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)

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions, such as sin⁻¹(x), are used to find angles when the value of a trigonometric function is known. For example, sin⁻¹(1/x) gives the angle whose sine is 1/x. Understanding how to interpret these functions is crucial for solving problems involving angles and their relationships in right triangles.

Right Triangle Relationships

In a right triangle, the relationships between the angles and sides are defined by trigonometric ratios: sine, cosine, and tangent. For instance, if θ is an angle, sin(θ) is the ratio of the opposite side to the hypotenuse, while cos(θ) is the ratio of the adjacent side to the hypotenuse. These relationships are essential for converting inverse trigonometric expressions into algebraic forms.

Pythagorean Theorem

The Pythagorean Theorem states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides (a² + b² = c²). This theorem is fundamental for finding unknown side lengths when working with trigonometric functions and their inverses, allowing for the calculation of expressions like cos(sin⁻¹(1/x)).