Equation sin α = √3/2

Statement

Find all $\alpha\in[0,2\pi)$ such that $\sin\alpha = \tfrac{\sqrt{3}}{2}$.

Solution

From the table of notable values, $\sin\!\left(\tfrac{\pi}{3}\right) = \tfrac{\sqrt3}{2}$: one solution is $\alpha = \tfrac{\pi}{3}$ (first quadrant). Since the sine is positive in the second quadrant as well, the other solution is the supplementary arc: $\alpha = \pi - \tfrac{\pi}{3} = \tfrac{2\pi}{3}.$ In the third and fourth quadrants the sine is negative, so there are no further solutions in $[0,2\pi)$: $\boxed{\alpha = \tfrac{\pi}{3}\ \text{oppure}\ \alpha = \tfrac{2\pi}{3}}.$

Links

Topics: Goniometry
Concepts: Associated arcs · Trigonometric equation · Sine
Functions: Sine
Skills: Reasoning by cases · Solving equations
Exercise types: Solving an equation