How to reduce an angle to the first quadrant

The formulae for associated arcs are almost always used in the same way: to reduce the computation of a general angle to an angle in the first quadrant.

In brief — How to use them in practice

Given a general angle (even negative, or greater than $2\pi$, or in one of quadrants II, III, IV), to compute its $\sin/\cos/\tan$:

1. reduce it to the interval $[0,2\pi)$ by subtracting multiples of $2\pi$;
2. if it is in quadrant II/III/IV, apply the appropriate associated arc to bring it back to the first quadrant;
3. compute it using the table of special values or the definition.

Example — A second-quadrant angle

Compute $\sin\!\left(\tfrac{5\pi}{6}\right)$.

$\tfrac{5\pi}{6}$ is in the second quadrant ($\pi/2 < 5\pi/6 < \pi$). We use supplementary arcs: $\sin\!\left(\tfrac{5\pi}{6}\right) = \sin\!\left(\pi-\tfrac{\pi}{6}\right) = \sin\!\left(\tfrac{\pi}{6}\right) = \tfrac{1}{2}.$

Example — A negative angle

Compute $\cos\!\left(-\tfrac{7\pi}{4}\right)$.

First we use opposite arcs: $\cos\!\left(-\tfrac{7\pi}{4}\right) = \cos\!\left(\tfrac{7\pi}{4}\right)$. Then we note that $\tfrac{7\pi}{4} = 2\pi - \tfrac{\pi}{4}$, so it is in the fourth quadrant: $\cos\!\left(2\pi - \tfrac{\pi}{4}\right) = \cos\!\left(-\tfrac{\pi}{4}\right) = \cos\!\left(\tfrac{\pi}{4}\right) = \tfrac{\sqrt{2}}{2}.$

Links

Topics: Trigonometry
Concepts: Special angles · Associated arcs
Skills: Calculating · Reasoning by cases · Using formulae