Find the exact value of each real number y if it exists. Do not use a calculator.
y = arcsin (―√3/2)

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Recall that the function \(\arcsin(x)\) gives the angle \(y\) in the interval \([-\frac{\pi}{2}, \frac{\pi}{2}]\) such that \(\sin(y) = x\).

Identify the value inside the arcsin function: \(x = -\frac{\sqrt{3}}{2}\). We need to find an angle \(y\) where \(\sin(y) = -\frac{\sqrt{3}}{2}\).

Recall the common sine values: \(\sin\left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2}\). Since the sine is negative, the angle must be in the fourth or third quadrant, but \(\arcsin\) only returns values in \([-\frac{\pi}{2}, \frac{\pi}{2}]\) (first and fourth quadrants).

Since \(\sin(y) = -\frac{\sqrt{3}}{2}\) and \(y\) is in \([-\frac{\pi}{2}, 0]\) (fourth quadrant), the angle is \(y = -\frac{\pi}{3}\).

Therefore, the exact value of \(y\) satisfying \(y = \arcsin\left(-\frac{\sqrt{3}}{2}\right)\) is \(y = -\frac{\pi}{3}\).

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Definition of arcsin (Inverse Sine Function)

The arcsin function is the inverse of the sine function, returning the angle whose sine is a given number. Its output range is limited to [-π/2, π/2], meaning it only returns angles in the first and fourth quadrants. Understanding this range is crucial for finding the correct angle corresponding to a sine value.

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Inverse Sine

Exact Values of Sine for Special Angles

Certain angles have well-known sine values, such as π/6, π/4, and π/3. For example, sin(π/3) = √3/2. Recognizing these exact values helps in determining the angle when given a sine value like -√3/2, by considering the sign and the reference angle.

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Common Trig Functions For 45-45-90 Triangles

Sign and Quadrant Considerations for Inverse Trigonometric Functions

Since arcsin outputs angles only between -π/2 and π/2, negative sine values correspond to angles in the fourth quadrant (negative angles). For y = arcsin(-√3/2), the angle is negative and matches the reference angle π/3, resulting in y = -π/3.

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