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Chapter 4: Problem 42

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

Short Answer

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

Step by step solution

01

Define \(\theta\) using a known trigonometric function

The very first step to solve this problem is to use a known trigonometric function such as sine, cosine or tangent. For example, let \(\sin(\theta) = x\). Now \(\theta\) is expressed in relation to \(x\). This expression represents an angle \(\theta\) whose sine is \(x\).
02

Evaluate the inverse trigonometric function

To isolate \(\theta\), we will use the inverse of the sine function. This is done by using the inverse sine function, \(\arcsin()\), on both sides of the equation. Here, it is very important to remember that the inverse sine of the sine of \(\theta\) gives back \(\theta\). This yields: \(\theta = \arcsin(x)\).
03

Check the solution

As a final step, it is always good to verify the solution. In this problem, the result can be checked by substituting \(x\) with a known sine value from the unit circle. For example, if we choose \(x = 0\), \(\theta = \arcsin(0)\) should give \(0\) which is the angle whose sine is \(0\). Subsequently, if we choose \(x = 1\), \(\theta = \arcsin(1)\) should give \(\pi/2\) which is the angle whose sine is \(1\). Thus, we find that our solution is correct in both cases.

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Most popular questions from this chapter

Write an algebraic expression that is equivalent to the given expression. (Hint: Sketch a right triangle, as demonstrated in Example \(7.)\) \(\csc \left(\arctan \frac{x}{\sqrt{2}}\right)\)

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. \(\arccos 0.37\)

Define the inverse secant function by restricting the domain of the secant function to the intervals \([0, \pi / 2)\) and \((\pi / 2, \pi],\) and sketch the graph of the inverse trigonometric function.

Fill in the blank. If not possible, state the reason. As \(x \rightarrow 1^{-},\) the value of \(\arccos x \rightarrow\) ___.
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