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Chapter 4: Problem 41

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

Short Answer

Expert verified
\(\theta = \arccos(x)\)

Step by step solution

01

Determine the original function

Let's assume that \( x = \cos(\theta) \) to start with, as this is one of the most common trigonometric functions.
02

Use the inverse function

The inverse function of \( \cos \) is \( \arccos \), which is used when we want to write \( \theta \) as a function of \( x \). We rearrange \( x = \cos(\theta) \) to be \( \theta = \arccos(x) \).
03

Proof of result

For the majority of \( x \) in the domain of \( \arccos \), which is \([-1, 1]\), inputting \( x \) in the function \( \theta = \arccos(x) \) will give us a valid \( \theta \). Therefore, the transformation is correct.

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Most popular questions from this chapter

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

Use a graphing utility to graph the function. \(f(x)=-3+\arctan (\pi x)\)

Use a calculator to evaluate the expression. Round your result to two decimal places. \(\arccos 0.37\)

Use a calculator to evaluate the expression. Round your result to two decimal places. \(\tan ^{-1} \frac{19}{4}\)

Write an algebraic expression that is equivalent to the given expression. (Hint: Sketch a right triangle, as demonstrated in Example \(7.)\) \(\sin (\arccos x)\)
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