it will be enhanced until it does.

Sign up for the mailing list and I’ll let you know when the course preview is ready, in November.

Happy math.

Print Edition of “Math, Better Explained” Now Available

I’m thrilled to announce the print edition of Math, Better Explained is available on Amazon:

With the magic of print-on-demand, you can order the book with overnight shipping (Amazon Prime!), and be reading full-color insights tomorrow. Yowza.

I’ve often been asked if a print version can be made, and I’m beaming to say it’s now a reality:

  • 12 chapters (~100 pages) of full-color explanations
  • Professional-quality typesetting & layout
  • Gorgeous, high-resolution text and diagrams
  • Compact, easy-to-carry size with comfortable margins (7″ x 10″)

The best part? There’s no garish marketing fluff needed by traditional books that compete for shelf space (testimonials, callouts, those can go in the Amazon description!). The book is my take on a simple, friendly presentation of the math essentials. It’s what I wish I had in high school (and college, and afterwards), and a tremendous value for the time and frustration it will save you.

Unlike a textbook you’re afraid to open, this book is meant to be accessible. Years later, flip back to that diagram that helped imaginary numbers click. Show that curious young student how the Pythagorean theorem goes way beyond triangles. Math is meant to be seen and felt, not just thought about.

The full-color format does increase the printing costs, but I wanted to share the highest-quality version I could (hey, I’m a reader too!). The introductory price (under $20) is heavily discounted and will change soon, so grab your copy today!

As always, happy math.

PS: Reviews are sincerely appreciated, and if you’re a math reviewer (or willing to be one!), contact me and I’ll get a copy your way. Thanks for your support!