Ch. 2 - Graphs of the Trigonometric Functions; Inverse Trigonometric Functions

Blitzer3rd EditionTrigonometryISBN: 9780137316601Not the one you use?Change textbook

Blitzer 3rd Edition

Ch. 2 - Graphs of the Trigonometric Functions; Inverse Trigonometric Functions

Problem 57

Chapter 2, Problem 57

In Exercises 55–62, use the properties of inverse functions f(f⁻¹ (x)) = x for all x in the domain of f⁻¹ and f⁻¹(f(x)) for all x in the domain of f, as well as the definitions of the inverse cotangent, cosecant, and secant functions, to find the exact value of each expression, if possible. sec(sec⁻¹ 7π)

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Recall the property of inverse functions: for any function \( f \) and its inverse \( f^{-1} \), we have \( f(f^{-1}(x)) = x \) for all \( x \) in the domain of \( f^{-1} \).

Identify the function and its inverse in the expression: here, \( \sec \) is the function and \( \sec^{-1} \) is its inverse, so \( \sec(\sec^{-1}(7\pi)) \) should simplify to \( 7\pi \) if \( 7\pi \) is in the domain of \( \sec^{-1} \).

Recall the domain of the inverse secant function \( \sec^{-1}(x) \): it is defined for \( |x| \geq 1 \), so check if \( 7\pi \) satisfies this condition (since \( 7\pi > 1 \), it is valid).

Confirm the range of \( \sec^{-1} \) to ensure the expression is valid: \( \sec^{-1}(x) \) returns an angle \( \theta \) such that \( \sec(\theta) = x \) and \( \theta \) lies in \( [0, \pi] \) excluding \( \frac{\pi}{2} \).

Therefore, by the inverse function property, \( \sec(\sec^{-1}(7\pi)) = 7\pi \).

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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 reverse the effect of their corresponding trigonometric functions, mapping values from the function's range back to its domain. For example, sec⁻¹(x) gives the angle whose secant is x. Understanding their domains and ranges is crucial to correctly evaluate expressions involving these inverses.

Properties of Inverse Functions

The key property f(f⁻¹(x)) = x holds for all x in the domain of f⁻¹, and f⁻¹(f(x)) = x for all x in the domain of f. This means applying a function and its inverse in succession returns the original input, provided the input lies within the appropriate domain. This property is essential for simplifying expressions like sec(sec⁻¹(7π)).

Domain and Range Restrictions of Secant and Its Inverse

The secant function, sec(θ) = 1/cos(θ), is defined where cos(θ) ≠ 0, and its inverse sec⁻¹(x) is defined for |x| ≥ 1. The principal value range of sec⁻¹(x) is [0, π] excluding π/2, ensuring sec(sec⁻¹(x)) = x for x in the domain. Recognizing these restrictions helps determine if expressions like sec(sec⁻¹(7π)) are valid or require further interpretation.

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