Table of contents

0. Review of College Algebra4h 43m
1. Measuring Angles40m
2. Trigonometric Functions on Right Triangles2h 5m
3. Unit Circle1h 19m
4. Graphing Trigonometric Functions1h 19m
5. Inverse Trigonometric Functions and Basic Trigonometric Equations1h 41m
6. Trigonometric Identities and More Equations2h 34m
7. Non-Right Triangles1h 38m
8. Vectors2h 25m
9. Polar Equations2h 5m
10. Parametric Equations1h 6m
11. Graphing Complex Numbers1h 7m

5. Inverse Trigonometric Functions and Basic Trigonometric Equations

Evaluate Composite Trig Functions

Trigonometry

5. Inverse Trigonometric Functions and Basic Trigonometric Equations

Evaluate Composite Trig Functions

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2:21 minutes

Problem 88

Textbook Question

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

Identify the inverse trigonometric function: \( \cos^{-1} \left( \frac{1}{x} \right) \). This represents an angle \( \theta \) such that \( \cos(\theta) = \frac{1}{x} \).

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 \( x \).

Use the Pythagorean theorem to find the opposite side: \( \text{opposite} = \sqrt{x^2 - 1^2} = \sqrt{x^2 - 1} \).

Now, find \( \sec(\theta) \). Recall that \( \sec(\theta) = \frac{1}{\cos(\theta)} \), which is also \( \frac{\text{hypotenuse}}{\text{adjacent}} \).

Substitute the values from the triangle: \( \sec(\theta) = \frac{x}{1} = x \).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.