Find the exact value of each real number y if it exists. Do not use a calculator.
y = cos⁻¹ (―1)

Verified step by step guidance

1. Understand that \( \cos^{-1}(x) \) is the inverse cosine function, which gives the angle whose cosine is \( x \).
2. Recognize that \( \cos^{-1}(-1) \) asks for the angle \( y \) such that \( \cos(y) = -1 \).
3. Recall that the range of \( \cos^{-1}(x) \) is \([0, \pi]\), meaning the angle \( y \) must be within this interval.
4. Identify that \( \cos(\pi) = -1 \), which is the only angle in the range \([0, \pi]\) where the cosine is \(-1\).
5. Conclude that \( y = \pi \) is the exact value of \( \cos^{-1}(-1) \).

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions, such as cos⁻¹ (arccos), are used to find the angle whose cosine is a given value. For example, if y = cos⁻¹(x), then cos(y) = x. These functions are defined within specific ranges to ensure they are one-to-one, which is crucial for determining unique angle values.

Range of the Arccosine Function

The range of the arccosine function, cos⁻¹(x), is from 0 to π radians (or 0 to 180 degrees). This means that when you find the angle whose cosine is a specific value, the result will always fall within this interval, which is important for identifying valid solutions.

Cosine Values

The cosine function outputs values between -1 and 1 for real angles. Specifically, cos(π) = -1, which is the only angle in the range of the arccosine function that corresponds to this value. Understanding the behavior of the cosine function helps in determining the exact angle when solving for y in the equation y = cos⁻¹(-1).