Textbook Question
Find the inverse of each function (on the given interval, if specified).
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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
Intersection problems Find the following points of intersection.
The point(s) of intersection of the parabolas y= x² and y= -x² + 8x
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Textbook Question
Simplify the difference quotients ƒ(x+h) - ƒ(x) / h and ƒ(x) - ƒ(a) / (x-a) by rationalizing the numerator.
ƒ(x) = √(1-2x)
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Textbook Question
Assume f is an odd function and that both f and g are one-to-one. Use the (incomplete) graph of f and the graph of g to find the following function values. <IMAGE>
f⁻¹ (10)
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Textbook Question
Working with composite functions
Find possible choices for outer and inner functions ƒ and g such that the given function h equals ƒ o g.
h(x) = (x³ - 5)¹⁰
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Textbook Question
Defining piecewise functions Write a definition of the function whose graph is given <IMAGE>
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