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

Inverse Sine, Cosine, & Tangent

Trigonometry

5. Inverse Trigonometric Functions and Basic Trigonometric Equations

Inverse Sine, Cosine, & Tangent

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Problem 6.91

Textbook Question

Use a calculator to find each value. Give answers as real numbers.
cos (tan⁻¹ 0.5)

Verified step by step guidance

Recognize that \( \tan^{-1}(0.5) \) is the angle whose tangent is 0.5.

Let \( \theta = \tan^{-1}(0.5) \). This means \( \tan(\theta) = 0.5 \).

Use the identity \( \cos(\theta) = \frac{1}{\sqrt{1 + \tan^2(\theta)}} \) to find \( \cos(\theta) \).

Substitute \( \tan(\theta) = 0.5 \) into the identity: \( \cos(\theta) = \frac{1}{\sqrt{1 + (0.5)^2}} \).

Simplify the expression to find the value of \( \cos(\theta) \).

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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 tan⁻¹ (arctangent), are used to find angles when the value of a trigonometric function is known. For example, tan⁻¹(0.5) gives the angle whose tangent is 0.5. Understanding how to interpret these functions is crucial for solving problems that involve finding angles from given ratios.

Trigonometric Ratios

Trigonometric ratios relate the angles of a triangle to the lengths of its sides. In the context of the tangent function, it is defined as the ratio of the opposite side to the adjacent side in a right triangle. Knowing these ratios helps in calculating other trigonometric values, such as cosine, based on the angle derived from the inverse function.

Calculator Functions

Using a calculator effectively requires understanding how to input functions and interpret the results. Most scientific calculators have specific buttons for trigonometric functions, including inverse functions. Familiarity with these functions allows for accurate calculations of trigonometric values, such as cos(tan⁻¹(0.5)), which involves first finding the angle and then calculating its cosine.