Find the exact value of each real number y. Do not use a calculator.
y = cos⁻¹ (―√2/2)

Verified step by step guidance

1. Recognize that \( \cos^{-1}(x) \) is the inverse cosine function, which gives the angle whose cosine is \( x \).
2. Identify the value \( x = -\frac{\sqrt{2}}{2} \) and recall that this corresponds to a specific angle on the unit circle.
3. Recall that \( \cos(\theta) = -\frac{\sqrt{2}}{2} \) at angles \( \theta = \frac{3\pi}{4} \) and \( \theta = \frac{5\pi}{4} \) in the unit circle.
4. Since \( \cos^{-1}(x) \) returns values in the range \([0, \pi]\), select the angle \( \theta = \frac{3\pi}{4} \).
5. Conclude that \( y = \frac{3\pi}{4} \) is the exact value of the angle whose cosine is \(-\frac{\sqrt{2}}{2}\).

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. In this case, we are looking for the angle whose cosine equals -√2/2. The output of the inverse function is typically restricted to a specific range to ensure it is a function, which for arccos is [0, π].

Unit Circle

The unit circle is a fundamental concept in trigonometry that helps visualize the values of trigonometric functions. It is a circle with a radius of one centered at the origin of a coordinate plane. The coordinates of points on the unit circle correspond to the cosine and sine of angles, making it easier to determine the angles associated with specific cosine values, such as -√2/2.

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 values of trigonometric functions for angles in different quadrants. For the cosine value of -√2/2, the reference angle is π/4, and since cosine is negative in the second quadrant, the corresponding angle is 3π/4.