Chapter 5: Trigonometry
Ratios of Right-Angled Triangles
Given a point in the $xy$-plane, we can create a right-angled triangle by considering the diagram below. The angle $\theta$ is called a reference angle. If we compare any two side lengths of this triangle, we can form six different ratios. These ratios are called trigonometric ratios and are defined below.
$
\begin{align*}
\displaystyle \cos(\theta) &= \frac{x}{h}\\
\\
\displaystyle \sin(\theta) &= \frac{y}{h}\\
\\
\displaystyle \tan(\theta) &= \frac{\sin(\theta)}{\cos(\theta)} = \frac{y}{x}
\end{align*}
$
$
\begin{align*}
\displaystyle \sec(\theta) &= \frac{1}{\cos(\theta)} = \frac{h}{x}\\
\\
\displaystyle \csc(\theta) &= \frac{1}{\sin(\theta)} = \frac{h}{y}\\
\\
\displaystyle \cot(\theta) &= \frac{1}{\tan(\theta)} = \frac{x}{y}
\end{align*}
$
