Introduction

The last thing we need to talk about before we can actually start on our physics content is trigonometry. If you're unfamiliar with what that is, here's the basic breakdown: trigonometry describes the relationships between the sides of right triangles by using the angles of the triangles. Advanced trigonometry can be used to describe the relationships between the sides of any triangle, but that's more of a math thing than a physics thing.

Basic trigonometry is used to find the lengths of sides of right triangles, and the angles of right triangles. The most common functions used in basic trigonometry are sine, cosine, and tangent. Each of these is actually a ratio of two sides on a right triangle, and they are functions of the angles of the triangle. This is why we will often write them as $\sin(\theta)$, $\cos(\theta)$, and $\tan(\theta)$, where $\theta$ is the angle of the triangle.

Enough basic explanation. What do each of these functions actually mean? Let's look at a right triangle to get started:

I've labelled what I'm going to call each side, and I'll refer to them as such from now on. Let's talk about the sine function first. The sine of an angle is defined as the ratio of the opposite side to the hypotenuse. In other words:

$$\sin(\theta) = \frac{\text{opposite} }{\text{hypotenuse} }$$
This means that if you know the angle and the length of the hypotenuse, you can find the length of the opposite side by multiplying the hypotenuse by the sine of the angle. This is pretty useful for things later on. Now, let's move on to the cosine function.

The cosine of an angle is defined as the ratio of the adjacent side to the hypotenuse. In other words:

$$\cos(\theta) = \frac{\text{adjacent} }{\text{hypotenuse} }$$
This means that if you know the angle and the length of the hypotenuse, you can find the length of the adjacent side by multiplying the hypotenuse by the cosine of the angle. I'm starting to sound like a broken record, aren't I? But this is important, so bear with me. Finally, we have the tangent function.

The tangent of an angle is defined as the ratio of the opposite side to the adjacent side. In other words:

$$\tan(\theta) = \frac{\text{opposite} }{\text{adjacent} }$$
This means that if you know the angle and the length of the adjacent side, you can find the length of the opposite side by multiplying the adjacent side by the tangent of the angle. Final function down! Now, you're a master at the basics of trig!

There are some common angles that you should know the sines and cosines of. Of course, if you have access to a calculator it's not strictly essential, but I still think that these three angles are very good to know. I'm of course talking about $30°$, $45°$, and $60°$.

These values are worth memorizing as they come up frequently in physics problems. Notice that the sine and cosine values for 30° and 60° are swapped - this is because these angles are complementary (they add up to 90°). This is generally true for all complementary angles due to how the right triangle works. That's because the complementary angle of a right triangle is the third angle of the triangle (the other two are the angle of interest and the right angle). This means that the sine of one angle is equal to the cosine of the other angle, and vice versa.

Now, we can also find angles when given two sides of any right triangle by using the inverse functions of sine, cosine, and tangent. The actual process is a little bit more complicated, but for this course we don't need to know the intricacies of how the functions work. We just need to know that we can find the angle by using the inverse functions by plugging in the value of the trig function into the inverse function in our calculator.

So, if you have a ratio that represents the sine of an angle, you want to plug that ratio into the inverse sine function, which is usually written as $\sin^{-1}\theta$ or $\arcsin\theta$. The other two are written as $\cos^{-1}\theta$ or $\arccos\theta$ and $\tan^{-1}\theta$ or $\arctan\theta$.

This is just surface-level trigonometry, and there is a world of other topics to learn about angles and triangles and triangular angles. This is good enough of an introduction for our purposes, however. Let's try a few very simple practice problems now to test your understanding of this topic.