4.4 | Trigonometric Functions of Any Angle


Let \(\theta\) be an angle in standard position. Its reference angle is the acute angle \(\theta’\) formed by the terminal side of \(\theta\) and the horizontal axis.
The right-triangle definition, unlike the unit circle, is not defined for values of \(\theta \geq \pi/2\) or \(\theta \leq 0\). We extend this definition to allow for such angles by defining
$$
\sin \theta =
\begin{cases}
\sin \theta’, \; & \text{ if } \theta \in Quad. I \text{ or } II \\
-\sin \theta’, \; & \text{ if } \theta \in Quad. III \text{ or } IV
\end{cases}
$$
and
$$
\cos \theta =
\begin{cases}
\cos \theta’, \; & \text{ if } \theta \in Quad. I \text{ or } IV \\
-\cos \theta’, \; & \text{ if } \theta \in Quad. II \text{ or } III
\end{cases}
$$
The remaining four basic trigonometric functions are defined in terms of these two.

E 4.4 Exercises