We will start with a simple right triangle. The longest side of a right triangle is known as the hypotenuse and the other two sides are called legs. The Pythagorean Theorem states that the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the legs. But that’s not typically how people remember it. If we were to label the lengths of the legs as \(a\) and \(b\text{,}\) and the length of the hypotenuse as \(c\text{,}\) then we would have the relationship \(a^2 + b^2 = c^2\text{,}\) which is the way most people remember it.

The core trigonometric relationship we need for physics are the relationships for the sine and cosine functions for right triangles. If we draw a right triangle and label one of the acute angles \(\theta\text{,}\) we can label the sides according to the following DIAGRAM and have the following relationships:

The sine and cosine functions are mathematical functions whose values you will get from a scientific calculator. Most of the work we will do at this level will have the angles measured in degrees, and you will want to make sure that your calculator is in the correct mode.

Here are the graphs of the sine and cosine functions. There is a lot that could be said about these functions, but the main thing to focus on is the shape of these graphs between 0 degrees and 90 degrees. The sine function starts at 0 and increases to 1, while the cosine function starts at 1 and decreases to 0, and the two graphs have the same value at 45 degrees.

GRAPHS

The tangent function is another trigonometric function. It’s defined by \(\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}\text{.}\) Since this function is defined in terms of sine and cosine, we technically don’t even need it. However, this combination ends up being quite useful, and so it’s worth knowing about it. The tangent function is usually found in the context of slopes, and it has the property that its value is 0 when the angle is 0, and this value gets infinitely large as the angle approaches 90 degrees. At this point, we won’t say anything more about it.

The basic applications of trigonometry in physics are to determine the length of the side of a triangle and to determine an angle. The most common configuration that we see is that we know the length of the hypotenuse of the triangle and we know one of the angles, and our task is to determine the lengths of the two legs. In the context of vectors, this is often referred to as vector decomposition.

The following diagram is useful enough that it’s worth memorizing. The assumptions here are that you know the angle \(\theta\) and that the length of the hypotenuse is \(L\text{.}\) Then by using the formulas for sine and cosine, you can determine the lengths of the two legs. This triangle shows up in many different orientations, and so you won’t want to remember it in terms of the words "horizontal" and "vertical." That will lead to many errors. It is better to remember that the cosine function is adjacent to the angle and the sine function is opposite the angle.