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. sec (cos⁻¹ 1/x)

Verified step by step guidance

1. Recognize that the expression is \( \sec(\cos^{-1}(1/x)) \). Let \( \theta = \cos^{-1}(1/x) \), which means \( \cos(\theta) = \frac{1}{x} \).
2. Since \( \theta \) is an angle in a right triangle, draw a right triangle where the adjacent side to \( \theta \) is 1 and the hypotenuse is \( x \) (because \( \cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} \)).
3. Use the Pythagorean theorem to find the length of the opposite side: \( \text{opposite} = \sqrt{x^2 - 1^2} = \sqrt{x^2 - 1} \).
4. Recall that \( \sec(\theta) = \frac{1}{\cos(\theta)} = \frac{\text{hypotenuse}}{\text{adjacent}} \). Using the triangle, \( \sec(\theta) = \frac{x}{1} = x \).
5. Therefore, \( \sec(\cos^{-1}(1/x)) \) can be expressed algebraically as \( x \).

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions, like cos⁻¹ (arccos), return an angle whose trigonometric ratio matches the given value. For cos⁻¹(1/x), the output is an angle θ such that cos(θ) = 1/x, with θ typically in the range [0, π]. Understanding this helps translate the expression into a geometric context.

Right Triangle Definitions of Trigonometric Ratios

Trigonometric ratios such as secant are defined using right triangles: sec(θ) = hypotenuse/adjacent. By interpreting θ = cos⁻¹(1/x), we can construct a right triangle where the adjacent side is 1 and the hypotenuse is x, allowing us to express sec(θ) algebraically in terms of x.

Algebraic Manipulation Using the Pythagorean Theorem

To find the missing side of the triangle, use the Pythagorean theorem: hypotenuse² = adjacent² + opposite². Given hypotenuse = x and adjacent = 1, the opposite side is √(x² - 1). This enables expressing sec(θ) = hypotenuse/adjacent = x/1 = x, or other related expressions, purely algebraically.