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. csc (cot⁻¹ x)

Verified step by step guidance

1. Recognize that the expression is \( \csc(\cot^{-1} x) \). Let \( \theta = \cot^{-1} x \), which means \( \cot \theta = x \).
2. Recall that \( \cot \theta = \frac{\text{adjacent}}{\text{opposite}} \). Since \( x > 0 \), we can represent the right triangle with adjacent side = \( x \) and opposite side = 1.
3. Use the Pythagorean theorem to find the hypotenuse: \( \text{hypotenuse} = \sqrt{x^2 + 1^2} = \sqrt{x^2 + 1} \).
4. Recall that \( \csc \theta = \frac{\text{hypotenuse}}{\text{opposite}} \). Substitute the values from the triangle: \( \csc \theta = \frac{\sqrt{x^2 + 1}}{1} \).
5. Therefore, \( \csc(\cot^{-1} x) = \sqrt{x^2 + 1} \), which is the algebraic expression in terms of \( x \).

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions, like cot⁻¹(x), return an angle whose trigonometric ratio equals x. Understanding how to interpret these functions is essential for converting expressions involving inverse trig functions into geometric or algebraic forms.

Right Triangle Definitions of Trigonometric Ratios

Trigonometric ratios such as sine, cosine, cotangent, and cosecant can be represented as ratios of sides in a right triangle. Using a right triangle helps visualize and rewrite expressions like csc(cot⁻¹ x) in terms of side lengths and algebraic expressions.

Relationship Between Cotangent and Cosecant

Cotangent is the ratio of adjacent to opposite sides, while cosecant is the reciprocal of sine (hypotenuse over opposite). Understanding how to express csc(θ) when θ = cot⁻¹(x) involves using the Pythagorean theorem to find the hypotenuse and then forming the correct ratio.