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.
$$
\sin \theta =
\begin{cases}
\sin \theta’, \; & \text{ if } \theta \in Quad. I \text{ or } II \\
-\sin \theta’, \; & \text{ if } \theta \in Quad. III \text{ or } IV
\end{cases}
$$
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.