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

Verified step by step guidance

1. Understand that \( y = \arccos(-\sqrt{3}/2) \) means we are looking for an angle \( y \) whose cosine is \(-\sqrt{3}/2\).
2. Recall that the range of the \( \arccos \) function is \([0, \pi] \), which means \( y \) must be within this interval.
3. Identify the reference angle whose cosine is \( \sqrt{3}/2 \). This angle is \( \pi/6 \) or 30 degrees.
4. Since the cosine is negative, \( y \) must be in the second quadrant where cosine values are negative.
5. Determine the angle in the second quadrant by subtracting the reference angle from \( \pi \), i.e., \( y = \pi - \pi/6 \).

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions, such as arccos, are used to find the angle whose cosine is a given value. For example, if y = arccos(x), then cos(y) = x. Understanding these functions is crucial for solving problems that involve finding angles from known trigonometric ratios.

Range of the arccos Function

The arccos function has a specific range of values, which is [0, π] for real numbers. This means that when you calculate arccos for a value, the result will always be an angle between 0 and π radians. Recognizing this range helps in determining the possible values of y when solving equations involving arccos.

Trigonometric Values of Special Angles

Certain angles have known trigonometric values, which are often derived from the unit circle. For instance, cos(5π/6) = -√3/2. Familiarity with these special angles allows for quick identification of exact values when solving trigonometric equations, such as the one presented in the question.