Find the degree measure of θ if it exists. Do not use a calculator.
θ = arcsin (-√3/2)

Verified step by step guidance

Recall that \(\theta = \arcsin(x)\) means \(\sin(\theta) = x\) and \(\theta\) lies within the range \([-\frac{\pi}{2}, \frac{\pi}{2}]\) or \([-90^\circ, 90^\circ]\) when working in degrees.

Identify the value inside the arcsin function: \(x = -\frac{\sqrt{3}}{2}\). We need to find an angle \(\theta\) such that \(\sin(\theta) = -\frac{\sqrt{3}}{2}\) within the restricted range of arcsin.

Recall the common sine values: \(\sin(60^\circ) = \frac{\sqrt{3}}{2}\). Since the sine is negative, \(\theta\) must be in the fourth quadrant (between \(-90^\circ\) and \(0^\circ\)) for arcsin.

Therefore, the angle \(\theta\) is the negative of \(60^\circ\), so write \(\theta = -60^\circ\) as the solution within the principal range of arcsin.

Express the final answer clearly as \(\theta = -60^\circ\) without using a calculator, confirming it satisfies \(\sin(\theta) = -\frac{\sqrt{3}}{2}\) and lies in the correct domain.

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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. Its output range is limited to angles between -90° and 90° (or -π/2 and π/2 radians), ensuring a unique solution for each input within [-1, 1].

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 helps identify the angle corresponding to a given sine value without a calculator.

Sign and Quadrant Considerations for arcsin

Since arcsin outputs angles only in the first and fourth quadrants (between -90° and 90°), a negative sine value corresponds to an angle in the fourth quadrant (negative angle). This restricts the solution to a specific range.

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