Textbook Question
Find the degree measure of θ if it exists. Do not use a calculator.
θ = arcsin (-√3/2)
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5. Inverse Trigonometric Functions and Basic Trigonometric Equations
Inverse Sine, Cosine, & Tangent
Problem 6.43
Textbook Question
Solve each equation for exact solutions.
sin⁻¹ x - 4 tan⁻¹ (-1) = 2π
Verified step by step guidance1
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 \( \tan^{-1}(-1) \) is an angle in the fourth quadrant where the tangent is \( -1 \), which is \( -\frac{\pi}{4} \).
Substitute \( \tan^{-1}(-1) = -\frac{\pi}{4} \) into the equation: \( \sin^{-1}(x) - 4(-\frac{\pi}{4}) = 2\pi \).
Simplify the equation: \( \sin^{-1}(x) + \pi = 2\pi \).
Solve for \( \sin^{-1}(x) \) by isolating it: \( \sin^{-1}(x) = 2\pi - \pi \).
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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(x) and tan⁻¹(x), are used to find angles when given a ratio. For example, sin⁻¹(x) gives the angle whose sine is x, while tan⁻¹(x) gives the angle whose tangent is x. Understanding how to manipulate these functions is crucial for solving equations involving them.
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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, angle sum and difference identities, and double angle formulas. These identities can simplify complex equations and help in finding exact solutions.
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Periodic Nature of Trigonometric Functions
Trigonometric functions are periodic, meaning they repeat their values in regular intervals. For instance, the sine and tangent functions have periods of 2π and π, respectively. This periodicity is essential when solving equations, as it allows for the identification of all possible solutions within specified intervals.
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