Chapter 1: Trigonometry

1.1 Trig Functions

For any right triangle, there are specific mathematical relationships between the angles and the lengths of the sides.

A right triangle oriented with the 90 degree angle at the bottom left. The other corner on the bottom of the triangle is labeled with the Greek letter theta. The vertical side is labeled "opposite," the bottom side is labeled "adjacent," and the long side extending from upper left to lower right is labeled "hypotenuse."
A right triangle, labeled with an angle and corresponding sides

In the figure above, the sides “opposite” and “adjacent” are relative to the angle that is labeled as 𝜃 (Greek lowercase theta). The hypotenuse is the longest side. It is also the side opposite from the right angle.

The relationships are called trigonometric (or trig) functions, and are

\[
\begin{align}
\sin(\theta) &= \frac{\text{opposite}}{\text{hypotenuse}} \tag{1.1} \\ \\
\cos(\theta) &= \frac{\text{adjacent}}{\text{hypotenuse}} \tag{1.2} \\ \\
\tan(\theta) &= \frac{\text{opposite}}{\text{adjacent}} \tag{1.3}
\end{align}
\]

These are all mathematical functions with input and output values. For the trig functions, the input values are lengths of the sides, and the output is given by dividing one length by the other. You read the functions as “sine of theta,” “cosine of theta,” and “tangent of theta.” (Sometimes when saying this out loud, the “of” gets dropped, e.g., “sine theta.”)

You may find using a mnemonic, a phrase where the first letter of each word in the phrase corresponds to the first letter in each word of the definitions above, helps you remember these. You can use whatever mnemonic works for you; my personal favorite is Some Old Hippie Caught Another Hippie Tripping On Acid. (The S in Some stands for the s in sine, the O in Old stands for the o in opposite, and so on.)

It is more common that you will use variables to represent the lengths of the sides. Consider the triangle in the figure below, where the side opposite from the angle \(\theta\) is labeled as side \(a\), the side adjacent to the angle is side \(b\), and the hypotenuse is side \(c\). The trig functions would then be:

\[ \sin(\theta) = \frac{a}{c} \qquad \cos(\theta) = \frac{b}{c} \qquad \tan(\theta) = \frac{a}{b} \]

A right triangle oriented with the 90 degree angle at the bottom left. The other corner on the bottom of the triangle is labeled with the Greek letter theta. The vertical side is labeled "a," the bottom side is labeled "b," and the long side extending from upper left to lower right is labeled "c."
Right triangle labeled with variables

Example

Practice

Practice

Practice

Practice