How To Learn Trigonometry Intuitively

Trig mnemonics likeSOH-CAH-TOAfocus on computations, not concepts:

TOA explains the tangent about as well as $x^2 + y^2 = r^2$ 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 arepercentagesto 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 typicalbody proportions:

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

How is this helpful?

嗯，当鲍勃找到一件夹克时，他可以把它捡起来，伸出手臂，然后估计主人的身高。和头部大小。和眼睛的宽度。一个事实与各种结论有关。

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鲍勃是外星人吗，在数学世界研究生物!

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 andgamsin, like biologists who find a link between your tibia and clavicle.

And because triangles show up in circles…

…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

与其像被冻在冰里的穴居人一样盯着三角形看，不如想象他们置身于一个场景中，猎捕猛犸象。

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.

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’sbigdagnabbit.

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.

假设我付了\ 14美元的税。你不知道那有多贵。但如果我说我付了95%的税，你就知道我被宰了。

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’swhy 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.)

让我们假设我们在单位圆上(半径为1)来简化计算。现在我们可以跳过除1，只说sin = 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:

sin的增长模式不是一条偶数线。前45度覆盖了70%的高度，最后10度(从80到90)只覆盖了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.

切/Secant: The Wall

One day your neighbor puts up a wallright nextto your dome. Ack, your view! Your resale value!

But can we make the best of a bad situation?

当然。我们把电影屏幕挂在墙上怎么样?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

我们有一些新奇的词汇。想象一下，看到投影在墙上的维特鲁威“谭先生”。你爬上梯子，确保你能“看见，不是吗?”(是的，他是裸体的……你现在不会忘记这个比喻了吧?)

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

它从0开始，无限高。你可以一直指向更高的墙，以获得无限大的屏幕!(That’ll cost ya.)
tan是sin的放大版!它永远不会变小，而当sin随着穹顶的弯曲而“顶部关闭”时，切线一直在增长。

How about secant, the ladder distance?

Secant starts at 1 (ladder on the floor to the wall) and grows from there
sec总是比tan长。用来搭屏风的倾斜的梯子一定比屏风本身要长，对吧?(在巨大的尺寸下，当梯子接近垂直时，它们就很接近了。But secant is always a smidge longer.)

Remember, the values arepercentages. 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…)

是时候修个通向天花板的坡道了，然后聊聊天。You pick an angle to build and work out:

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

切/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 hadzero关于余割的"直观结论"But with the dome/wall/ceiling metaphor, here’s what we see:

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:

From thePythagorean Theorem($a^2 + b^2 = c^2$) we see how the sides of each triangle are linked.

And fromsimilarity, 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.)

这就是我们求sin /cos = tan /1的方法。

我总是试图记住这些事实，当它们在脑海中浮现时。SOH-CAH-TOA是一个很好的捷径，但首先要真正理解!

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:

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

你不需要记住$1^2 + \cot^2 = \csc^2$，除了一些愚蠢的测试，错误的琐事。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 acheatsheetfor yourself.

In a follow-up, we’ll learn about graphing, complements, and usingEuler’s Formulato 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).

正如所料，在圆(x=90)的顶部，切线永远达不到x轴，并且是无限长。

I like this intuition because it helps us remember the name “tangent”, and here’s a niceinteractive trig guideto explore:

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

三角函数取一个角度，返回一个百分比。$\sin(30) = .5$表示30度角是最大高度的50%。

The inverse trig functions let us work backwards, and are written $\sin^{-1}$ or $\arcsin$ (“arcsine”), and often writtenasinin various programming languages.

如果高度是穹顶的25%角度是多少?

Plugging asin(.25) into a calculator gives an angle of 14.5 degrees.

那么奇异的东西呢，比如逆sec ?通常情况下，它不能作为计算器功能使用(即使是我创建的计算器，唉)。

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:

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

假设割线为3.5，即单位圆半径的350%。和墙的角度是多少?

$\begin{aligned} \frac{\sec}{1} &= \frac{1}{\cos} = 3.5 \\ \cos &= \frac{1}{3.5} \\ \arccos(\frac{1}{3.5}) &= 73.4 \end{aligned}$

Appendix: A Few Examples

Example: Find the sine of angle x.

哎呀，多无聊的问题啊。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

$\begin{aligned} 3^2 + 4^2 &= \text{hypotenuse}^2 \\ 25 &= \text{hypotenuse}^2 \\ 5 &= \text{hypotenuse} \end{aligned}$

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

Follow-up: Find the angle.

当然可以。我们有几种方法。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?

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 isHubert’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’dliketo 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

$\begin{aligned} \sec &= \frac{1}{\cos} = 8 \\ \cos &= \frac{1}{8} \\ \arccos(\frac{1}{8}) &= 82.8 \end{aligned}$

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.

还不错吧?在穹顶/墙壁/天花板的类比之前，我将淹没在一堆乱七八糟的计算中。把这个场景可视化会让它变得简单，甚至有趣，看看哪个三角函数伙伴可以帮助我们解决问题。

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 ofGrey Mattersput together interactive diagrams for the analogies (drag the slider on the left to change the angle):

Thanks!