Values using associated arcs

Statement

Compute (without a calculator): $\sin\!\left(\tfrac{2\pi}{3}\right)$, $\cos\!\left(\tfrac{5\pi}{4}\right)$, $\tan\!\left(\tfrac{7\pi}{6}\right)$, $\sin\!\left(-\tfrac{\pi}{3}\right)$.

Solution

Reduce each angle to the first quadrant using associated arcs.

• $\tfrac{2\pi}{3} = \pi - \tfrac{\pi}{3}$ (second quadrant, supplementary arcs): $\sin\!\left(\tfrac{2\pi}{3}\right) = \sin\!\left(\tfrac{\pi}{3}\right) = \boxed{\dfrac{\sqrt3}{2}}$.
• $\tfrac{5\pi}{4} = \pi + \tfrac{\pi}{4}$ (third quadrant, arcs differing by $\pi$): $\cos\!\left(\tfrac{5\pi}{4}\right) = -\cos\!\left(\tfrac{\pi}{4}\right) = \boxed{-\dfrac{\sqrt2}{2}}$.
• $\tfrac{7\pi}{6} = \pi + \tfrac{\pi}{6}$ (third quadrant): $\tan\!\left(\tfrac{7\pi}{6}\right) = \tan\!\left(\tfrac{\pi}{6}\right) = \boxed{\dfrac{\sqrt3}{3}}$.
• Opposite arcs: $\sin\!\left(-\tfrac{\pi}{3}\right) = -\sin\!\left(\tfrac{\pi}{3}\right) = \boxed{-\dfrac{\sqrt3}{2}}$.

Links

Topics: Goniometry
Concepts: Notable angles · Associated arcs · Cosine · Sine · Tangent
Functions: Cosine · Sine · Tangent
Skills: Reasoning by cases · Using formulae
Exercise types: Evaluating expressions