Sine waves confused me. Yes, I can mumble "SOH CAH TOA" and draw lines within triangles. **But what does it mean**?

I was stuck thinking sine had to be extracted from other shapes. A quick analogy:

You: Geometry is about shapes, lines, and so on.

Alien: Oh? Can you show me a line?

You (looking around): Uh... see that brick, there? A line is one edge of that brick.

Alien: So lines are part of a shape?

You: Sort of. Yes, most shapes have lines in them. But a line is a basic concept on its own: a beam of light, a route on a map, or even--

Alien: Bricks have lines. Lines come from bricks. Bricks bricks bricks.

Most math classes are exactly this. "Circles have sine. Sine comes from circles. Circles circles circles."

Argh! No - circles are *one example* of sine. In a sentence: **Sine is a natural sway, the epitome of smoothness: it makes circles "circular" in the same way lines make squares "square".**

Let's build our intuition by seeing sine as its own shape, and *then* understand how it fits into circles and the like. Onward!

## Sine vs Lines

Remember to separate an *idea* from an *example*: squares are *examples* of lines. Sine clicked when it became its own idea, not "part of a circle."

Let's observe sine in a simulator (Email readers, you may need to open the article directly):

Hubert will give the tour:

**Click start**. Go, Hubert go! Notice that smooth back and forth motion? That's Hubert, but more importantly (sorry Hubert), that's sine! It's natural, the way springs bounce, pendulums swing, strings vibrate... and many things move.**Change "vertical" to "linear"**. Big difference -- see how the motion gets constant and robotic, like a game of pong?

Let's explore the differences with video:

**Linear motion**is constant: we go a set speed and turn around instantly. It's the unnatural motion in the robot dance (notice the linear bounce with no slowdown at 0:07, the strobing effect at :38).**Sine**changes its speed: it starts fast, slows down, stops, and speeds up again. It's the enchanting smoothness in liquid dancing (human sine wave at 0:12 and 0:23, natural bounce at :47).

Unfortunately, textbooks don't show sine with animations or dancing. No, they prefer to introduce sine with a timeline (try setting "horizontal" to "timeline"):

(source)

Egads. This is the schematic diagram we've always been shown. Does it give you the feeling of sine? Not any more than a skeleton portrays the agility of a cat. Let's watch sine move and *then* chart its course.

## The Unavoidable Circle

Circles have sine. Yes. But seeing the sine inside a circle is like getting the eggs back out of the omelette. It's all mixed together!

Let's take it slow. In the simulation, set Hubert to vertical:none and horizontal: sine*. See him wiggle sideways? That's the motion of sine. There's a small tweak: normally sine starts the cycle at the neutral midpoint and races to the max. This time, we start at the max and fall towards the midpoint. Sine that "starts at the max" is called cosine, and it's just a version of sine (like a horizontal line is a version of a vertical line).

Ok. Time for both sine waves: put vertical as "sine" and horizontal as "sine*". And... we have a circle!

A horizontal and vertical "spring" combine to give circular motion. Most textbooks draw the circle and try to extract the sine, but I prefer to build up: start with pure horizontal or vertical motion and add in the other.

## Quick Q & A

A few insights I missed when first learning sine:

**Sine really is 1-dimensional**

Sine wiggles in one dimension. Really. We often graph sine over time (so we don't write over ourselves) and sometimes the "thing" doing sine is also moving, but this is optional! A spring in one dimension is a perfectly happy sine wave.

(Source: Wikipedia, try not to get hypnotized.)

**Circles are an example of two sine waves**

Circles and squares are a combination of basic components (sines and lines). The circle is made from two connected 1-d waves, each moving the horizontal and vertical direction.

(Source http://1ucasvb.tumblr.com/)

But remember, circles aren't the *origin* of sines any more than squares are the origin of lines. They're examples, not the source.

**What do the values of sine mean?**

Sine cycles between -1 and 1. It starts at 0, grows to 1.0 (max), dives to -1.0 (min) and returns to neutral. I also see sine like a percentage, from 100% (full steam ahead) to -100% (full retreat).

**What's is the input 'x' in sin(x)?**

Tricky question. Sine is a cycle and x, the input, is **how far along we are in the cycle**.

Let's look at lines:

- You're traveling on a square. Each side takes 10 seconds.
- After 1 second, you are 10% complete on that side
- After 5 seconds, you are 50% complete
- After 10 seconds, you finished the side

Linear motion has few surprises. Now for sine (focusing on the "0 to max" cycle):

- We're traveling on a sine wave, from 0 (neutral) to 1.0 (max). This portion takes 10 seconds.
- After 5 seconds we are... 70% complete! Sine rockets out of the gate and slows down. Most of the gains are in the first 5 seconds
- It takes 5 more seconds to get from 70% to 100%. And going from 98% to 100% takes almost a full second!

Despite our initial speed, sine slows so we gently kiss the max value before turning around. This smoothness makes sine, sine.

For the geeks: Press "show stats" in the simulation. You'll see the percent complete of the total cycle, mini-cycle (0 to 1.0), and the value attained so far. Stop, step through, and switch between linear and sine motion to see the values.

Quick quiz: What's further along, 10% of a linear cycle, or 10% of a sine cycle? Sine. Remember, it barrels out of the gate at max speed. By the time sine hits 50% of the cycle, it's moving at the average speed of linear cycle, and beyond that, it goes slower (until it reaches the max and turns around).

**So x is the 'amount of your cycle'. What's the cycle?**

It depends on the context.

- Basic trig: 'x' is degrees, and a full cycle is 360 degrees
- Advanced trig: 'x' is radians (they are more natural!), and a full cycle is going around the unit circle (2*pi radians)

Play with values of x here:

But again, cycles depend on circles! Can we escape their tyranny?

## Pi without Pictures

Imagine a sightless alien who only notices shades of light and dark. Could you describe pi to it? It's hard to flicker the idea of a circle's circumference, right?

Let's step back a bit. Sine is a repeating pattern, which means it must... repeat! It goes from 0, to 1, to 0, to -1, to 0, and so on.

**Let's define pi as the time sine takes from 0 to 1 and back to 0.** Whoa! Now we're using pi without a circle too! Pi is a concept that *just happens* to show up in circles:

- Sine is a gentle back and forth rocking
- Pi is the time from neutral to max and back to neutral
- n * Pi (0 * Pi, 1 * pi, 2 * pi, and so on) are the times you are at neutral
- 2 * Pi, 4 * pi, 6 * pi, etc. are full cycles

Aha! That is why pi appears in so many formulas! Pi doesn't "belong" to circles any more than 0 and 1 do -- **pi is about sine returning to center**! A circle is an *example* of a shape that repeats and returns to center every 2*pi units. But springs, vibrations, etc. return to center after pi too!

**Question: If pi is half of a natural cycle, why isn't it a clean, simple number?**

Let's answer a question with a question. Why does a 1x1 square have a diagonal of length $\sqrt{2} = 1.414...$ (an irrational number)?

It's philosophically inconvenient when nature doesn't line up with our number system. I don't have a good intuition. My hunch is simple rules (1x1 square + Pythagorean Theorem) can still lead to complex outcomes.

## How fast is sine?

I've been tricky. Previously, I said "imagine it takes sine 10 seconds from 0 to max". And now it's pi seconds from 0 to max back to 0? What gives?

- sin(x) is the
*default*, off-the-shelf sine wave, that indeed takes pi units of time from 0 to max to 0 (or 2*pi for a complete cycle) - sin(2x) is a wave that moves twice as fast
- sin(x/2) is a wave that moves twice as slow

So, we use sin(n*x) to get a sine wave cycling as fast as we need. Often, the phrase "sine wave" is referencing the general shape and not a specific speed.

## Part 2: Understanding the definitions of sine

That's a brainful -- take a break if you need it. Hopefully, sine is emerging as its own pattern. Now let's develop our intuition by seeing how common definitions of sine connect.

## Definition 1: The height of a triangle / circle!

Sine was first found in triangles. You may remember "SOH CAH TOA" as a mnemonic

- SOH: Sine is Opposite / Hypotenuse
- CAH: Cosine is Adjacent / Hypotenuse
- TOA: Tangent is Opposite / Adjacent

For a right triangle with angle x, sin(x) is the length of the opposite side divided by the hypotenuse. If we make the hypotenuse 1, we can simplify to:

- Sine = Opposite
- Cosine = Adjacent

And with more cleverness, we can draw our triangles with hypotenuse 1 in a circle with radius 1:

Voila! A circle containing all possible right triangles (since they can be scaled up using similarity). For example:

- sin(45) = .707
- Lay down a 10-foot pole and raise it 45 degrees. It is 10 * sin(45) = 7.07 feet off the ground
- An 8-foot pole would be 8 * sin(45) = 5.65 feet

These direct manipulations are great for construction (the pyramids won't calculate themselves). Unfortunately, after thousands of years we start thinking the *meaning* of sine is the height of a triangle. No no, it's a shape that *shows up* in circles (and triangles).

Realistically, for many problems we go into "geometry mode" and start thinking "sine = height" to speed through things. That's fine -- just don't get stuck there.

## Definition 2: The infinite series

I've avoided the elephant in the room: **how in blazes do we actually calculate sine!?** Is my calculator drawing a circle and measuring it?

Glad to rile you up. Here's the circle-less secret of sine:

**Sine is acceleration opposite to your current position**

Using our bank account metaphor: Imagine a perverse boss who gives you a raise the exact *opposite* of your current bank account! If you have \$50 in the bank, then your raise next week is \$50. Of course, your income might be \$75/week, so you'll still be earning some money \$75 - \$50 for that week), but eventually your balance will decrease as the "raises" overpower your income.

But never fear! Once your account hits negative (say you're at \$50), then your boss gives a legit \$50/week raise. Again, your income might be negative, but eventually the raises will overpower it.

This constant pull towards the center keeps the cycle going: when you rise up, the "pull" conspires to pull you in again. It also explains why neutral is the max speed for sine: If you are at the max, you begin falling and accumulating more and more "negative raises" as you plummet. As you pass through then neutral point you are feeling all the negative raises possible (once you cross, you'll start getting positive raises and slowing down).

By the way: since sine is acceleration opposite to your current position, and a circle is made up of a horizontal and vertical sine... you got it! Circular motion can be described as "a constant pull opposite your current position, towards your horizontal and vertical center".

## Geeking Out With Calculus

Let's describe sine with calculus. Like e, we can break sine into smaller effects:

- Start at 0 and grow at unit speed
- At every instant, get pulled back by negative acceleration

How should we think about this? See how each effect above changes our distance from center:

- Our initial kick increases distance linearly: y (distance from center) = x (time taken)
- At any moment, we feel a restoring force of -x. We integrate twice to turn negative acceleration into distance:

Seeing how acceleration impacts distance is like seeing how a raise hits your bank account. The "raise" must change your income, and your income changes your bank account (two integrals "up the chain").

So, after "x" seconds we might guess that sine is "x" (initial impulse) minus x^3/3! (effect of the acceleration):

Something's wrong -- sine doesn't nosedive! With e, we saw that "interest earns interest" and sine is similar. The "restoring force" changes our distance by -x^3/3!, which creates *another* restoring force to consider. Consider a spring: the pull that yanks you down goes too far, which shoots you downward and creates *another* pull to bring you up (which again goes too far). Springs are crazy!

We need to consider every restoring force:

- y = x is our initial motion, which creates a restoring force of impact...
- y = -x^3/3!, which creates a restoring force of impact...
- y = x^5/5!, which creates a restoring force of impact...
- y = -x^7/7! which creates a restoring force of impact...

Just like e, sine can be described with an infinite series:

I saw this formula a lot, but it only clicked when I saw sine as a *combination of an initial impulse and restoring forces*. The initial push (y = x, going positive) is eventually overcome by a restoring force (which pulls us negative), which is overpowered by its own restoring force (which pulls us positive), and so on.

A few fun notes:

- Consider the "restoring force" like "positive or negative interest". This makes the sine/e connection in Euler's formula easier to understand. Sine is like e, except sometimes it earns negative interest. There's more to learn here :).
- For very small angles, "y = x" is a good guess for sine. We just take the initial impulse and ignore any restoring forces.

## The Calculus of Cosine

Cosine is just a shifted sine, and is fun (yes!) now that we understand sine:

- Sine: Start at 0, initial impulse of y = x (100%)
- Cosine: Start at 1, no initial impulse

So cosine just starts off... sitting there at 1. We let the restoring force do the work:

Again, we integrate -1 twice to get -x^2/2!. But this kicks off another restoring force, which kicks off another, and before you know it:

## Definition 3: The differential equation

We've described sine's behavior with specific equations. A more succinct way (equation):

This beauty says:

- Our current position is y
- Our acceleration (2nd derivative, or y'') is the opposite of our current position (-y)

Both sine and cosine make this true. I first hated this definition; it's so divorced from a visualization. I didn't realize it described the essence of sine, "acceleration opposite your position".

And remember how sine and e are connected? Well, e^x can be be described by (equation):

The same equation with a positive sign ("acceleration equal to your position")! When sine is "the height of a circle" it's really hard to make the connection to e.

One of my great mathematical regrets is not learning differential equations. But I want to, and I suspect having an intuition for sine and e will be crucial.

## Summing it up

The goal is to move sine from some mathematical trivia ("part of a circle") to its own shape:

- Sine is a smooth, swaying motion between min (-1) and max (1). Mathematically, you're accelerating opposite your position. This "negative interest" keeps sine rocking forever.
- Sine
*happens to appear*in circles and triangles (and springs, pendulums, vibrations, sound...). - Pi is the time from neutral to neutral in sin(x). Similarly, pi doesn't "belong" to circles, it just happens to show up there.

Let sine enter your mental toolbox (*Hrm, I need a formula to make smooth changes...*). Eventually, we'll understand the foundations intuitively (e, pi, radians, imaginaries, sine...) and they can be mixed into a scrumptious math salad. Enjoy!

## Appendix

Using this approach, Alistair MacDonald made a great tutorial with code to build your own sine and cosine functions.

## Other Posts In This Series

- A Visual, Intuitive Guide to Imaginary Numbers
- Intuitive Arithmetic With Complex Numbers
- Understanding Why Complex Multiplication Works
- Intuitive Guide to Angles, Degrees and Radians
- Intuitive Understanding Of Euler's Formula
- An Interactive Guide To The Fourier Transform
- Intuitive Understanding of Sine Waves
- An Intuitive Guide to Linear Algebra
- A Programmer's Intuition for Matrix Multiplication
- Imaginary Multiplication vs. Imaginary Exponents

uweApril 18, 2011 at 9:30 amThis was excellent! Well done.

KalidApril 18, 2011 at 9:58 am@Uwe: Thanks!

D-POWERApril 18, 2011 at 11:33 amAnother great article from master Kalid, I’m really happy :D.

ErichApril 18, 2011 at 11:38 amExcellent work! Thank you.

I particularly enjoyed having the infinite series model click intuitively, and seeing that the unit circle contained all possible right triangles. Why, yes, yes it does!

AnonymousApril 18, 2011 at 12:22 pmBrilliant. I must agree with Erich, the infinite series visualization is wonderfully intuitive.

KalidApril 18, 2011 at 4:52 pm@D-POWER: Awesome ;)

@Erich/Anonymous: Thanks for letting me know what made it click! I’m working on an idea to make it easier to share these types of aha moments.

PolyergicApril 18, 2011 at 8:16 pmYour graph “Better Models of Sine” illustrating the successive series approximations of sine has an error: it indicates that sin(x) = x – x³/3! + x⁵/5! – x⁷/y! + x⁹/9! – x⁹/9! + x¹¹/11! …, which includes x⁹ twice rather than once. With as many terms, it should be sin(x) = x – x³/3! + x⁵/5! – x⁷/y! + x⁹/9! – x¹¹/11! + x¹³/13! …

FayleahMay 3, 2016 at 2:53 amI guess this would be the numbers aspect of what i saw won’t with it lol

I mentioned this in my comment, please check it out! Lemme know if this relates to what i was talking about. I’m really not good with numbers, so.

KalidApril 18, 2011 at 9:36 pm@Polyergic: Doh! Great catch — it should be fixed now :).

AnonymousApril 19, 2011 at 8:58 amWow! Awesome Mr Kalid . You really should have been my teacher :)

AnonymousApril 19, 2011 at 11:03 pmYou have no idea how happy you just made me.

AnonymousApril 19, 2011 at 11:07 pmYou just made my brain do this. http://kissmyblackads.blogspot.com/2011/02/mercedes-benz-left-brain-right-brain.html

KalidApril 19, 2011 at 11:36 pm@Anon1: Thank you!

@Anon2: Glad it helped — sine has bugged me for so long.

@Anon3: Love those pictures! Our brains need both :).

AnonymousApril 20, 2011 at 12:00 amOne thing I still don’t understand is why S=O/H, C=A/H, T= O/A.

??

KalidApril 20, 2011 at 12:08 am@Anonymous: That’s just the names we’ve given to those ratios, like saying perimeter = 4 * side [in a square].

But as it turns out, sine isn’t limited to triangles — that is just the first place it was noticed.

AnonymousApril 20, 2011 at 4:24 pmI haven’t made the connection between sine as an idea and why the ratios in SOHCAHTOA are what they are. Am I making sense?

nschoeApril 21, 2011 at 10:16 amWow, thanks once again Kalid. Your explanatiosn are truly wonderful, just how do you come to such a level of knowledge and how do you manage to explain it so easily?

I wish you were my teacher.

Every article is just magic, please keep writing it’s a real relief every time you release another article.

loimprevistoApril 23, 2011 at 4:12 pmThank you!

That was a fantastic lesson. Since I left school I’ve come back to math every few years to try and remember everything I’d forgotten. The best feeling in the world (yep, even better than *that* one…) is the “Eureka!” moment when everything just makes sense. Your article gave me two of those, from watching Hubert move in his circle and from seeing the derivitave definition of sine and how it related to e. You have a gift for teaching and writing, thank you for sharing it.

KalidApril 24, 2011 at 12:04 am@Anonymous: I put an answer at http://www.reddit.com/r/learnmath/comments/guyik/why_do_the_ratios_in_sohcahtoa_work/, let me know if it helps!

@nschoe: Thanks for dropping by! I appreciate the kind words — I don’t think I really understand that much, it’s more my lack of understanding/satisfaction which drives me to seek simpler explanations. The notion that sine is this cyclical wave that we all see just didn’t click deeply with me, I needed something deeper. Many ideas are like that (e, imaginaries, etc.) so I start trying to find analogies that might fit better :).

I’ll definitely keep writing, appreciate the support!

@loimprevisto: You’re welcome! You got it, those Eureka moments are so incredibly fulfilling. It’s what I strive for when writing, I just want to share what clicked hoping it clicks for other people too. Thanks for sharing what aspects helped (Hubert / derivative definition), I have a project in the works to make these insight exchanges easier & more community driven :).

Information Overload 2011-04-24 « citizen428.blog()April 24, 2011 at 9:33 am[…] Intuitive Understanding of Sine WavesI’m generally a fan of Better Explained, but this is an especially good article. […]

AnonApril 25, 2011 at 10:49 amSince my engineering studies I always liked Euler’s formula, connecting sine and cosine to the unit circle in the complex number plane. That’s what Hubert’s sine-sine setting reminded me of.

KalidApril 25, 2011 at 2:54 pm@Anon: Thanks for sharing — yes, Euler’s formula definitely makes it all click.

JoeApril 26, 2011 at 5:50 pmI’ve been reading these for a while, I have to say I think this is the best one yet. We did Taylor series a month or so ago in my Calc class, the end of this article aided my comprehension a whole lot more than any of the class work ever did. Keep ’em coming, please!

KalidApril 26, 2011 at 8:11 pm@Joe: Thanks — this was one of the longer ones to write so glad it was helpful… I’ll keep cranking :)

mark ptakApril 30, 2011 at 7:36 amYou are a master. At some point I would love to hear your take on why the pattern that emerges in transformation matrices

cos -sin

sin cos

changes for rotations about the y axis. For now I”m feeling hungry for a salad….

i❤computersMay 1, 2011 at 11:49 am“Sine waves…psssh, I know that!”

No…apparently I didn’t and you just made my life easier :D

KalidMay 1, 2011 at 4:23 pm@mark: Thanks — transformations matrices would be a fun addition.

@iheartcomputers: Exactly! I was the same way, I thought I understood them too :).

Dan FinkelMay 18, 2011 at 5:34 pmKalid–great article! Dan Meyer just added a very nice modelling of a sine curve that fits with your description as sine as a smooth back and forth (or up and down). The video is here: http://vimeo.com/23798213

KalidMay 19, 2011 at 5:38 pm@Dan: Thanks for the pointer! I love seeing more examples in the real world.

simoneJune 11, 2011 at 10:46 amHi Kalid, great article and great site! But I haven’t understand a thing: the opposite acceleration of x is the double integral of (-x). So we get that, just a moment after the beginning, sin(x) = x – x^3/3!. That’s fine.

Now we have another opposite acceleration so we have to integrate twice -(x – x^3/3!), don’t we? we then get an acceleration of -x^3/3! + x^5/5!. Summing this to the previous result of sin(x) we get sin(x) = x – 2*x^3/3! + x^5/5!. But this is obviously wrong, as the series has -1*x^3/3! instead of -2*x^3/3!.

It seems like we don’t have to integrate -(x – x^3/3!) but only x^3/3!; Why? if the acceleration must be the opposite of the current value, I expect to integrate all of its members.

Thank you very much

KalidJune 11, 2011 at 6:40 pm@simone: Great question! You might take a look at the diagrams here:

http://betterexplained.com/articles/an-intuitive-guide-to-exponential-functions-e/

Basically, each motion (-x) creates interest (-x^3/3!), which creates s interest (+x^5/5!), which creates interest…

My intuitive understanding is that the initial motion (-x) begins a chain reaction, but the chain goes forward — it doesn’t pull back and change the original. I.e., the interest doesn’t go back and change the original… it just generates its own interest (this might be more clear in the diagrams on that article… Mr. Blue creates Mr. Green who creates Mr. Red… Mr. Blue doesn’t even know about Mr. Red!).

CJune 25, 2011 at 2:30 pmI struggled to teach sine to my friend, but now I realize that I didn’t understand it myself. Not intuitively anyway. Can’t wait for your next post!

KalidJune 25, 2011 at 4:42 pm@C: Thanks! Yep, often times I don’t get an intuition until visiting the topic a 2nd (or 3rd) time :).

TheoJuly 3, 2011 at 11:13 amKalid, you are god of mathematical explanation.I have been visiting this site and I will always visit it.So many things I was taught in college is just making sense now. Thank you!

MichaelJuly 4, 2011 at 4:32 pmYour website is brilliant. I am 37 and have always struggled with mathematics. Reading your site makes me feel like I suffered a form of child abuse the way I was taught at school.

I left high school with the impression that sine “came from”

triangles. University classes left me none the wiser. It was only when I bought and read books which were written fairly well that I understood sine to come from circles. It was an epiphany. I was in my mid 20s.Now you give me another epiphany, that circles come from sine. Brilliant, but 20 years late! I wish you had been around all those years ago.

Keep it up.

JayJuly 6, 2011 at 2:37 amAwesome!!. This changes everything!!

ZackJuly 17, 2011 at 3:07 amIf there is a Chinese version, it will be better for me.

MatthewJuly 20, 2011 at 6:34 amI noticed your Sine wave simulator is gone….can you find another one we can link to?

PeterJuly 20, 2011 at 8:51 am“pi is about sine returning to center! A circle is an example of a shape that repeats and returns to center every 2*pi units. But springs, vibrations, etc. return to center after pi too!”

Even Feynman never figured this out:

“About a half year later, I found another book which gave the inductance of round coils and square coils, and there were other pi’s in those formulas. I began to think about it again, and I realized that the pi did not come from the circular coils. I understand it better now; but in my heart I still don’t know where that circle is, where that pi comes from.”

http://www.fotuva.org/feynman/what_is_science.html

KalidJuly 20, 2011 at 6:27 pm@Matthew: Hrm, I think the website I linked to may have been down for a bit — it should be back in the article now. Thanks for the note though!

@Peter: Wow, thanks for the reference! The “pi must be about circles” mantra has been pounded into all of us for a long time :).

KalidAugust 6, 2011 at 5:23 pm@Theo: Wow, thanks for the kind words! I’m basically doing the same as you — going back to relearn what I thought I learned in college :).

@Michael: Thanks for the kind words and encouragement! There are so many misconceptions that I’m only beginning to unravel (this whole sine business only started clearing up in the last few months). I love that epiphany feeling. Sorry for the delay in reply, I was on vacation when these earlier comments were posted.

@Jay: You’ll never look at sine the same again :).

@Zack: Don’t know Chinese unfortunately… Google translate?

werterberAugust 11, 2011 at 11:18 pmIt hypnotizes me this picture > http://goo.gl/U1DJ8

KalidAugust 12, 2011 at 8:45 am@werterber: Sine has that rhythmic sway, right?

werterberAugust 14, 2011 at 10:06 amyeah. thx for great articles. I hope I can learn more about math, because i will study informatics after summer.

KalidAugust 14, 2011 at 9:07 pm@werterber: Thanks!

AshleyOctober 3, 2011 at 3:05 amYou are awesome!!! Seeing sine as motion and not part of a static geometric diagram is so new! I’ve been really curious about how to understand math in different ways. I see at uni that the concepts learned in school are gussied up in different disguises, based on the discipline.

I just had a question to clarify what you said about pi as being a notion of time. I didn’t quite understand, given that I’ve always thought of pi as distance. But this is sort of a two-sides of a coin thing…do you mean that distance=rate x time, and just assume that the unit for the rate of sine to go from neutral to neutral is pi/seconds? So that the units cancel out and thus distance is equal to time?

Also, could you just briefly clarify how you got the successive terms in “The Better Models of Sine” section? x is the initial impulse, and I think I follow your double integration of the opposite position, but then where do you get x5/5! and so forth?

KalidOctober 3, 2011 at 3:41 am@Ashley: Thanks for the note! Yep, a key to math is trying to see it from different angles — some click better than others.

1) Yep, pi is like time if you assume rate = “1 unit per second” (or really, 1 unit per unit time). So then you have distance = rate * time, or distance = time.

If we’re talking about radians (distance traveled along the outside of the circle) then if we go for 2*pi units of time, we’ve traveled a distance of 2*pi and we’re all the way around. If you see sine as a “living, moving” process, the 2*pi is how often the process is back to its initial position. pi by itself is the time from neutral to neutral (middle-top-middle or middle-bottom-middle).

2) Great question. With e, we’ve seen that our “interest earns interest”. In the same way, the “restoring force” of sine creates its own restoring force, which creates another restoring force… and so on.

The thing is, these happen _simultaneously_, we don’t need to wait for the first force to happen (or maybe another way, it happens infinitely quickly so we can’t tell). So when we compute sine, we have to account for as many restoring forces as possible to be accurate.

So, our original force is x (call this A). This creates a restoring force B (pulling us to center) of x^3/3! (the double integral of x). That creates force C which tries to balance the pull center with a push away (double integral of x^3/3! is x^5/5!).

In this way, A creates B, which creates C, which creates D… to infinity :). Actually, each restoring force is in the opposite direction (x has restoring force -x^3/3! because the rule for sine is your acceleration, your second derivative, is the opposite your current position).

Hope this helps!

TaraNovember 19, 2011 at 9:19 amWoah! This is awesome! I can’t believe I found this website yesterday and I was like “Too bad I didn’t find this website some time before my exam, would’ve made my preparations much more fun and versatile”. But then I go to the exam, and it turns out I got the wrong date- it’s actually tomorrow. So I have this whole day free to read a bunch of your articles. They make me all excited about math again! Yeepeee! :D

Thank you sir, you are a blessing!

KalidNovember 20, 2011 at 11:49 am@Tara: Awesome! Really glad you’ll be able to make use of it :). Good luck on that exam.

ChristianNovember 23, 2011 at 9:15 amCorrection: sine is bounded between -1 and 1, inclusively, if the argument is a real number. In fact, there are (infinitely many) complex solutions for sin(x) = 2. For example,

sin(π/2 + i*ln(2 + √3)) = 2

DmitryNovember 24, 2011 at 11:48 pm@Christian: Although it is true that sine takes on values outside of [-1,1] for complex arguments – and in fact takes on *all* complex values (more generally, this is true, with the exception of at most 1 value, for *any* non-constant analytic complex functions, by Picard’s theorem http://en.wikipedia.org/wiki/Picard_theorem), I would argue that this critique is misplaced here as it refers to a mathematical extension of the definition of sine to the complex plane, a space of consideration that was never once intended to be mentioned here. Sine was originally constructed to carry a certain *meaning* – to represent the path of harmonic motion (modulo amplitude) (which, by definition, doesn’t “leave” its interval of oscillation) and parameterize the unit circle (which, again by definition, cannot have a distance from the center greater than 1).

This lesson aimed to explain the origin of, intuition behind, and exposition of: the sine wave, not the general rigorous properties of the sine function over the complex field (which belongs in a class on complex analysis, not in a blog post on high school algebra).

In addition, @Kalid only mentions these “bounds” (he, in fact, never uses this more formal term that carried the very baggage that you intended to dispose of) within the language of “moving” and “swaying” – these already imply that we are on the real line: intuitively this is because motion implies time (or at the very least – 1 dimensional travel) which, for all intents and purposes, is “real” (as C is 2 dimensional – we cannot “trace” across the entire vector field of sine over C).

Often concepts originally defined to mean something spacial/physical get generalized with mathematical language, and along the way lose some of the properties that originally motivated their definition in the first place – this does not mean that the explanation (or even conception) of them in those original terms is “incorrect” – there is no “one true sine function” but rather a general notion of sine with distinct appropriate definitions for different contexts. So, correction: your comment was just an interesting addendum.

ArmitageFebruary 29, 2012 at 10:44 amHere’s your chance. :)

http://ocw.mit.edu/courses/mathematics/18-03sc-differential-equations-fall-2011/

Note that it’s one of the much more thorough OCW Scholar courses.

Anonymous cowardMay 22, 2012 at 12:15 amSine is the point that rotates around another fixed point which causes motion in a linear fashion on a line bisecting the fixed point? Like the arm affixed to the wheels of a steam locomotive running along a track?

I don’t get it still… Worse yet is how, from this, am I to make heads or tails about trig identities? They seem to be everywhere…

kalidMay 22, 2012 at 12:38 pm@Anonymous: Great question. I see sine as a general sway back and forth. We can notice this sway on a circle by realizing the height of a point sways up and down if we just look at that axis. It’s a bit like the arm on the wheels on a locomotive, yep. (But again, that’s just an example — sine is a general concept which shows up everywhere! Some of the confusion is around the definition of sine (it’s a sway!) vs. examples (it shows up in circles, and pendulums, and so on).

Trig identities are another beast entirely. Basically, it turns out that one “sway” (sine) can have relationships to other sways (cosine), for example sin^2 + cos^2 = 1. I’d like to cover this more!

s m garadAugust 12, 2012 at 4:27 amgood explanation!!

kalidAugust 16, 2012 at 10:49 pm@s m: thanks!

RamAugust 27, 2012 at 7:16 pmThis is great! Appreciate you taking so much effort to put it all together!

mraAugust 29, 2012 at 8:19 pmHow fast is sine?

I have a question and a comment.

I get what you are saying about sin(x) going from 0 to max in pi units of time. But that is only true when you are using radians as your measure of time. And radians are defined (I think) by the angle you can get around a circle by traveling along the circumference a distance of 1 radius.

So is it really fair to say that pi doesn’t belong to circles? Without radians, there’s no pi. Without circles, there are no radians. So without circles, there’s no pi. Is that not so?

Also, you say: Circular motion can be described as “a constant pull opposite your current position, towards your horizontal and vertical center”. I don’t think that’s what you meant to say, is it? Didn’t you mean to say that circular motion is a constant pull perpendicular to current velocity? A planet orbiting on a circular path experiences a force and an acceleration that is always perpendicular to its path (and has the same strength throughout the orbit, else the orbit is an elipse).

Likewise, I think to be precise you don’t want to say sine is acceleration opposite to your curent position. Sine (one dimensional) is acceleration toward the origin, in proportion to the distance from the origin.

YatharthROCKSeptember 18, 2012 at 9:57 amI don’t get how you got this part:

I guess I need to learn calculus (BTW, when are you planning to start the series you promised), but can you try explaining what’s happening?

YatharthROCKSeptember 18, 2012 at 10:19 amSo I was seriously taking on trig (apparently it’s a prerequisite for calculus), and I came upon trig and family. 3 sides of a triangle having 6 relationships called Sine, Cosine, Tangent, Cotangent, Secant, Cosecant. Is that it? Is that all there is to trig apart from learning to manipulate them?

I read your article a couple of times, but I still didn’t get a few things. You said the `x` in `sin(x)` was how far along it was in a wave. But then you said the `x` meant how fast in relation to the ‘normal’ wave which took `π` seconds to return to neutral. What gives?

And why can’t we calculate all of these by hand? I mean, sine and cosine and all are functions, right? They must have an exact definition. Why can’t we figure them out by hand? Why do we need these ‘log books’? That reminds me. Is there a better way to find logs than tables? Some sort of a approximation method maybe…

podAhmadSeptember 26, 2012 at 11:54 amg88888888888888 post as usual specially the series explanation is awesome.

But here comes a question!!!!!!

From 0 to pi there are only four reactions or reactions to reactions…but shouldnt it be a continuous chain of reaction, i mean there should be infinite terms between 0 to pi.

kalidSeptember 28, 2012 at 12:07 am@Ram: Thanks!

@mra:

1) Asking “How fast is sine?” is like saying “How fast is a circle?”. It’s a general shape, which you can traverse as quickly or slowly as you need. By “default” we use radians to measure angle, and get through the entire neutral-max-neutral-low-neutral cycle in 2*pi radians (6.28).

2) “Is it really fair to say that pi doesn’t belong to circles?”. It depends on your point of view. I could imagine a world where sine was discovered first (from the motion of springs, let’s say), then pi was discovered, and later on, the shape of a circle was discovered.

3) I think we’re saying the same thing. A pull opposite your current position vector is towards the center, which the direction of the pull. If you are at (1,0) and moving in a circular path, then your position vector is pointing East, your velocity is pointing North, and your acceleration is pointing West [towards the origin].

@Yatharth: Yep, you’ll need calculus to decipher that :). It’s basically saying “Sine accelerates your opposite of your position (if your position is x, your acceleration is -x).” To find the total distance that this negative acceleration will impact you, you integrate twice, and get -x^3/3!

Trig is basically the anatomy for circles and triangles. Learning every part of them, how they’re connected, how to find the sizes of one part given a different one.

To clarify: the variable is how fast you are along in your wave. If I write sin(pi) I mean “I am pi units along in the wave which takes 2*pi units total”, which means I’m at the halfway point. If I write sin(2x), then I am going to travel the wave twice as fast as the regular sin(x) [since I’ll be twice as far along for the same x value].

You can find sine/cosine by hand, but it’s painful. You plug in values of “x” in that infinite equation [but only take as many terms as your sanity can handle]. There are shortcuts for finding logs, sine, etc. by hand but are no longer really used, for obvious reasons. The first log tables took dozens of man-years to make.

@podAhmad: I’m not really sure what you mean by only 4 reactions or reactions to reactions. There’s an infinite sequence of them, but I only showed a few terms (with … for the continuation) in the equations.

YatharthROCKSeptember 29, 2012 at 6:28 am@kalid Thanks. That made sense. But how inblazes do calculators calculate sine? I bet they don’t use an infinite series and stop when they’re tired…

WilliamOctober 26, 2012 at 10:20 amHey Kalid, thanks for your great and inspiring posts.

I would like to have a question here. You mentioned that 2*pi is the time it takes to travel back and forth. I wonder if there is any way to deduce this? (i’m thinking of using the relation acceleration y”=-y). Thanks so much :)

kalidOctober 26, 2012 at 10:17 pmHi William, great question. It’s a bit circular [pardon the pun] because 2*pi is defined to be the distance around the unit circle, aka one full cycle of sine. You can “compute” the value of pi using successive approximation (see http://betterexplained.com/articles/prehistoric-calculus-discovering-pi/). Hope this helps!

AjitNovember 8, 2012 at 7:38 amIts really very usefull, and very nicely explained… Thanx!

kalidNovember 19, 2012 at 10:30 pmThanks Ajit.

An Interactive Guide To The Fourier Transform | BetterExplainedDecember 20, 2012 at 3:43 am[…] "sinusoid" is a specific back-and-forth pattern (a sine or cosine wave), and 99% of the time, it refers to motion in one […]

JulianJanuary 2, 2013 at 4:02 pmA few years ago I took a basic AC/DC class where I remember that we made and measured sine waves on an oscilloscope and my partner and I after we finished our project made this https://sphotos-b.xx.fbcdn.net/hphotos-ash4/292060_548653444059_1236366013_n.jpg Any chance you could tell me what we did cause we could just keep adding them on and our professor was pretty confused. It’s been bugging me for a while now since I can’t find anything similar online.

RobertFebruary 23, 2013 at 8:34 amExcellent explanation of the sine wave, but you seem to quickly jump through the meaning of sine in a unit circle.

avenidagezJune 14, 2013 at 4:39 pmFrom atoms that shape spheres, to earth, to the moon, to the sea waves, sines is no other thing than the projection of the circular motion, stable, in decay or increasing, being it light, sound, radio waves, all are sinusoidal, in fact sine math as a two planes is unrealistic since most waves are tridimensional, So the circle is not a matter of geometry, geometry is one of the results of the study of circles. Men invented math not the circle, neither the waves.

If you love math, what I have always needed is an equation for the linearity of a sphere, because it would explain much of the quantum behaviour. Imagine an sphere, must be an equation that starting from any point it follows a path such that forms a perfect balance sphere. That line may cross itself but if it does you must balance the exact opposite site if there is any at all. If there is no way to do it, then my question stills. What is the balance of a particle all scientific draws as spheres, even ignoring the particles, they translate the same for atoms and talk about electron spin, and I know you know what happens when something unbalance spins. Is it he reason of the quantum behaviour?

ZenoJune 30, 2013 at 2:06 pmHey, Kalid!

First of all, thanks for the whole site. It’s really changed my definition of “knowing” something, and it’s given me a new zest for learning!

I had a question about the pi without pictures part. I’m really interested in pi not being related to circles, but circles being related to pi. Problematically, though, I feel like I could show that sightless alien a perfectly smooth oscillation that had a period that _wasn’t_ related to pi. I feel like a perfectly smooth shade-oscillation with period 2 seconds isn’t any less smooth than the pi version.

If the smoothness of oscillation defines pi, what makes regular sin(x) any more special than sin(x*pi) for example?

Avenida GezJune 30, 2013 at 2:59 pmZeno, there is no pi not being related to circles, since pi is a constant which equals the number of times the length of the radius of ANY circle size, fits in half around the circle.

pi is the result of having found that relation between the radius and the perimeter of a circle is always the same, That is whythe whole perimeter of a circle = 2*pi*R

You may draw a series of points in space and calculate its position from an origin based on the sine, and the final figure or equation may not result in a sine wave, then what is sine?

sine is not related to circles, but to triangles rectangles, where one angle is 90 degrees, there is a constant relation between the 3 sides, csquare = asquare + bsquare the sum of the squares of the shorter side equals the square of the longer side, no matter the size of the triangle

So the same as pi, that relation in triangles got related to angles in a 360 degrees cycle (read it cycle, not circle, this is a whole turn around something) so you can imagine something doing a whole cycle which might not be the route of a circle (radius variation if you want) so the cycle refers to 360 degrees, and sine refers to a triangle rectangle side relations, and as you can see we are not tallking neither refering to circles which in fact do not participate.

AnonymousJuly 7, 2013 at 9:47 pmthanks it helped me alot :)

AnonymousAugust 28, 2013 at 11:22 amWow…My mind is blown.No one ever teaches students these things,and they can make a world of a difference.

Your articles are phenomenal and so is your attitude.

kalidSeptember 3, 2013 at 4:33 pm@Anon: Thanks, really appreciate it; my goal is to teach the way I wish I was taught.

manibharathySeptember 10, 2013 at 7:10 ambetter explained!!!!!

Colin RobinsonOctober 22, 2013 at 3:26 pmI agree that defining sine in terms of its second derivative, or “acceleration opposite to position”, is an excellent alternative to the triangle and circle definitions, and makes the connection between sin x and much clearer.

Have you looked at the hyperbolic sine function (sinh)? It’s like a half-way house between sin x and , and its graph is distinctly wave-like, though a surreal wave! Its first derivative is the hyperbolic cosine (cosh), but its second derivative is itself. Acceleration is equal to position, as in the case of .

kalidOctober 23, 2013 at 1:43 amThanks Colin! Great point with the hyberbolic versions — I’ve only dabbled in them very briefly. I like that they can be defined in terms of exponentials… they’d be a fun topic to get into :).

DanielNovember 14, 2013 at 2:36 amThanks, you rock

StephenFebruary 1, 2014 at 1:20 pmHey Kalid,

Keep up the great work! It’s really amazing.

I was wondering if you might be able to offer any intuition as to why sine of an angle is also the ratio of the opposite side to the hypotenuse in a right triangle? I get that right triangles would have consistent ratios for the different side lengths, but how is sine (and cosine, tangent) able to describe these ratios?

kalidFebruary 1, 2014 at 10:54 pm@Stephen: Thanks! Great question on the connection between sine and triangles. I actually prefer starting with circles, and then seeing how triangles fit in.

Let’s say we’re traveling around the unit circle (radius 1). As we go around, we have some height above the x-axis. From this discussion, we can see that a circle is made up of two “sways”, one controlling the vertical position (call that sway sine), and another controlling the horizontal position (call that sway cosine).

The only difference between the sways is where they start: sine starts neutral, and starts moving up. Cosine starts at its max value, and starts moving towards center. For now, let’s think about sine, our vertical position.

If we take our spot on the circle and extend a line down, and to the center of the circle, we end up making a right triangle! The height, our sine, is the side “opposite” to the angle, and the line to the center is the hypotenuse. The resulting triangle must be a right triangle, because we drop a line straight down from our current position.

Now, by an accident of history, we started working with triangles before circles, so we found this “vertical pattern” (sine) in triangles, then built up to circles. But again, I’d prefer to start with circles and work down to the triangles buried inside.

Another way to put it: the swaying vertical motion of a circle, called sine, can also be seen in the swaying heights of the right triangles formed along our circular path.

(The last step is to allow for any hypotenuse, not just the unit circle, so we scale by “H” to show that sine is present in any circle no matter how large, similar to how pi is present in any circle, no matter the radius.)

StephenFebruary 2, 2014 at 9:10 pmHi Kalid,

Thank you so much for your speedy reply. I really appreciate it.

I really see what you mean when it comes to thinking of sin and cos as ‘swaying’. I think I’m starting to really internalize that way of looking at it, which definitely is helping to develop my intuitive grasp of these functions.

However, I think my intuition is still at the stage where I begin with the assumption that circles are made up of a perfect balance of two complementary sways as described by sin and cos, and THEN work backwards to intuitively confirm this initial assumption.

Ideally, I would like my intuition to see how sin and cos connect to circles without first assuming that they do, and then finding different ways to confirm that this is in fact the case.

For example, your article on e delved into the math in an incredibly intuitive way, to the point where I saw exactly how the general concept of growth fell directly out of the details of the equations. Or with your article on Pythagorean theorem, where you explicitly highlighted the physical areas that were implicitly being added, which accounted for the squared terms in the equation.

So I’m wondering if you think it’s possible to approach sin and cos the same way, where circles can kind of ‘fall out’ of the math. Because right now I’m at the stage where my intuition is only confirming, rather than deriving. Perhaps if you know any of the history of how these functions were first discovered…that is often very helpful in developing intuition that derives rather than confirms.

Please let me know if this isn’t making sense and I will try and elaborate more. I’m being a little bit vague because I’m still trying decide if I’m at the point where I need to accept something as empirically true, or if I can reduce my understanding to more fundamental principles.

Thanks so much for bearing with me, Kalid. I appreciate your work so much

Best,

Stephen

kalidFebruary 4, 2014 at 3:06 pmHi Stephen, great comment. To be honest, my intuition for sine/cosine isn’t as strong as the ones for e, Pythagorean Theorem, etc. so I’m a bit in the “discovering” vs “deriving” mode myself.

I might say something like this: suppose we have a concept of perfectly smooth growth, which is epitomized by e^x.

If we combine the idea of perfectly smooth growth (e^x) with rotations (imaginary numbers), we get e^ix.

Intuitively, before graphing anything, we should imagine that e^ix results in something along the lines of “perfectly smooth rotation”.

What would this shape be? Well, it should be symmetrical (why would it favor one side over the other?). It should embody the essence of rotation, spinning. And so on.

Pretty soon, we might see that a circle is the shape which satisfies this intuition. Now the question of sine/cosine comes in.

This circle exists in 2d: if we analyze each dimension independently, it seems like each dimension should be moving perfectly smoothly as well (again, why would one be favored over the other?). We can’t rotate in a single dimension, but whatever motion we have, should be smooth.

That pattern of motion, the smooth sway in a single dimension, can be called sine. And we can work out that

e^ix [perfectly smooth rotation] = cos(x) [smooth sway in the horizontal direction] + i*sin(x) [smooth sway in the vertical direction]

We’ve separated our 2d motion into a combination of two 1d trajectories. Getting into even more nitty gritty, the series expansion of e^ix = series expansion of cos(x) + series expansion of i*sin(x)

That is… cosine and sine can literally be “factored” out of the combined circular path we see in e^ix.

Hopefully that helps?

StephenFebruary 6, 2014 at 10:28 pmHi Kalid,

Thanks again for some more great insights. Very helpful and definitely adding to my understanding of sin.

So I think I see everything up to the point where we make the leap from rectangular values (x’s and y’s) to polar values (theta). I think the gap in my intuitive understanding of sin stems from not being able to see where the Taylor series of sin comes from.

I think your article is great at teasing out the corroborative intuition on the Taylor series: given the existence of the series itself, you demonstrate that the alternating positive and negative terms are corrective tugs in opposite directions, which gives rise to the wave appearance. I get that, and I think it’s a great insight.

But do you think there is a way to draw a diagram with some triangles or circles or something, so we can start with rectangular x’s and y’s, and derive the Taylor series intuitively, which we THEN define to be the function called “sin(x)”? (rather than say the commonly used circular derivation, where the definition of sin and cos is a given, and based on the fact that each function’s derivative is the negative of the other, you come up with a Taylor series to satisfy that condition. That’s a corroborative approach to be)

I like how you point out that something more ‘fundamental’ is going on with the trig functions that goes beyond circles: periodicity. But at the end of the day, are the Taylor series of these functions capable of being derived without using the visualization of a circle?

The person who asked this question on math.stackexchange is getting at what I’m looking for:

http://math.stackexchange.com/questions/185356/rigorous-proof-of-the-taylor-expansions-of-sin-x-and-cos-x

However, the most up-voted explanation here is, to me, circular and un-intuitive. It employs the definition of sin and cos and some trig identities to show where the Taylor series came from, which is to me unhelpful.

Let me know if I’m chasing ghosts. Thanks again for your time, Kalid

Best,

Stephen

StephenFebruary 6, 2014 at 10:36 pm…..in the same way the you very intuitively explained the Taylor series for e. In that case, I could visualize what each term corresponded to.

MonqMarch 17, 2014 at 7:57 pmFirstly, I’d like to thank you for putting in the time & the work to even be able to explain math in this way. It helped me very much in a time of need. :)

Secondly, I apologize for not going through all your comments to see if this question had already been answered, but could you possibly explain tangent, secant and cosecant wave patterns on a graph? Or if you’ve already answered that somewhere, post the link to the page.

Much appreciated.

Cheers.

kalidMarch 18, 2014 at 8:04 am@Stephen: Whoops, I lost track of this comment, I’d like to cover it, probably in a follow-up article ;)

@Monq: I don’t have anything on the graphs, but http://betterexplained.com/articles/intuitive-trigonometry/ covers the meaning of the various functions. I’d like to do a follow-up on the graphs as well.

GVApril 30, 2014 at 4:43 pmYour explanation of how SINE originated is brilliant. My husband (has PhD) and I have Master’s degrees and never thought of this explanation until our highschool daughter asked me this. I am very glad that you have this site.

kalidMay 2, 2014 at 8:32 amThanks GV, really glad you enjoyed it! I didn’t start seeing the meaning of Sine until maybe a decade after learning it “officially”.

Fractals, Phase Conjugation, Electromagnetic Pulses, and Sine Waves…. Oh My!! | The Shift of Time and Energy!May 12, 2014 at 6:45 am[…] yesterday, we start off with the energy of a sine wave, coming out of the deep west of my lady’s field, stretching across to the center of may. […]

Lisa Gawlas – Fractals, Phase Conjugation, Electromagnetic Pulses, And Sine Waves…. Oh My!! – 12 May 2014 | Lucas 2012 InfosMay 12, 2014 at 7:45 am[…] yesterday, we start off with the energy of a sine wave, coming out of the deep west of my lady’s field, stretching across to the center of may. The […]

Lisa Gawles – The Shift of Time and Energy! – Fractals, Phase Conjugation, Electromagnetic Pulses, and Sine Waves…. Oh My!! – 5-12-14 | Higher Density BlogMay 12, 2014 at 10:25 pm[…] yesterday, we start off with the energy of a sine wave, coming out of the deep west of my lady’s field, stretching across to the center of may. The […]

AnonymousJune 26, 2014 at 9:15 amYou are a hero for sharing this explanation. Thank you so much.

MichaelJuly 24, 2014 at 6:38 pmYou might like my Sin wave visualization (about half way down on this page). Please feel free to do whatever you want with it (and if you make it better, let me know!)

http://www.gravitypersists.com/mathprototypes

STEPHEN VIELAN HAWKINGAugust 29, 2014 at 8:22 amExcellent

please explain pi in Terms of fundemental concepts

AnonymousSeptember 4, 2014 at 2:13 pmThank you!!!!

AnonymousSeptember 4, 2014 at 2:47 pmThis is theeeeeee best explanation I have ever had, even after taking two semesters of Calculus!!!!!

kooshaSeptember 12, 2014 at 3:24 pmAbout 10 years of studding (from the first day sine was introduced to us in high-school) and ‘today’ I actually learned what sine really is. Dear kalid, you are the best teacher ever. I wish others were like you too.

deviSeptember 13, 2014 at 9:24 pmSeriously this math better explained.. Thank you for the time and efforts :)

Tim McGrathOctober 14, 2014 at 10:24 pmTo all the other great examples of periodic motion, may I add the trampoline? In my mind, I’ve been having fun bouncing from pi to pi.

Rajender JatanaOctober 18, 2014 at 6:48 amYou are rocking and adding new dimensions to maths

Tim McGrathOctober 19, 2014 at 2:44 pmLet me see if I have this right. Sin(x) is a position function. Sin(x)=x at first, a linear relationship, because there is little or no resistance around the neutral position. But then as the distance increases, so does the resistance, and a restoring force must be added to x, which alters the position. This restoring force, -x, is negative acceleration, a pull back towards the neutral position. We know that acceleration is the second derivative of the position function. Thus, to determine the new position, we double integrate -x to arrive at -x^3/3! and add it to the series. However, this being the inexact world of harmonic motion, the new term overshoots the mark and requires a correction. So we then double integrate -x^3/3!, being sure to cancel the negatives first, which gives us x^5/5!, which also overshoots true sine and must be corrected with another term, and so on.

This is the most dynamic view of both sine and the Taylor series that I’ve ever seen.

kalidOctober 19, 2014 at 11:04 pmHi Tim! Yes, that’s exactly it. As the restoring force slows you down (changing your distance by -x^3/3!), it means you aren’t going along your original “x” trajectory any more, and should not feel the full brunt of -x^3/3! — you get a bonus of x^5/5!.

But, that bonus means you’re going further along than you expected, so feel an extra restoring force of -x^7/7!. And the cycle keeps going :). Basically, the longer you want to model sine, the more levels of restoring forces you need to stay accurate [and if we’re on a very small timescale, we just assume sin(x) = x and ignore the impact of even the first restoring force].

Glad you like the perspective, I need to see/visualize things to make sure they’re really clicking.

Swati M. KelkarNovember 11, 2014 at 9:05 pmYour clarity of thought is a gift. I teach college and often draw upon your analogies. Thank you for sharing your insights generously.

Bilal AhmadNovember 30, 2014 at 3:41 pmGreat article! Thanks a lot for sharing your insights!

I remember another mnemonic: Some People Have, Curly Brown Hair, Through Proper Brushes: Sine=Perpendicular/Hypotenuse, Cos=Base/Hypotenuse, Tan=Perpendicular/Base. I like it better :)

AnonymousDecember 14, 2014 at 6:02 amThis brought me to tears

luluDecember 22, 2014 at 6:51 amWOW ..amazing! Good job

luluDecember 22, 2014 at 6:52 amthank you very much sir

DevJanuary 19, 2015 at 10:05 amGreat explanation. Easy to understand.

Can you explain what is phase angle or phase ofset (it cofused me when I was learning about deriving the progressive wave equation in physics)

kalidJanuary 22, 2015 at 2:43 amHi Dev, great question. If you have a circular path, the phase angle (aka phase offset) is where the path starts. Instead of starting at 0 degrees and spinning around, the pattern might start at 90 degrees (top of the circle) and spin around from there. There’s a bit more in this article: http://betterexplained.com/articles/an-interactive-guide-to-the-fourier-transform/

Ben GeistJanuary 28, 2015 at 12:59 amThis is wonderful! I’m so happy to finally have some independent concept of what sine is! And even if it doesn’t make Trig easier it does make a lot more logical sense now. Thank you for making my day=D

VishnuMarch 20, 2015 at 8:10 amSeriously i have no words man. . I have always hated math though i scored centum in my schools coz of sin and cos wondering what the hell do they mean “Really” . Just happened to read this and its like some mysterious secret unvieling before my eyes. Thanks a lot . Wish a i had a teacher like you. :) Share more of ur intuitions.

schroedingercatMarch 31, 2015 at 6:16 pmThis was exactly what I needed. It was so beautiful to me that the “better graph” at the bottom in the calculus stuff I dont understand (yet) actually made me tear up. I can actually feel what I am doing now when I do trig and that helps me understand it so much more deeply. THANK YOU.

kalidApril 1, 2015 at 9:56 am@Ben, @Vishnu: Really glad it helped!

@schroedingercat: That’s wonderful to hear, thank you!

Paul NelsonApril 16, 2015 at 2:34 pmI am a 67 superannuated person that has used PC programs for a number of years with only High School basic knowledge of trig: sines and cosines and eternity. Recently I’ve tried to explore electromagnetic science for some unknown reason. Although I have learned a great deal from your creative attempt to picture radio sinusoidal “lines”, I still cannot conceptualize how these infinite things, particulates/energy, can spread from a finite source in all directions and still fit within my brain. I can sort’v feel a singular string, an expanding and detracting coil, passing through the firmament, A string expanding in all infinite directions radio wave from an antenna doesn’t exactly fit my limited view of a sinusoidal

Can you try to give me some non-nerd picture of how a sinusoidal wave projectile is is expanded, or morphed, so it travels in all directions. Do you understand my conflict?

Very much appreciate your out-of-the-box 2 dimensional to 3 dimensional excursions. Thanx pgn

AnonymousApril 16, 2015 at 8:55 pmPaul Nelson: If I can butt in here, I may be able to help. Kalid can correct me if I’m wrong.

To make a conceptual switch from a two-dimensional ray to a three-dimensional emanation, think in terms of a field instead of a wave. Any radiating object, whether it’s the sun or an electron, emits a field of energy, and the field travels in all directions from the source. If you drop a rock into a pond, you get the same effect. The disturbance ripples in all directions from the point of impact.

Tim McGrathApril 16, 2015 at 8:56 pmPaul Nelson: If I can butt in here, I may be able to help. Kalid can correct me if I’m wrong.

To make a conceptual switch from a two-dimensional ray to a three-dimensional emanation, think in terms of a field instead of a wave. Any radiating object, whether it’s the sun or an electron, emits a field of energy, and the field travels in all directions from the source. If you drop a rock into a pond, you get the same effect. The disturbance ripples in all directions from the point of impact.

sophyyApril 26, 2015 at 12:50 amThis is called smashing illusion of complexity to the grounds of simplicity.!the bestestt version .thankyu kalid:)

MARCIO SIQUEIRAMay 27, 2015 at 11:57 amI’m yet thinking about acceleration and Sine!!!

It’s hard to absord that!

But I got almost everything!

Thank you!

vivekJuly 19, 2015 at 11:38 amThis is really great. Unable to understand the factorial formula. But one thing I learnt is mutualfund interests are like sine waves sometimes

NaveedNovember 10, 2015 at 11:37 pmAmazing Khalid

Had you been born earlier, I am sure there would have been much less “math haters” in this world.

Keep it up

Joris van de BrandeNovember 25, 2015 at 4:24 amHey Khalid,

I don’t get your bank account metaphor

In bank 50, income 75/week

Raise opposite your bank account

W0 50

W1 50+75-50 = 75

W2 75+75-75 = 75

W3 75+75-75 = 75

It does not seem to decrease this way :)

HeatherApril 16, 2016 at 7:07 pmYes. Thank you so much!!! Ever since “learning” about sine I have been asking “but what IS sine?” I feel so much better after hearing your explanation. I could do the problems, but without understanding what sine really is, it was hard to understand what i was doing as anything other than a series of steps. Much thanks!!!

kalidApril 24, 2016 at 4:45 pmThanks, glad it helped!

FayleahMay 3, 2016 at 2:50 amHow is sine one dimensional if it moves in two directions? Up-down and then left-right. In that sense, since a circle is two-dimensional, and sins is two-dimensionsl, then sine IS the circle, but one half of the circle would be flipped to the space next to it to make a horizontal “s” shape. So sine is a sort of unfolded circle. Which means it IS a circle. Saying they’re two different things doesn’t sound accurate to me. The brick analogy compares a totally different shape, a rectangular one, with lines that stop and start in order to make the shape. Sine is continuous, so is the circle. So while a couple of straight lines would make up a square, a circle/sine would be different because it’s continuous. The endpoint is the same as the start point, unlike a square. We can find a midpoint through pi, but a WHOLE sine curve goes from -1 to 1, just because you CAN cut it in half to make it 0 to 1 or 0 to -1 doesnt mean that’s the full endpoint, or that it stops after one hump. Sure, sine uses two humps, but what I’m saying is that there’s no legitimate endpoint between those, so a circle isn’t made up of two separate humps, the two separate humps are one continuous thing: a circle. I think we just unfolded it so we could graph it better and make connections with it.

Also, you mentioned that pi is an infinite number, and you said it was a “random” thing because pi represents the halfway points in a circle, and halfway isn’t an infinite concept. But aren’t lines technically made up of an infinite number of points, no matter how long they are? So it would make sense that the halfway point being infinite would be represented by an infinite number, pi, because halfway is still infinite on it’s own.

And in your “better models of sine” picture, wouldn’t the larger curves need to be similar to the smaller ones on the opposite side of the sine line? They look a little off to me. So like, the larger curves represent a force stronger than the smaller ones on the opposite side, which is why it isn’t a straight line. If the force was equal on both sides it would be straight, which, to explain further, would mean the curves curving away from the sine curve would be flipped over on both sides equally. Sine curve curves, so one side will be smaller. Bigger force pushes away, which makes the line go in the opposite direction. I think you got this, but i think the graph is a bit off, because i think the smaller curves would be the same shape but smaller, thus “similar.”

Also, going back to the topic of different directions, you’d need force coming from 2 directions to make a circle, which is why it’s 2-D. And force coming from 3 directions would make a sphere. And, to digress, 4 directions would make a 4-dimensional sphere, which is a difficult concept, but you could double the points of a sphere and connect them to get a 3-D (or technically 2-D, since you’re drawing it on paper, but we can only see and comprehend 3-D at most) representation of that.

When it comes to the numbers and equations though, you lost me for the most part lol. I really only think this through spatially/visually.

Feel free to email me a reply to this comment if i made a mistake! I want to understand this fully, so if there’s error in my logic please let me know.

DanielJune 8, 2016 at 12:38 pmOne dimension has two directions.

Jeff SmithJune 5, 2016 at 3:26 pmHi. In the equation for drawing a sine wave — y = A sin x — what is the numerical value of “sin”? You have to know that, right? How do you figure that out to get x and y? Could you send an example with value for “sin” plugged in? Thanks.

ShirleyJune 28, 2016 at 11:56 amKalid,

This is enlightening. When I was looking at some formulas, inventions, I always want to know: what the hell makes these inventors think this way?

So when I look at Fourier transform,

1) it is relatively easy to understand that you want to decompose a complex time based signal into the recipes. (because it is always to good to know the recipes so that you can easily recreate anytime anywhere easily).

2) It is not too difficult to imagine somehow you would think the underlying recipes are repeating themselves (so, the frequency based signal).

3) Then it kind of make sense to see the sines and cosines since essentially you want to represent things that are repeating themselves.

4) And somehow the complex number jump in because they represent things that are repeating themselves in two dimensions: circular path.

I guess I am seeing some dots, but still, I feel like big gaps still missing from above 1,2,3,4.

You have very very nice explanation for individual concepts like sines, e, euler formula, Fourier series…..

I would really love to see them closed connected so that I can give myself a satisfying answer: why complex number is introduced, how it can be used to tackle real-life problems. And the answer I am seeking is not simply: see, there is complex number in fourier, so this is how they are used, and Fourier is very important in solving many problems like digital processing, heat transfer….. blah, blah, blah….

I would love to understand, why, what if without it?

Mohammad AlhashashJuly 20, 2016 at 7:31 amthat makes a lot of mental difference ! thanks mate

Shelby IrvinSeptember 4, 2016 at 5:28 amOH MY GOD

this has made my week. you don’t even understand…. sine was one of those niggling questions at the back of my head – ‘yeah, yeah, i know how to calculate it, but what does it *mean*??’

AND NOW I KNOW

AND IT’S AWESOME.

…..mindblown.

SachinNovember 3, 2016 at 1:57 amWhat a beautiful way to describe so many “simple” concepts in one post! This is one of your best posts Kalid. Keep ’em coming!

Tim McGrathNovember 3, 2016 at 8:54 pmI was just reviewing this today. And Sachin’s right. It is one of your best posts. A beautiful job on a beautiful equation.

kalidNovember 11, 2016 at 5:17 pmThanks Tim and Sachin!

HasanNovember 21, 2016 at 10:57 amDear Khalid. Thanks for such a beautiful piece of writing.

Can you explain the presence of sine in Snell’s law?

Juichia CheDecember 12, 2016 at 12:39 pmSo the perverted boss is the bank charging me bank fees.

gwen robertsDecember 12, 2016 at 1:00 pmWonderful!

PoomathiDecember 20, 2016 at 11:54 amWow!! I had that Ahah moment. Thanks for the wonderful article.

what a cool website!January 1, 2017 at 5:55 pmThanks what a cool website

Jamie PowersJanuary 1, 2017 at 5:56 pmThanks! What a great website.

SikanderJanuary 13, 2017 at 1:37 amExcellent

dmsApril 27, 2017 at 11:25 amThank you!

F1LT3RMay 14, 2017 at 8:24 pmThis was really helpful, thank you! One question I have at the moment, is how you calculate sin/cos for larger numbers? I noticed that the larger number you are calculating, the more number of steps you need in the “infinite” series to pull the value back inline with the sine shape. Is there a trick to knowing what precision to calculate the value based on the number’s size?

F1LT3RMay 15, 2017 at 4:13 amI realized last night that because 2 x PI has the same sin value as 0, that I could just use the remainder of x / 2PI to calculate the value. Duh. :)

kalidMay 16, 2017 at 11:05 amNice! Was going to write that you only need the remainder after a full rotation :). It’s an easy way to get the approximation really accurate, vs. walking forward with a zillion terms in the series.

Graphing Functions | IB PrepJuly 19, 2017 at 8:14 pm[…] Read to understand more on these functions: Article on sine and cos functions. […]

LeahOctober 9, 2017 at 1:11 pmKalid! This is FANTASTIC! I was having trouble relating sin functions (y=sin(x)) to waves and triangles simultaneously. It all makes so much sense now, thank you!

JPedroMarch 2, 2018 at 8:50 pmThis is what I had been searching for, thank you :]

Sine Wave Generation - TechBestFebruary 8, 2019 at 5:53 am[…] https://betterexplained.com/articles/intuitive-understanding-of-sine-waves/ […]