In Exercises 29–51, find the exact value of each expression. Do not use a calculator. _ cos(sin⁻¹ √2/2)

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions, such as sin⁻¹, are used to find the angle whose sine is a given value. In this case, sin⁻¹(√2/2) yields an angle in the range of -π/2 to π/2, specifically π/4, since sin(π/4) = √2/2. Understanding how to interpret these functions is crucial for solving the problem.

Cosine Function

The cosine function relates the angle of a right triangle to the ratio of the length of the adjacent side to the hypotenuse. For any angle θ, cos(θ) can be found using the unit circle, where the x-coordinate of the point on the circle corresponding to θ gives the cosine value. This concept is essential for evaluating cos(sin⁻¹(√2/2)).

Pythagorean Identity

The Pythagorean identity states that for any angle θ, sin²(θ) + cos²(θ) = 1. This identity allows us to find the cosine of an angle if we know the sine. In this problem, once we determine sin(θ) = √2/2, we can use the identity to find cos(θ) by calculating cos(θ) = √(1 - sin²(θ)).