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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 principal values of the inverse cosine function, \(\cos^{-1}x\), lie in the interval \([0, \pi]\) because the cosine function is periodic and must be restricted to a unique interval to be bijective. The interval \([0, \pi]\) is chosen as it covers all values of the cosine function exactly once and is centered around \(0\), which is convenient for most real-world applications.

Step by step solution

01

Understanding the cosine function

The cosine function is a periodic function that describes a specific relationship between the angular measure of a triangle (in radians) and its sides. It is defined for all real numbers and takes values in the range \([-1, 1]\). The cosine function repeats its values in intervals of \(2\pi\), which means \(\cos(x) = \cos(x+2\pi n)\), where n is an integer.
02

Defining inverse functions

An inverse function is a function that reverses the effect of the original function, sending the output of the original function back to the input. Inverse functions can only exist for bijective functions, meaning that each input and output is paired with one and only one value. However, since the cosine function is periodic, it cannot be bijective unless we restrict its domain.
03

Choosing a domain for the cosine function

In order to create an inverse function for the cosine function, we must restrict its domain to a unique interval, so that it becomes bijective. This restricted interval must be chosen in such a way that the cosine function takes all values from \([-1, 1]\) exactly once. Moreover, it is usually convenient to choose an interval centered around \(0\) (since most real world applications use angles close to \(0\)). The interval that satisfies these conditions is \([0, \pi]\).
04

Defining the inverse cosine function

Given the domain restriction of the cosine function mentioned in step 3, we can now define the inverse cosine function, \(\cos^{-1}x\). The inverse cosine function, denoted as \(\cos^{-1}x\) or \(arccos(x)\), is the function that gives the angle whose cosine is \(x\). It is defined for values in the range \([-1, 1]\), and its range is the restricted interval for the cosine function - \([0, \pi]\).
05

Concluding with the interval of \(\cos^{-1}x\)

In conclusion, the principal values of the inverse cosine function, \(\cos^{-1}x\), lie in the interval \([0, \pi]\). This is because the cosine function is periodic and must be restricted to a unique interval so it can be made bijective. The chosen interval for the restricted domain is \([0, \pi]\), which allows the inverse function to exist and covers all the values of the cosine function exactly once.

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