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Chapter 1: Problem 10

Why do the values of \(\cos ^{-1} x\) lie in the interval \([0, \pi] ?\)

Short Answer

Expert verified
Answer: The values of the inverse cosine function, \(\cos^{-1}x\), lie in the interval \([0, \pi]\) because the cosine function is restricted to this interval to make it one-to-one and onto, ensuring it passes the horizontal line test. Our chosen interval captures all output values in \([-1, 1]\) exactly once. When the inverse function \(\cos^{-1}(x)\) is realized, input values are in the interval \([-1, 1]\), and the resulting values lie within the interval \([0, \pi]\).

Step by step solution

01

Review the Cosine Function

The cosine function \(\cos(x)\) is a periodic function with period \(2\pi\), defined for all real numbers. It is an even function, with a range of \([-1,1]\). Meaning the maximum value of cosine function is 1 and the minimum value is -1. The cosine function evaluates the horizontal component of a point on the unit circle.
02

Inverse Function Basics

An inverse function reverses the input and output of an original function. In other words, if we have a function \(f(x)\) and its inverse function \(f^{-1}(y)\), for any input \(x\) and output \(y\), we get the mapping \((x, y) \to (y, x)\). But not all functions have inverse functions. For a function to have an inverse, it must be bijective (both injective and surjective) - every value of the given domain is mapped to a unique value from the codomain.
03

Finding Why Inverses of Trigonometric Functions Need Restrictions

Periodic trigonometric functions, like cosine, are not bijective over their entire domain since they do not pass the horizontal line test (multiple values of \(x\) will yield the same \(y\)). We need to restrict the domain of the original function, such that every value of the range corresponds to only one unique input value in the domain.
04

Restricting the Domain of the Cosine Function

To restrict the cosine function such that it is one-to-one and onto, we need to choose a suitable interval of \(x\) to capture all output values in \([-1, 1]\) exactly once. One interval that satisfies this constraint is \([0, \pi]\). In this interval, the cosine function is monotonically decreasing, and thus, passes the horizontal line test.
05

Establishing the Range of the Inverse Cosine Function

After restricting the cosine function to \([0, \pi]\), the inverse function, \(\cos^{-1}(x)\), is realized. The inverse cosine function reverses the mappings - input values are in the interval \([-1, 1]\), and the resulting values lie within the interval \([0, \pi]\). That is why the values of \(\cos^{-1}x\) lie in the interval \([0, \pi]\).

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