In Exercises 11–24, find all solutions of each equation. __ √ 3 sin x = ------- 2

Verified step by step guidance

1. Recognize that the equation \( \sin x = \frac{\sqrt{3}}{2} \) is a standard trigonometric equation.
2. Recall that \( \sin x = \frac{\sqrt{3}}{2} \) corresponds to specific angles on the unit circle, namely \( x = \frac{\pi}{3} \) and \( x = \frac{2\pi}{3} \) in the interval \([0, 2\pi)\).
3. Since the sine function is periodic with a period of \(2\pi\), the general solutions can be expressed as \( x = \frac{\pi}{3} + 2k\pi \) and \( x = \frac{2\pi}{3} + 2k\pi \), where \( k \) is any integer.
4. Consider the periodic nature of the sine function to ensure all solutions are captured by adding integer multiples of the period \(2\pi\).
5. Verify the solutions by substituting back into the original equation to ensure they satisfy \( \sin x = \frac{\sqrt{3}}{2} \).

Key Concepts

Trigonometric Functions

Trigonometric functions, such as sine, cosine, and tangent, relate angles to the ratios of sides in right triangles. The sine function, specifically, represents the ratio of the length of the opposite side to the hypotenuse. Understanding these functions is crucial for solving equations involving angles and their corresponding values.

Inverse Trigonometric Functions

Inverse trigonometric functions, such as arcsin, arccos, and arctan, are used to find angles when given a ratio. For example, if sin(x) = a, then x can be found using x = arcsin(a). This concept is essential for determining the angle solutions in trigonometric equations.

Periodic Nature of Trigonometric Functions

Trigonometric functions are periodic, meaning they repeat their values in regular intervals. For sine, the period is 2π, indicating that sin(x) = sin(x + 2πn) for any integer n. This property is important when finding all solutions to trigonometric equations, as it allows for the identification of multiple angle solutions within specified ranges.