1.1 Trig Functions
For any right triangle, there are specific mathematical relationships between the angles and the lengths of the 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:
\[ \sin(\theta) = \frac{a}{c} \qquad \cos(\theta) = \frac{b}{c} \qquad \tan(\theta) = \frac{a}{b} \]
Example 1.1
Find sin(𝜃), cos(𝜃), and tan(𝜃) for the diagram below.
First, identify the hypotenuse. Then, identify each side’s relationship to the angle. The longest side is the hypotenuse, so side with a length of 17 units is the hypotenuse. The side opposite from the angle 𝜃 has a length of 8 units, and the side adjacent to the angle 𝜃 has a length of 15 units.
\begin{align}
\sin(\theta) &= \frac{\text{opposite}}{\text{hypotenuse}} & \cos(\theta) &= \frac{\text{adjacent}}{\text{hypotenuse}} & \tan(\theta) &= \frac{\text{opposite}}{\text{adjacent}} \\
&= \frac{8}{17} & &= \frac{15}{17} & &= \frac{8}{15} \\
&= 0.471 & &= 0.882 & &= 0.533
\end{align}
In physics class, you’ll almost always give your answer in decimal form, rather than as an exact fraction. It is common practice to round to two or three decimal places. You’ll learn more about rounding in chapter 3.