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

Identify the given equation and interval: We need to find all values of \(s\) in the interval \([0, 2\pi)\) such that \(\sin s = -\frac{\sqrt{3}}{2}\).

Recall the reference angle: The value \(\frac{\sqrt{3}}{2}\) is a common sine value corresponding to an angle of \(\frac{\pi}{3}\). Since the sine is negative, we look for angles where sine is negative.

Determine the quadrants where sine is negative: Sine is negative in the third and fourth quadrants. So, the solutions will be angles in these quadrants with reference angle \(\frac{\pi}{3}\).

Write the general solutions for \(s\): In the third quadrant, \(s = \pi + \frac{\pi}{3}\). In the fourth quadrant, \(s = 2\pi - \frac{\pi}{3}\).

Simplify the expressions for \(s\) and verify they lie within the interval \([0, 2\pi)\) to find the exact solutions.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Unit Circle and Angle Measurement

The unit circle is a circle with radius 1 centered at the origin of the coordinate plane. Angles measured in radians correspond to points on the circle, where the x-coordinate is cos(θ) and the y-coordinate is sin(θ). Understanding the unit circle helps identify angles with specific sine values within a given interval.

Sine Function and Its Values

The sine function gives the y-coordinate of a point on the unit circle and ranges between -1 and 1. Knowing the exact sine values for common angles, such as sin(π/3) = √3/2, allows us to find angles where sine equals a specific value, including negative values by considering the appropriate quadrants.

Solving Trigonometric Equations in a Given Interval

To solve equations like sin s = -√3/2 within [0, 2π), identify all angles where the sine equals the given value. Since sine is negative in the third and fourth quadrants, use reference angles and quadrant signs to find all solutions within the specified interval.

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