Textbook Question
Find the inverse of each function (on the given interval, if specified).
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 1.76
Inverse sines and cosines Evaluate or simplify the following expressions without using a calculator.
cos (cos⁻¹ ( -1 ))
Verified step by step guidance1
Recognize that \( \cos^{-1}(-1) \) is asking for the angle whose cosine is \(-1\).
Recall that the range of \( \cos^{-1}(x) \) is \([0, \pi]\).
Determine the angle within this range where the cosine value is \(-1\).
The angle that satisfies \( \cos(\theta) = -1 \) within the range \([0, \pi]\) is \( \pi \).
Substitute \( \pi \) back into the original expression: \( \cos(\pi) \).
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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 cos⁻¹(x), are used to find angles when given a trigonometric ratio. For example, cos⁻¹(-1) gives the angle whose cosine is -1, which is π radians (or 180 degrees). Understanding these functions is crucial for evaluating expressions involving them.
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Derivatives of Other Inverse Trigonometric Functions
Cosine Function
The cosine function, denoted as cos(x), relates the angle x in a right triangle to the ratio of the length of the adjacent side to the hypotenuse. It is periodic and ranges from -1 to 1. Knowing the values of cosine at key angles (like 0, π/2, π, etc.) is essential for simplifying expressions involving cosine.
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5:53Graph of Sine and Cosine Function
Composition of Functions
Composition of functions involves applying one function to the result of another. In this case, evaluating cos(cos⁻¹(-1)) means finding the cosine of the angle whose cosine is -1. This concept is fundamental in calculus and algebra, as it allows for the simplification of complex expressions by breaking them down into manageable parts.
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3:48Evaluate Composite Functions - Special Cases
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
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)
Textbook Question
Simplify the difference quotient (ƒ(x)-ƒ(a)) / (x-a) for the following functions.
ƒ(x) = (1/x) - x²
Textbook Question
Inverse sines and cosines Evaluate or simplify the following expressions without using a calculator.
cos⁻¹ √3/2
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Textbook Question
Solving equations Solve the following equations.
3(ˣ³⁻⁴) = 15