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 by 1 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{\arcsin(.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{\arctan(.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):



Other Posts In This Series

  1. How To Learn Trigonometry Intuitively
  2. Easy Trig Identities With Euler's Formula
  3. Intuition For The Law Of Sines
  4. Intuition For The Law Of Cosines

Questions & Contributions


  1. In your example you need to specify which angle you want the sine of because at the moment it is ambiguous.

  2. Hi Khalid,
    My daughter is in high school. I want her to score good in SAT exam. Do you have any package or suggestion.
    Appreciated very much for your response.


  3. Kalid, you did it again! As an engineer and programmer I use those trig identities all the time but never have they been made so succinctly clear to me. Huge aha-moment with the dome-wall-ceiling analogy.
    Such a pity of all the wasted time I’ve wrestled with trig in high school :-)

    Please all teachers of the world use this!

  4. @Pravin: I don’t really have many specific test prep recommendations, unfortunately. At a high level, my approach is to gain a solid intuition for the ideas + do practice exams to make sure things are clicking. If you’re having difficulties with a certain type of problem, it’s important to look for an analogy/explanation that builds deep understanding.

    @Luke: Awesome, thanks! I’d used trig a lot in school, and didn’t have the identities come together until recently (argh).

  5. My grad stat prof said…it takes a brilliant person to see a simple concept…

    Your work is brilliant, thank you.

  6. Thanks Patrick, really appreciate it — I think there always has to be a simple explanation beneath the surface complexity. (One of my favorite Einstein quotes is that unless you can explain a topic clearly, you don’t really understand it :))

  7. hi khalid,

    A good explanation indeed. An innovative and creative presentation. Can you plz do the same for hyperbolic trigonometric functions????? Plz plz plz ….. I look forward for it

  8. RE: 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 hypotenuse.

    Are you sure? Should it be….
    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 Wall Distance (Radius).

  9. @rn: Hyperbolic trig functions would be a nice follow-up :). I’m hoping to explore the implications of Euler’s formula.

    @Hans: Whoops, I should have clarified — the “hypotenuse” was meant to refer to the unit circle (radius = hypotenuse = 1) but this was unclear. I’ll fix up the phrasing, thanks!

  10. Why oh why oh why oh why don’t they teach it like this in the classroom??? Thank you so much for sharing your intuitive connections to these concepts. You would think that by now, the standard curriculum would be focused around visual learning, since every human being is a visual learner, rather than teaching concepts in a fashion designed for a robot. I am showing these pages to everyone I know that has trouble with math. There’s no reason to be afraid of the subject if it’s taught like this.

  11. Thanks Doug, really glad it helped. Math (and any subject, really) can be so much easier to learn when we look for an approach that gets things to click deep down. Often times we try to trudge through, which works in the short term, but doesn’t build lasting understanding or enthusiasm. Just as you say, the fear of learning even “difficult” subjects can be removed — it’s a blast when things to click.

  12. Khalid ,
    Trigonometry is used to be Tricknometry but now I find it is as simple as eating banana .Great job well done.I love your creative thinking.

  13. Wow! This is the best I’ve ever seen. I’ve never thought in terms of percentages. It’s is little bit lengthy though, but your’re a star Kalid!

  14. I’m a programmer and I love algebra, but I gave up hope on ever understanding trigonometry beyond simply memorizing percentages. I never realized it was this simple.

    Thank you.

  15. Hi Khalid,
    Keep helping the world in this Wonderful way.
    Thank you very much.
    Always waiting for ur new ideas in simplifying the concept.

  16. Khalid,

    Thanks for the lovely explanation.

    Even with the “wall” concept, it’s not intuitive why Tan(x) is a positive function in the 3rd Quadrant. That Tan(x) = Y component / X component or Sine(x)/Cos(x) kinda explains it – that both quantities are negative in the 3rd quadrant and hence the Tan function, which is a ratio, is positive. However, when you visualise the Tan function in the 3rd Quadrant, intuitively it feels like it should be negative.

    Added complication is that if you take the word TANGENT literally as a slope of the circle then at 90 degrees the slope of the tangent should be zero. But Tan function is undefined at 90 degrees (division by zero at this point).

    While if you visualize Tan as a magnitude/ length of the tangent, then it ought to be negative in the 3rd Quadrant. How can I visualize this better?

  17. @Kai: Always interested in checking out resources, though I’ll have to brush up (i.e. learn) some German!

    @krishnamoorthy: Thanks so much. I do think most ideas can be as simple as falling off a log if seen the right way.

    @Johan: Thanks! Hah, you should have seen the original post, which was about twice as big :). I’ll be doing a follow-up with some of that content.

    @Tom: Thanks. I was in the same boat, thinking I had to memorize everything. It’s almost like refactoring ugly code, sometimes there’s a simpler way to think about an existing problem which makes everything snap together.

    @Bhadrasheel: Appreciate it!

    @Ellie: Great question — how the trig angles behave in other quadrants is something I’d like to cover in the follow-up. (Article was getting big, something good for the follow-up!)

    Using the percentage analogy, tangent is the height relative to the wall distance, but each component can have a sign:

    Is the wall in front (positive) or behind (negative)

    Is the height above ground (positive) or below ground (negative)

    For example,

    * For x = -30, we are pointing “underground” so the tangent is negative.
    * For x = 120, we are pointing “backward”. The height is positive, but we are on the “back wall” so it’s negative.
    * For x = 210, we are pointing underground AND backward. So this is negative height on the back wall, which counts as positive :)

    This matches the signs for cosine (front wall / back wall) and sine (above ground / underground) so the calculations are the same :).

  18. @kalid: Actually I am doing the same that you are doing, breaking everything down, not taking formula as-is, trying to find the insights behind… but just in German and a tiny bit more animated 😉

    Some English speakers have asked me already to transfer my videos into English. I think I will give it a try this year, if I find time. I will send you a message as soon as the first video is ready. Just remember ‘Echt Einfach TV’ (which means Real Simple TV).

    Kind regards!

  19. Thank you for this wonderful intuitive explanation, Kalid! You’ve done it again!

    While I could follow the explanations, I did want to follow your advice and not get too hung up on an individual diagram. I also wanted to play around with the concepts, so I put together the following demos on the online Desmos calculator:

    Sine/Cosine: The Dome:

    Tangent/Secant: The Wall:

    Cotangent/Cosecant: The Ceiling:

    Visualize The Connections:

    Putting these demos together and seeing the results also helped make everything clearer, and I thought others might find these useful.

    Thanks again, Kalid!

  20. hi, thanks for another great article. my only suggestion would be the large triangle in ‘visualising connections’ and the others is that it (can) look as though each label is a bit of the line and hence the total length is the sum of all the functions – i take it you mean ‘when you go to this point, you use this function to give you the length’ ie, step up from say, 1 to csc to sec NOT (1+csc+sec) = lenth of line. not a biggy but if it caught me out for a couple minutes, it might end up a road-block for someone else.

  21. I saw this last night before going to bed, and used it this morning with my Geometry class as we began our Trig unit today.

    After reading it last night, presenting it to the kids this morning, and reading through this again, trigonometry finally makes intuitive sense to me.

    I am confident that this will help my students see this in a clearer light, and hopefully the handout that I put together to introduce sine and cosine today is helping them make meaningful connections.

  22. @kai: Sounds great, let me know and I’ll check them out.

    @joe: Great feedback, I’ll see if I can add a note to clarify. When you’re making the diagrams you tend to have all sorts of unstated assumptions which aren’t there for other people :).

    @Chris: That’s so awesome, I love it when the analogies come in handy for teaching. I really like how you’ve worked percentages into the worksheet, it puts a meaning behind the calculation (3/5… oh, that’s 60%!).

    @Ellie: Happy to help!

  23. This comment is to Ellie: why tangent is positive in the 3rd quadrant.

    Note that tangent is NOT the slope of the circle but the slope (=rise/run=sine/cosine) of the radius extending from the center to the unit circle. When that radius is extending to the 3rd quadrant, the slope remains the same (sign and size).

    Similarly, tangent is negative in the 2nd and 4th quadrants.

  24. Kalid, you have a beautiful way of explaining things. Your illustrations, intuition buildup and Aha! moments produce a snapping feeling in my brain. Everything just falls into place never to be forgotten again.

    I think that your articles are an invaluable gift to mankind. Keep it up! All the best.

  25. I am confused about the ceiling diagram. How come height traversed is always 1? Seems height can be bigger as the line extends beyond the dome.

  26. Hi vinay, try this interactive calculator for an example:

    When building a ramp up to the ceiling, the distance we travel depends on the angle we pick. However, the ceiling itself is always 1 unit above the ground. (In a building the ceiling is always a constant height, no matter how steep the stairs are to get there.)

  27. Dear friend, Thank you for this precious point of view about trigonometry.
    It was a pleasure reading your article. It was amazing!
    Congratulations! Thanks a lot!

  28. @Scott: Thanks so much for getting the interactive diagrams together, I’ve put them onto the post. (My apologies for the delay on approving that comment, it was stuck in my moderation queue because of an overactive spam filtering rule.)

    @Hector: Glad you enjoyed it!

  29. “Every circle is really the unit circle, scaled up or down to a different size.” This one did it for me. :)

  30. First off…I am completely in agreement with Alecks on the insight. As I spend time presently on the calculus of integration of trigonometric functions I can’t wait to hear the insight on that one. (Found it ….the downside of this site… is that I am starting to wait for Kalid to provide me the insight….must work on that).

  31. Thanks Mark — hah, don’t want anyone getting dependent! =)

    I’d like to do a follow-up on the calculus properties of the trig functions, now that I have a better understanding of them myself…

  32. Kalid, your website has added an immense amount of intuition to my understanding of mathematics. Thank you for your fresh approach to the topics you cover. After reading this I began getting into hyperbolic trig functions. Most of which, I can’t find anything that provides much intuition on the subject. I think, since you love e so much, you could provide a lot of intuition on these functions since their definitions involve 2 terms of e. Thanks for your time.

  33. Somehow last night I went from the triangles in the circles to the wedges formed by the secant and tangent lines. With that, the area of the 30 60 90 triangle with one leg length of 1 becomes 1/(2V3) [ one over 2 root three]. That is the beauty of the insights you provide they build up our own abilities to make new connections.

  34. Thanks Mark, that’s a cool extension. I’ve been milling about, thinking of other intuitions that can pop out (such on the Law of Sines), hope to share them down the road too.

  35. Thanks Andrew, I’d like to do a follow-up on hyperbolic trig functions — their connection to e is pretty neat. Also, we can even define the regular trig functions in terms of e as well :).

  36. Kalid if u remember I messaged you regarding trigzz !! .. and now seeing it … huge whoo moment you re better than my science teachers better say crammer robozz :00 !! U re genius indeed

  37. Marvelous!

    Being a Physics teacher I have to give an insight into basic mathematics(trigs, calculus, probability, complex nos etc) to my students and I so far for trigs I use the circle/tangent analogy. But your dome analogy is far more efficient and natural.

    Thanks for this wonderful insight!

  38. Thanks Harish, really glad it helped :). [I love being able to provide teachers with new analogies to try out.]

    It was only recently (i.e. a few months ago) that trig started clicking this way, I wish I’d had it as a student too!

  39. Hey K–

    This is beautiful, but how about adding, while you’re at it, the derivatives of the sin and cos functions. The explanations I’ve seen are understandable but more elaborate than intuitive.


  40. Hey Tim! Great idea, I’d like to cover the derivatives of sine/cosine in a follow-up. I’m still working on a solid intuition beyond the definitions/calculus reasoning :).

  41. Thanks Kalid,

    Your triple triangle diagram and the ‘tanned gent you can see’ tip certainly clarifies the relationships of the ratios – when interpreted correctly. Also the fact that the unknown sides are percentages of the known sides is seriously illuminating.

    You could shorten the explanation by cutting some of the anatomy content as well as the higher Trig references. But well done for this explanation which I am unlikely to forget anytime soon.


  42. Thanks Alan, glad the analogies helped. The anatomy part helps me realize the role of trig (way to explore an alien shape) but everyone has a different takeaway :).

  43. I just love how you explained that SUC-A-TOE(A) does not work(as I call it). I have been battling this issue with teachers for years now. And yes, I do know that it’s SOH-CAH-TOA
    I prefer to teach “CONCEPTS” then to just give “quick” ways to “memorize” a formula. No learning occurs!

  44. Thanks Diana! I agree, memorizing acronyms is a poor substitute for internalizing the actual concept (they should serve as reminders, not lessons).

  45. This is my first comment on your site & wanted to express my gratitude. I just hate how all my math teachers except for one(alg2 teacher) teach math in time-consuming, unnecessary, & confusing manner. I like your use of “thought experiments” & explaining the underlying concepts. It makes Math extremely simple & helps with more advanced topics that use the ones you teach as the basis. I honestly do not know of another site dedicated to teaching the underlying concepts as a means to understand the topic overall. You have been a godsend for me in math. I cant believe you aren’t way way way more popular bc of how good you are in “deciphering” the “encryption” the majority of math teachers place on math topics/concepts.

  46. Hi Kalid,
    Once again, thank you for helping everyone see how the trig functions can be applied in real life. I often feel there’s a disconnection between the contents learned in maths and my actual life, and now I can happily link them together in a more intuitive sense.

    However, I do believe there’s still a fair bit of rote required despite the intuitiveness of your explanation. It’s not immediately apparent which terms equal to ONE and which ones are free to extend beyond the unit circle e.g. what I mean is that it’s not immediately obvious to associate tangent and secant to the wall example, and COtangent and COsecant to the ceiling. Perhaps you can share some insight as to how you came about the two above examples. Your epiphany or ‘aha’ moments that led you to write these examples or the thought process you went through to get to your wonderful analogy, so that as learners, we’re not overly reliant on others to come up with an effective method of learning what you’ve dubbed as the anatomy of math , as was mentioned earlier on by @Mark Ptak.

  47. Hi Flora, thanks for the comment and great feedback.

    I probably wasn’t clear enough in the analogy. I imagine the dome (distance of 1) as a type of boundary, where sine/cosine are stuck in the dome (their max value is 1, min value -1 when facing backwards) and all other trig functions exist outside the dome and can take on nearly any value. (Technically, sec and cosec have a minimum distance of 1, so can take any value from 1 to infinity, or -1 to -infinity when facing backwards). I’d like to do a follow-up analyzing some more of the behavior, as students are often forced to graph the values of trig functions (it’s better to visualize what values they can take on).

    For learning, I’d like to describe the process a bit more as well. I have a general article on my strategy (, for this example I started thinking about what circular objects in the real world might represent the unit circle. I thought of a dome (after too many IMAX movies maybe?) and then the screen was a natural way to represent some partial height (sine) and distance away (cosine). From there, I was able to imagine other “buildings” around me that might represent the screen in other positions — it turned out the trig functions showed up there too. A lot of it is trial and error where you hunt around for an analogy that seems to cover a few use cases (it doesn’t have to cover them all). I hope to write more about this too.

  48. Hi Kalid,

    Excellent article, thank you VERY much for sharing. My life would be so much easier in school if teachers made these relations clear.

    I got confused in the “Tangent/Secant: The Wall” section, where in the diagram you say “secant = ladder “hypotenuse””, but in the embedded calculator you say “% of hyp.” where “hyp.” means the radius. So, the hypotenuse is the secant or the radius?

  49. Hi Aurelio, really glad it was helpful.

    Good question on the meaning of the hypotenuse. In my head, I see the hypotenuse as the “longest side of the triangle”.

    When we’re in the dome, the hypotenuse [longest side] touches the dome, and is equal to the radius. When we have the secant, we’re drawing a line all the way to the wall [which extends outside the dome]. In this case, the secant is written in terms of the hypotenuse, such as 2.5 or 10. [So the secant is 250% or 1000% larger than the original hypotenuse which is stuck inside the dome. This is also the radius of the dome.]

  50. You want to rewrite the science of Trigonometry?
    you are confusing the trig relationship with unite circle!
    you seem to be a theatre lover and only thinking of screen and your place in the theatre ! well not everyone !!

    to me you seem to measure water by Grains size !!

  51. Anonymous or what do you call yourself, we know you are a genius but keep it to yourself. We are satisfied with what Khalid is givining. If you so much know trig as you claimed, why have we not heard about you uptil now? Are you not aware that people don’t know mathematics? Are you telling me that that the traditional teaching style has helped maths? Now, someone comes up to help majority of us, you are not included, you are scathingly criticising. Please, go and sit down.

  52. My guess is he or she is either one of those teachers who have been pursuing their students to memorize trig formulae OR one of those purists who make it their goal of life to keep things ‘as they were’ and so are blind to actual logic.

  53. Hi Kalid,

    I really like your articles. This is a second thank-you note from me.

    I keep aside a few hours every week for exploring math, and I read this article last week during that time. Now, today, during one of those hours, I was amazed to find that the explanations were still clear in my brain. I mean, I can really visualize the sin/cos etc. Thanks a lot.

  54. Hi Kalid,
    Longtime lurker of your site, admirer of your ability to intuit.I appreciate examples like the hellish voyage on the appendix. I found your site way after leaving college and things are finally becoming clear to me. Thanks for your passion.

  55. Identifying pause and digest moments.. there are several:
    11 min video broken down
    0 Motiv/intro
    0:38 Trig as anatomy… what are the ratios
    2:00 Analogy of the Dome, Screen, Distance to Screen
    3:27 Think of in terms of percents
    4:46 Projecting on a wall
    6:17 Ramp to the ceiling
    7:25 Connections: Putting it all together
    8:38 “Facts” or relationships we get
    10:05 Wall or Ceiling first
    For those learning this, realizing utility of pausing at particular moments could be helpful.

    Great video!

  56. Kalid:

    The fact that trig relations are percentages is a very good insight! I must compliment you.

    I would have a modest(?) proposal: I am of an older generation, did my studies in the 50’s. We had a visual tool always at our side, an analogue calculator, a thing called a SLIDE RULE. It is based on logarithms, and you multiply on it by SLIDING one log number line scale against another. My big slide rule has trig functions and exponential functions in the form of number lines that refer to the basic log scale. It’s all out there in front, right before your eyes. I think my generation and those before us had a better visual and intuitive grasp of math because of the slide rule. Almost all calculations on a slide rule are based on proportions. They become natural if you work with a slide rule. And: you have to keep orders of magnitude straight in your head. A slide rule user must always know where they are in a calculation. You might make little mistakes in the 3rd or 4th significant digit, but never a BIG electronic calculator type of mistake.

    I have never stopped using a slide rule and never really graduated to electronic calculators. I think it is worth anybody’s while who is interested in an intuitive feel for math to look into slide rules. Their are lots of them floating around and the web has lots of info on how to use them.

  57. Hi Leo, thanks for the comment, glad the percentage analogy worked!

    The slide rule is a really interesting device (I’ve only used it in online simulations, never in real life) but I think there’s cool insights within as well. Having a tangible representation of what logs are doing is another way to build an intuition, might be a fun article!

  58. Kalid:

    One might get a taste by using an online simulation, but there is no substitute for holding one your hands. They are precision instruments, the final product of 300 years of development and evolution. And they were expansive. My high school slide rule cost $55 in 2014 dollars, and my big scientific-engineering one was $250. Both are in working order today after 55-60 years. Back then, a person bought usually one high quality one in a lifetime. The German-American rocket engineer Werner von Braun worked on the Apollo moon shots with one he bought in the early 1920’s.

    (I have never replaced a set of batteries…)

    I am not suggesting one really learns to use one seriously. With all the relations there in front of you, they are great objects for meditations on mathematics. Most of your insights in this site could be arrived at by figuring out precisely how a slide rule works. If a person uses a slide rule a lot, you can estimate the result of a computation by closing your eyes, picturing a slide rule, and operating it in your imagination. Calculators are of course remarkable, but they are “black boxes”. Nobody knows what’s going on inside. The result pops out magically. They are highly anti-intuitive.

    Get your hands on a physical slide rule. Beware: they cast a spell.

    I am working my way through your entire site. Great stuff, new little things at every point.

  59. Thank you! This helps a lot! I am on winter break, and am studying math on my own to ” understand” it instead of just memorizing it. I’ve had two physics classes and up to Calculus I but I am very weak at Trig because I have no idea, it just doesn’t click. Calculus was way easier, but trig popped up everywhere in it and in physics too and I was just at its mercy! The diagram was the best… I always wondered where the other identies were located on the triangles/ circle dome. the most useful diagram next to the original unit circle I’ve ever come across!

  60. @Leo: Thanks for the info, glad you’re enjoying the site :).

    @Sarah: Awesome to hear! Trig bugged me for so long as well until I found a way to have the relationships click.

  61. Salam wa alay kum ….Mr. Khalid and very very thanx for providing such sites …I think this is the best site for math seekers….

  62. Hi.
    This posting is great! Makes it a lot easier to understand. I knew how to do Trigonometry but realized just how tricky it was when I had to try & explain it to my teenage son. This certainly made it easier. Thank you.

  63. Just a small tip for anyone out there using Google as a calculator for any of these functions, beware that it returns results in radians. I was puzzled for a few minutes on Kalid’s example to solve for sine of angle x until I realised the answer Google was computing for arcsin(.60) [0.64] was in radians. Performing a conversion from that to degrees (by multiplying the radian value by 180/π) yielded the expected result of 36.9 degrees. Happy Math!

  64. I could always work with the formulas, but had a hard time visualizing them and understanding them intuitively. This really helped, thank you!

  65. Hey Khalid, it’s really nice to see how you demystify many of the concepts that we learned in high school. I’ve always tried to get some intuition but failed so far so I gave up on this quest. It’s refreshing to see how you’ve been able to make it work. However, I would love to see how you reasoned your way through to getting these 3 similar triangles all stacked up on top of one another. Can you please post an article or a reply to this comment? Would love to see more of your work!

  66. Thanks for the article – it’s great! One question though, is there a reason why the tangent is always vertical??

  67. Hi Geraint, the tangent doesn’t have to be vertical, but it’s lined up that way for simplicity in the diagram. See the section of the article Appendix: The Original Definition Of Tangent.

  68. Hi Ayubi, glad you enjoyed it! It might be a fun article, but I essentially look for connections between things wherever I can. All the trig functions seem to have similar relationships (something^2 + something^2 = something^2) which fits the Pythagorean model. Seeing everything as percentages seemed to help clarify as well (tan is just another percentage, except it can go to infinity). A lot of it is trial and error and a belief that things can be simple if we look at it the right way (and it may take a lot of time before it jumps out at us).

  69. Wow…I feel like my foot has been itching since high school, and it took me 18 years until I was finally able to take off my shoe and scratch it. I can’t believe how much sense this post makes. Kudos, Kalid, for making this subject so easy and understandable! If only all high school teachers everywhere would watch your videos.

  70. “[…]like biologists who find a link between your fibia and clavicle.”
    What in the world is a fibia? Of all my years in sports medicine and direct research in human performance, I have never heard of a fibia. I am assuming you either meant tibia or fibula. Thanks for the laugh.

  71. @sayed: The ceiling and wall are very similar. If you flip the wall and ceiling, the tangent/cotangent are switched as well. This is the same as swapping an angle for its complement, so tan(a) = cot(90 – a).

  72. Thank you so much for this Kalid! You are absolutely amazing!! :) It’s a real shame that maths isn’t taught this way at school. I am starting to enjoy maths so much more after reading your articles. Just in time, as I’m starting my computer science degree soon! ^_^

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