Find the exact values of s in the given interval that satisfy the given condition.


[0, 2π) ; sin s = -√3 / 2

Verified step by step guidance

1. Identify the reference angle for \( \sin s = -\frac{\sqrt{3}}{2} \). The reference angle for \( \sin \theta = \frac{\sqrt{3}}{2} \) is \( \frac{\pi}{3} \).
2. Since the sine function is negative, determine in which quadrants sine is negative. Sine is negative in the third and fourth quadrants.
3. For the third quadrant, use the reference angle to find \( s = \pi + \frac{\pi}{3} \).
4. For the fourth quadrant, use the reference angle to find \( s = 2\pi - \frac{\pi}{3} \).
5. Verify that both solutions are within the interval \([0, 2\pi)\).

Key Concepts

Unit Circle

The unit circle is a circle with a radius of one centered at the origin of a coordinate plane. It is fundamental in trigonometry as it provides a geometric representation of the sine and cosine functions. The angles measured in radians correspond to points on the circle, where the x-coordinate represents cosine and the y-coordinate represents sine.

Sine Function

The sine function, denoted as sin(θ), is a fundamental trigonometric function that relates the angle θ to the ratio of the length of the opposite side to the hypotenuse in a right triangle. In the context of the unit circle, it gives the y-coordinate of a point on the circle corresponding to the angle θ. Understanding the values of sine for specific angles is crucial for solving trigonometric equations.

Reference Angles

Reference angles are the acute angles formed by the terminal side of an angle and the x-axis. They help in determining the sine and cosine values for angles in different quadrants. For example, when sin(s) = -√3/2, the reference angle can be found in the first quadrant, and the corresponding angles in the third and fourth quadrants can be derived, which is essential for finding all solutions within a specified interval.