Textbook Question
Solve each equation for exact solutions.
cos⁻¹ x + tan⁻¹ x = π/2
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5. Inverse Trigonometric Functions and Basic Trigonometric Equations
Inverse Sine, Cosine, & Tangent
Problem 6.5
Textbook Question
Which one of the following equations has solution π?
a. arccos (―1) = x
b. arccos 1 = x
c. arcsin (―1) = x
Verified step by step guidance1
Step 1: Understand the problem by identifying the trigonometric functions involved: arccos and arcsin.
Step 2: Recall the definitions: \( \arccos(x) \) is the angle whose cosine is \( x \), and \( \arcsin(x) \) is the angle whose sine is \( x \).
Step 3: Evaluate each option: For option (a), \( \arccos(-1) \) is the angle whose cosine is \(-1\).
Step 4: Evaluate option (b): \( \arccos(1) \) is the angle whose cosine is \(1\).
Step 5: Evaluate option (c): \( \arcsin(-1) \) is the angle whose sine is \(-1\).
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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 arcsin and arccos, are used to find angles when given a trigonometric ratio. For example, if sin(x) = y, then arcsin(y) = x. These functions have specific ranges: arcsin outputs values between -π/2 and π/2, while arccos outputs values between 0 and π.
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Values of Inverse Functions
Each inverse trigonometric function corresponds to specific values of angles. For instance, arccos(-1) equals π because the cosine of π is -1. Understanding these key angle values is essential for solving equations involving inverse trigonometric functions.
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Trigonometric Identities
Trigonometric identities are equations that involve trigonometric functions and are true for all values of the variables involved. They help simplify expressions and solve equations. Familiarity with these identities aids in recognizing solutions to equations involving sine and cosine, particularly when determining angles like π.
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