Question

In Exercises \(27-30\) , give the measure of the angle in radians and degrees. Give exact answers whenever possible. $$\sin ^{-1}\left(-\frac{\sqrt{2}}{2}\right)$$

Short answer

The measure of the angle in degrees is either \(225°\) or \(315°\), and in radians it is either \(5\pi/4\) or \(7\pi/4\).

Step-by-step solution

Set Up the Equation

Use the knowledge of inverse trigonometric functions to set up the equation: \(\sin ^{-1}\left(-\frac{\sqrt{2}}{2}\right) = θ\)

Determine the Quadrant

Because sin is negative, the angle, \(θ\), must be in either the third or fourth quadrant.

Find the Reference Angle

The reference angle corresponding to \(\sqrt{2}/2\) is \(45°\) or \(\pi/4\) radians.

Apply Symmetry of Sine Function

Use the sine function's symmetry properties to find the required angle in the standard \(0-360°\) or \(0-2\pi\) range. It will be \(180° + 45° = 225°\) or \(360° - 45° = 315°\) in degrees, and \(\pi + \pi/4 = 5\pi/4\) or \(2\pi - \pi/4 = 7\pi/4\) radians.