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

[0, 2π) ; sin s = -√3 / 2

Guida verificata passo dopo passo

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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