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

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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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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].

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Inverse Sine

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.

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Sine, Cosine, & Tangent of 30°, 45°, & 60°

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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