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

Verified step by step guidance

1. Understand that \( y = \arcsin(-\frac{\sqrt{3}}{2}) \) means we are looking for an angle \( y \) whose sine is \(-\frac{\sqrt{3}}{2}\).
2. Recall that the range of the \( \arcsin \) function is \([-\frac{\pi}{2}, \frac{\pi}{2}]\), which means \( y \) must be within this interval.
3. Identify the reference angle whose sine is \( \frac{\sqrt{3}}{2} \). This angle is \( \frac{\pi}{3} \) because \( \sin(\frac{\pi}{3}) = \frac{\sqrt{3}}{2} \).
4. Since we need \( \sin(y) = -\frac{\sqrt{3}}{2} \), and \( y \) must be in the range \([-\frac{\pi}{2}, \frac{\pi}{2}]\), the angle \( y \) is in the fourth quadrant.
5. Determine the angle in the fourth quadrant by taking the negative of the reference angle: \( y = -\frac{\pi}{3} \).

Key Concepts

Arcsine Function

The arcsine function, denoted as arcsin or sin⁻¹, is the inverse of the sine function. It takes a value from the range of -1 to 1 and returns an angle in radians or degrees whose sine is that value. For example, if y = arcsin(x), then sin(y) = x. Understanding this function is crucial for solving problems involving angles derived from sine values.

Unit Circle

The unit circle is a circle with a radius of one centered at the origin of a coordinate plane. It is fundamental in trigonometry as it provides a geometric interpretation of the sine, cosine, and tangent functions. The coordinates of points on the unit circle correspond to the cosine and sine of angles, making it essential for determining the values of trigonometric functions and their inverses.

Reference Angles

Reference angles are the acute angles formed by the terminal side of an angle and the x-axis. They help in determining the sine, cosine, and tangent values for angles in different quadrants. For the arcsine function, knowing the reference angle is important because it allows us to find the exact angle corresponding to a given sine value, especially when dealing with negative values or angles greater than 90 degrees.