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Chapter 4: Problem 45

Use an inverse trigonometric function to write \(\theta\) as a function of \(x .\)

Short Answer

Expert verified
Without additional context or information, \(\theta\) can be written as a function of \(x\) using an inverse trigonometric function as follows: \(\theta = \arcsin(x)\), \(\theta = \arccos(x)\), or \(\theta = \arctan(x)\).

Step by step solution

01

Choose an inverse function

Firstly, you must pick an inverse trigonometric function to express \(\theta\) in terms of \(x\). You have a choice between \(\arcsin(x)\), \(\arccos(x)\), \(\arctan(x)\), \(\arccsc(x)\), \(\arcsec(x)\), or \(\arccot(x)\). These functions can be selected depending on what more specifications you have about \(x\) and \(\theta\). Without additional information, we will use \(\arcsin(x)\), \(\arccos(x)\), and \(\arctan(x)\) in this discussion, as these are the primary inverse trigonometric functions.
02

Express \(\theta\) as \(\arcsin(x)\)

Use the inverse sine function to write \(\theta\) as a function of \(x\). That is \(\theta = \arcsin(x)\) or \(\theta = \sin^{-1}(x)\). This implies that \(x\) is the sine value of an angle and \(\theta\) is that angle.
03

Express \(\theta\) as \(\arccos(x)\)

You can also use the inverse cosine function to express \(\theta\) as a function of \(x\). That is \(\theta = \arccos(x)\) or \(\theta = \cos^{-1}(x)\). Here, \(x\) is the cosine value of an angle and \(\theta\) is that angle.
04

Express \(\theta\) as \(\arctan(x)\)

Or, you can use the inverse tangent function to write \(\theta\) as a function of \(x\). That is \(\theta = \arctan(x)\) or \(\theta = \tan^{-1}(x)\). In this sense, \(x\) is the tangent value of an angle and \(\theta\) is that angle.

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Most popular questions from this chapter

Evaluate the expression without using a calculator. \(\sin ^{-1}\left(-\frac{\sqrt{3}}{2}\right)\)

Use a calculator to evaluate the expression. Round your result to two decimal places. \(\arcsin (-0.125)\)

Use a calculator to evaluate the expression. Round your result to two decimal places. \(\tan ^{-1}(-\sqrt{2165})\)

Use a graphing utility to graph \(f\) \(g,\) and \(y=x\) in the same viewing window to verify geometrically that \(g\) is the inverse function of \(f .\) (Be sure to restrict the domain of \(f\) properly.) \(f(x)=\cos x, \quad g(x)=\arccos x\)

Sketch a graph of the function. \(y=2 \arccos x\)
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