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 6.39

Textbook Question

Solve each equation for exact solutions.
sin⁻¹ x - tan⁻¹ 1 = -π/4

Verified step by step guidance

Recognize that \( \sin^{-1} x \) represents the angle whose sine is \( x \), and \( \tan^{-1} 1 \) represents the angle whose tangent is 1.

Recall that \( \tan^{-1} 1 = \frac{\pi}{4} \) because the tangent of \( \frac{\pi}{4} \) is 1.

Substitute \( \tan^{-1} 1 \) with \( \frac{\pi}{4} \) in the equation: \( \sin^{-1} x - \frac{\pi}{4} = -\frac{\pi}{4} \).

Add \( \frac{\pi}{4} \) to both sides of the equation to isolate \( \sin^{-1} x \): \( \sin^{-1} x = 0 \).

Determine the value of \( x \) such that \( \sin^{-1} x = 0 \), which means \( x = \sin(0) \).

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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 sin⁻¹ (arcsin) and tan⁻¹ (arctan), are used to find angles when given a ratio of sides in a right triangle. For example, sin⁻¹(x) gives the angle whose sine is x, while tan⁻¹(1) gives the angle whose tangent is 1, which is π/4. Understanding these functions is crucial for solving equations involving angles.

Trigonometric Identities

Trigonometric identities are equations that involve trigonometric functions and are true for all values of the variables involved. Key identities include the Pythagorean identity, sin²(θ) + cos²(θ) = 1, and angle addition formulas. These identities can simplify complex equations and help in finding exact solutions for trigonometric equations.

Solving Trigonometric Equations

Solving trigonometric equations involves finding the values of the variable that satisfy the equation. This often requires isolating the trigonometric function and using inverse functions or identities to find the angle solutions. In the given equation, manipulating the terms and applying the properties of inverse functions will lead to the exact solutions.