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. Step 1: Understand the problem. We need to express \( \csc(\cot^{-1}(x)) \) as an algebraic expression.
2. Step 2: Recognize that \( \cot^{-1}(x) \) is the angle whose cotangent is \( x \). This means \( \cot(\theta) = x \), where \( \theta = \cot^{-1}(x) \).
3. Step 3: Recall the definition of cotangent: \( \cot(\theta) = \frac{\text{adjacent}}{\text{opposite}} \). So, if \( \cot(\theta) = x \), we can consider a right triangle where the adjacent side is \( x \) and the opposite side is \( 1 \).
4. Step 4: Use the Pythagorean theorem to find the hypotenuse of the triangle: \( \text{hypotenuse} = \sqrt{x^2 + 1^2} = \sqrt{x^2 + 1} \).
5. Step 5: Recall the definition of cosecant: \( \csc(\theta) = \frac{\text{hypotenuse}}{\text{opposite}} \). Therefore, \( \csc(\cot^{-1}(x)) = \frac{\sqrt{x^2 + 1}}{1} = \sqrt{x^2 + 1} \).

Key Concepts

Inverse Trigonometric Functions

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

Cosecant Function

The cosecant function, denoted as csc(θ), is the reciprocal of the sine function, defined as csc(θ) = 1/sin(θ). In the context of a right triangle, it relates the length of the hypotenuse to the length of the opposite side. Recognizing how to express csc in terms of other trigonometric functions is essential for simplifying expressions.

Right Triangle Relationships

In a right triangle, the relationships between the angles and sides are governed by trigonometric ratios. The sides are typically labeled as opposite, adjacent, and hypotenuse, which correspond to the angles. Understanding these relationships allows for the conversion of trigonometric expressions into algebraic forms, facilitating the solution of problems involving angles and side lengths.