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 6.27

Textbook Question

Evaluate each expression without using a calculator.
tan⁻¹ (tan (π/4))

Verified step by step guidance

Recognize that \( \tan^{-1} \) is the inverse function of \( \tan \), meaning \( \tan^{-1}(\tan(x)) = x \) for values of \( x \) within the range of \( \tan^{-1} \).

Identify the range of \( \tan^{-1} \), which is \( \left(-\frac{\pi}{2}, \frac{\pi}{2}\right) \).

Check if \( \frac{\pi}{4} \) is within the range of \( \tan^{-1} \).

Since \( \frac{\pi}{4} \) is within the range, \( \tan^{-1}(\tan(\frac{\pi}{4})) = \frac{\pi}{4} \).

Conclude that the expression evaluates to \( \frac{\pi}{4} \) because \( \tan^{-1} \) and \( \tan \) cancel each other out within the specified range.

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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 the angle whose tangent is a given number. They essentially reverse the action of the standard trigonometric functions. For example, tan⁻¹(tan(x)) will return x if x is within the principal range of the arctangent function, which is typically (-π/2, π/2).

Tangent Function

The tangent function, denoted as tan(x), is a fundamental trigonometric function defined as the ratio of the opposite side to the adjacent side in a right triangle. It is periodic with a period of π, meaning that tan(x) = tan(x + nπ) for any integer n. At specific angles, such as π/4, the tangent value is well-known, which simplifies calculations.

Principal Value

The principal value of an inverse trigonometric function is the unique output angle that lies within a specified range. For tan⁻¹, the principal value is restricted to the interval (-π/2, π/2). This means that when evaluating expressions like tan⁻¹(tan(π/4)), the result will be the angle within this range that corresponds to the tangent value, ensuring consistency in the output.