In Exercises 63–82, use a sketch to find the exact value of each expression.cos (sin⁻¹ 4/5)

Verified step by step guidance

1. Start by understanding that \( \sin^{-1} \left( \frac{4}{5} \right) \) represents an angle whose sine is \( \frac{4}{5} \).
2. Visualize a right triangle where the opposite side to the angle is 4 and the hypotenuse is 5.
3. Use the Pythagorean theorem to find the adjacent side: \( a^2 + 4^2 = 5^2 \).
4. Solve for the adjacent side \( a \) to find \( a = \sqrt{5^2 - 4^2} \).
5. Now, use the definition of cosine: \( \cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} \) to find \( \cos(\sin^{-1}(\frac{4}{5})) \).

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions, such as sin⁻¹ (arcsin), are used to find the angle whose sine is a given value. In this case, sin⁻¹(4/5) gives an angle θ such that sin(θ) = 4/5. Understanding how to interpret these functions is crucial for solving problems involving angles derived from trigonometric ratios.

Right Triangle Relationships

Trigonometric functions are often defined in the context of right triangles. For sin(θ) = opposite/hypotenuse, if sin(θ) = 4/5, we can visualize a right triangle where the opposite side is 4 and the hypotenuse is 5. This relationship allows us to find the adjacent side using the Pythagorean theorem, which is essential for calculating other trigonometric values.

Cosine Function

The cosine function relates the adjacent side of a right triangle to its hypotenuse. Once we determine the lengths of the sides of the triangle from the sine value, we can find cos(θ) using the formula cos(θ) = adjacent/hypotenuse. This step is necessary to evaluate the expression cos(sin⁻¹(4/5)) and find its exact value.