Find all solutions of each equation. sin x = (√3)/2

Verified step by step guidance

1. Recognize that the equation is \(\sin x = \frac{\sqrt{3}}{2}\). This means we need to find all angles \(x\) where the sine value equals \(\frac{\sqrt{3}}{2}\).
2. Recall the unit circle values for sine. The sine of an angle is \(\frac{\sqrt{3}}{2}\) at specific standard angles. Identify these angles in the interval \([0, 2\pi)\), which are \(x = \frac{\pi}{3}\) and \(x = \frac{2\pi}{3}\).
3. Since sine is periodic with period \(2\pi\), write the general solutions by adding integer multiples of \(2\pi\) to each of the principal solutions. This gives \(x = \frac{\pi}{3} + 2k\pi\) and \(x = \frac{2\pi}{3} + 2k\pi\), where \(k\) is any integer.
4. If the problem domain is not specified, state that these general solutions cover all possible values of \(x\) for the equation \(\sin x = \frac{\sqrt{3}}{2}\).
5. Summarize the solution set as all angles \(x\) such that \(x = \frac{\pi}{3} + 2k\pi\) or \(x = \frac{2\pi}{3} + 2k\pi\), with \(k \in \mathbb{Z}\).

Key Concepts

Understanding the Sine Function

The sine function relates an angle in a right triangle to the ratio of the length of the opposite side over the hypotenuse. It is periodic with a period of 2π, meaning its values repeat every 2π radians. Knowing this helps find all possible angles x that satisfy sin x = √3/2.

Reference Angles and Unit Circle Values

Reference angles are acute angles used to determine sine values in different quadrants. The value √3/2 corresponds to a reference angle of π/3 radians (60°) on the unit circle. Understanding which quadrants have positive sine values (Quadrants I and II) is essential to find all solutions.

General Solution for Sine Equations

Since sine is periodic, the general solutions for sin x = a are x = θ + 2nπ and x = π - θ + 2nπ, where θ is the reference angle and n is any integer. This formula accounts for all angles with the same sine value across the unit circle.