In Exercises 83–94, use a right triangle to write each expression as an algebraic expression. Assume that x is positive and that the given inverse trigonometric function is defined for the expression in x. cos (sin⁻¹ 1/x)

Verified step by step guidance

1. Recognize that the expression is \( \cos(\sin^{-1}(1/x)) \). Let \( \theta = \sin^{-1}(1/x) \), which means \( \sin(\theta) = \frac{1}{x} \).
2. Since \( \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} \), represent this as a right triangle where the opposite side is 1 and the hypotenuse is \( x \).
3. Use the Pythagorean theorem to find the adjacent side of the triangle: \( \text{adjacent} = \sqrt{x^2 - 1^2} = \sqrt{x^2 - 1} \).
4. Recall that \( \cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} \). Substitute the values from the triangle to get \( \cos(\sin^{-1}(1/x)) = \frac{\sqrt{x^2 - 1}}{x} \).
5. Write the final algebraic expression for the original expression as \( \frac{\sqrt{x^2 - 1}}{x} \), assuming \( x > 0 \) to keep the square root defined and positive.

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions, like sin⁻¹(x), return the angle whose trigonometric ratio equals x. Understanding that sin⁻¹(1/x) gives an angle θ such that sin(θ) = 1/x is essential for rewriting expressions involving these functions.

Right Triangle Definitions of Trigonometric Ratios

Trigonometric ratios (sine, cosine, tangent) can be represented as ratios of sides in a right triangle. By constructing a right triangle with an angle θ where sin(θ) = opposite/hypotenuse = 1/x, we can find other ratios like cos(θ) in terms of x.

Pythagorean Theorem

The Pythagorean theorem relates the sides of a right triangle: (hypotenuse)² = (opposite)² + (adjacent)². This allows us to find the missing side length when two sides are known, which is crucial for expressing cos(sin⁻¹(1/x)) algebraically.