Skip to main content
Ch. 1 - Functions
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 1 - FunctionsProblem 89
Chapter 1, Problem 89

Express θ\(\theta\) in terms of xx using the inverse sine, inverse tangent, and inverse secant functions. <IMAGE>

Verified step by step guidance
1
Identify the trigonometric relationships given in the problem. Typically, this involves recognizing the sides of a right triangle or the coordinates of a point on the unit circle.
Express the angle \( \theta \) in terms of \( x \) using the inverse sine function. Recall that \( \theta = \sin^{-1}(x) \) when \( x \) is the opposite side over the hypotenuse in a right triangle.
Express the angle \( \theta \) in terms of \( x \) using the inverse tangent function. Remember that \( \theta = \tan^{-1}(x) \) when \( x \) is the opposite side over the adjacent side in a right triangle.
Express the angle \( \theta \) in terms of \( x \) using the inverse secant function. Note that \( \theta = \sec^{-1}(x) \) when \( x \) is the hypotenuse over the adjacent side in a right triangle.
Combine these expressions to find a consistent expression for \( \theta \) in terms of \( x \) using the given inverse trigonometric functions. Ensure that the domain and range of each function are considered to maintain validity.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
7m
Was this helpful?

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, arctan, and arcsec, are used to find angles when the values of the trigonometric ratios are known. For example, if sin(θ) = x, then θ can be expressed as θ = arcsin(x). These functions are essential for expressing angles in terms of their corresponding ratios, allowing for the conversion between angle measures and their sine, tangent, or secant values.
Recommended video:
06:35
Derivatives of Other Inverse Trigonometric Functions

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 the definitions of tangent and secant in terms of sine and cosine. Understanding these identities is crucial for manipulating and simplifying expressions involving trigonometric functions, especially when expressing one function in terms of another.
Recommended video:
7:17
Verifying Trig Equations as Identities

Domain and Range of Inverse Functions

The domain and range of inverse trigonometric functions are critical for determining valid inputs and outputs. For instance, the domain of arcsin is [-1, 1] and its range is [-π/2, π/2]. Knowing these constraints helps ensure that the expressions derived from inverse functions are valid and meaningful, particularly when solving for angles in terms of other variables.
Recommended video:
5:10
Finding the Domain and Range of a Graph
Related Practice
Textbook Question

{Use of Tech} Sum of squared integers Let T (n) = 1² + 2² + ... + n², where n is a positive integer. It can be shown that T (n) = n (n + 1) (2n + 1) / 8


c. What is the least value of n for which T(n) > 1000?

Textbook Question

Even and odd at the origin


a. If ƒ(0) is defined and ƒ is an even function, is it necessarily true that ƒ(0) = 0? Explain.

1
views
Textbook Question

Composition of even and odd functions from tables Assume ƒ is an even function, g is an odd function, and both are defined at 0. Use the (incomplete) table to evaluate the given compositions. <IMAGE>


g. ƒ (g(g(-2)))

4
views
Textbook Question

Composition of even and odd functions from tables Assume ƒ is an even function, g is an odd function, and both are defined at 0. Use the (incomplete) table to evaluate the given compositions. <IMAGE>


i. g(g(g(-1)))

5
views
Textbook Question

Composition of even and odd functions from tables Assume ƒ is an even function, g is an odd function, and both are defined at 0. Use the (incomplete) table to evaluate the given compositions. <IMAGE>



e. g(g(-1))

1
views
Textbook Question

{Use of Tech} Polynomial calculations

Find a polynomial ƒ that satisfies the following properties. (Hint: Determine the degree of ƒ; then substitute a polynomial of that degree and solve for its coefficients.)

ƒ(ƒ(x)) = 9x - 8

3
views