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)

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1. Identify the inverse trigonometric function: \( \cos^{-1} \left( \frac{1}{x} \right) \). This represents an angle \( \theta \) such that \( \cos(\theta) = \frac{1}{x} \).
2. Draw a right triangle where \( \theta \) is one of the angles. Since \( \cos(\theta) = \frac{1}{x} \), label the adjacent side as 1 and the hypotenuse as \( x \).
3. Use the Pythagorean theorem to find the opposite side: \( \text{opposite} = \sqrt{x^2 - 1^2} = \sqrt{x^2 - 1} \).
4. Now, find \( \sec(\theta) \). Recall that \( \sec(\theta) = \frac{1}{\cos(\theta)} \), which is also \( \frac{\text{hypotenuse}}{\text{adjacent}} \).
5. Substitute the values from the triangle: \( \sec(\theta) = \frac{x}{1} = x \).

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions, such as cos⁻¹ (arccos), are used to find the angle whose cosine is a given value. In this case, cos⁻¹(1/x) gives an angle θ such that cos(θ) = 1/x. Understanding how to interpret these functions is crucial for solving problems involving angles and their relationships to triangle sides.

Secant Function

The secant function, denoted as sec(θ), is the reciprocal of the cosine function, defined as sec(θ) = 1/cos(θ). In the context of the problem, once we determine the angle θ from the inverse cosine, we can find sec(θ) by calculating the reciprocal of cos(θ), which is essential for converting the expression into an algebraic form.

Right Triangle Relationships

In a right triangle, the relationships between the angles and sides are governed by trigonometric ratios. For example, if we let θ be the angle found from cos⁻¹(1/x), we can use the definitions of sine, cosine, and secant to relate the sides of the triangle to the angle. This understanding allows us to express trigonometric functions in terms of algebraic expressions involving x.