# 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.