Textbook Question
More composite functions Let ƒ(x) = | x | , g(x)= x² - 4 , F(x) = √x , G(x) = (1)/(x-2) Determine the following composite functions and give their domains.
G o G
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Ch. 1 - Functions
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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Table of contents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
Chapter 1, Problem 1.74
Inverse sines and cosines Evaluate or simplify the following expressions without using a calculator.
cos⁻¹ (- 1/2 )
Verified step by step guidance1
Understand that \( \cos^{-1}(x) \) represents the angle whose cosine is \( x \).
Recall that the range of \( \cos^{-1}(x) \) is \([0, \pi]\).
Identify the angle \( \theta \) in the range \([0, \pi]\) such that \( \cos(\theta) = -\frac{1}{2} \).
Recognize that \( \cos(\pi - \theta) = -\cos(\theta) \), and since \( \cos(\frac{\pi}{3}) = \frac{1}{2} \), then \( \cos(\pi - \frac{\pi}{3}) = -\frac{1}{2} \).
Conclude that \( \theta = \pi - \frac{\pi}{3} \) is the angle in the range \([0, \pi]\) such that \( \cos(\theta) = -\frac{1}{2} \).
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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 arcsine and arccosine, are used to find angles when given a ratio of sides in a right triangle. For example, cos⁻¹(x) gives the angle whose cosine is x. These functions are essential for solving problems where the angle is unknown, and they have specific ranges to ensure each output is unique.
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Derivatives of Other Inverse Trigonometric Functions
Range of Inverse Cosine
The range of the inverse cosine function, cos⁻¹(x), is restricted to the interval [0, π]. This means that when evaluating cos⁻¹(-1/2), the resulting angle must fall within this range. Understanding this range is crucial for correctly interpreting the output of the inverse cosine function.
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4:49Inverse Cosine
Unit Circle
The unit circle is a fundamental concept in trigonometry that helps visualize the values of trigonometric functions. It is a circle with a radius of one centered at the origin of a coordinate plane. By using the unit circle, one can determine the angles corresponding to specific cosine values, such as -1/2, which corresponds to angles in the second and third quadrants.
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5:10Evaluate Composite Functions - Values on Unit Circle
Related Practice
Textbook Question
Sketch the graph of the inverse of ƒ. <IMAGE>
Textbook Question
Find the inverse of the function ƒ(x) = 2x. Verify that ƒ(ƒ⁻¹(x)) = x and ƒ⁻¹(ƒ(x)) = x .
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Textbook Question
Convert the following expressions to the indicated base.
using basa e, for and
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Textbook Question
Simplify the difference quotient (ƒ(x)-ƒ(a)) / (x-a) for the following functions.
ƒ(x) = 4 - 4x + x²
Textbook Question
How do you obtain the graph of from the graph of ?