Give the degree measure of θ. Do not use a calculator.
θ = arcsin (―√3/2)

Verified step by step guidance

Recall that \(\theta = \arcsin\left(-\frac{\sqrt{3}}{2}\right)\) means we are looking for an angle \(\theta\) whose sine value is \(-\frac{\sqrt{3}}{2}\).

Identify the reference angle by considering the positive value \(\frac{\sqrt{3}}{2}\). From common special angles, \(\sin 60^\circ = \frac{\sqrt{3}}{2}\), so the reference angle is \(60^\circ\).

Since the sine value is negative, determine in which quadrants sine is negative. Sine is negative in Quadrants III and IV.

The range of the arcsin function is \([-90^\circ, 90^\circ]\) (or \([-\frac{\pi}{2}, \frac{\pi}{2}]\)), which corresponds to angles in Quadrants IV and I. Because the sine is negative, the angle must be in Quadrant IV within this range.

Therefore, the angle \(\theta\) is the negative of the reference angle, so express \(\theta\) as \(-60^\circ\).

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Key Concepts

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

Inverse Sine Function (arcsin)

The inverse sine function, arcsin, returns the angle whose sine value is a given number. It is defined for inputs between -1 and 1 and outputs angles in the range of -90° to 90° (or -π/2 to π/2 radians). Understanding arcsin helps find the angle θ when sin(θ) is known.

Sine Values of Special Angles

Certain angles have well-known sine values, such as 30°, 45°, and 60°. For example, sin(60°) = √3/2. Recognizing these values allows you to identify the angle corresponding to a given sine value without a calculator.

Sign and Quadrant Considerations for Inverse Trigonometric Functions

Since arcsin outputs angles only in the first and fourth quadrants, a negative sine value indicates an angle in the fourth quadrant (between -90° and 0°). This helps determine the correct angle measure for negative sine values like -√3/2.

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