I had a hard time understanding trigonometry in school. I recently checked out some textbooks and remembered why it was so tricky.

In those books, the basic trigonometric functions were introduced like this:

These statements are accurate, but they don’t really help you understand. It’s hard to feel how it works.

Let’s try to figure it out in a different way.

Here’s a simple angle tool. You can move the yellow line to create any angle, from 0° to 360°:

Let’s say = 1
(1 meter, 1 foot, or just 1 — whatever works for you)

Now we can measure the vertical position of :

We’ve just defined the sine!

Sine is all about the vertical position. It’s 0 when we start at 0°, and then it grows to 1 at 90°. Try different angles and see how it changes.

Let’s move on to the horizontal position:

The horizontal position of is the cosine. It changes in a different way: when we start at 0°, it equals 1, but as we move upwards, it shrinks to 0.

Let’s connect the sine and cosine lines. You can form plenty of right triangles:

You can see some interesting patterns there. But the most valuable insight is this:

We know how and are related to in any right triangle.

For any acute angle — be it 5°, 45°, or 89° — you’ll always get the same triangle shape, and the same relationship between its sides.

In any right triangle with a 30° angle, is always ½ of . Because sin 30° = 0.5.

In any right triangle with a 75° angle, is always about 0.26 × . Since cos 75° ≈ 0.26.

And so on!

How does it work if has a different length?

Not much more complex! Sine and cosine are just multiplied by that length.

(In the demo below, we’ll operate in the 0–90° range, where sine and cosine are non-negative.)

Sine and cosine just tell you what fraction of the hypotenuse each side is.

You don’t need to memorize those textbook definitions. If you remember what sine and cosine look like on the circle, everything else should follow.

To be continued: Tangent, and why they replace degrees with all those π and 2π things.