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Chapter 4: Problem 70

Find a positive angle less than \(360^{\circ}\) or \(2 \pi\) that is coterminal with the given angle. $$-\frac{38 \pi}{9}$$

Short Answer

Expert verified
The positive angle less than \(2\pi\), which is coterminal with \(-\frac{38\pi}{9}\), is \(\frac{16\pi}{9}\).

Step by step solution

01

Identify Coterminal Angles

Coterminal angles are angles that share the same terminal side. They could be found by adding or subtracting multiples of \(2\pi\) or \(360^{\circ}\) (a full revolution) to the given angle. So, we need to add multiples of \(2\pi\) to \(-\frac{38 \pi}{9}\) until we get a positive angle less than \(2\pi\).
02

Add Multiples of \(2\pi\)

Finding the proper value to add can be difficult so let's try to simplify the task. We know that \(2\pi \times 1 = 2\pi\) or \(\frac{18 \pi}{9}\), \(2\pi \times 2 = 4\pi\) or \(\frac{36 \pi}{9}\), \(2\pi \times 3 = 6\pi\) or \(\frac{54 \pi}{9}\) and so on until we find a positive angle less than \(2\pi\). Try to aim for a multiple that will result in an angle slightly larger than \(\frac{38\pi}{9}\) but remember the result should be less than \(2\pi\) or \(\frac{18 \pi}{9}\). This may take several trials and errors.
03

Compute the Positive Coterminal Angle

For example, it was found that 3 rotations or \(6\pi\) or \(\frac{54 \pi}{9}\) added to \(-\frac{38\pi}{9}\) gives a positive coterminal angle. When added together, we get \(-\frac{38\pi}{9} + \frac{54\pi}{9} = \frac{16\pi}{9}\). This is our positive coterminal angle that is less than \(2\pi\).

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Most popular questions from this chapter

Determine whether each statement makes sense or does not make sense, and explain your reasoning. After using the four-step procedure to graph \(y=-\cot \left(x+\frac{\pi}{4}\right),\) I checked my graph by verifying it was the graph of \(y=\cot x\) shifted left \(\frac{\pi}{4}\) unit and reflected about the \(x\) -axis.

Determine whether each statement makes sense or does not make sense, and explain your reasoning. When an angle's measure is given in terms of \(\pi,\) I know that it's measured using radians.

Graph one period of each function. $$y=\left|3 \cos \frac{2 x}{3}\right|$$

Solve: \(\quad \log _{2}(2 x+1)-\log _{2}(x-2)=1\) (Section 3.4, Example 7)

will help you prepare for the material covered in the next section. a. Graph \(y=-3 \cos \frac{x}{2}\) for \(-\pi \leq x \leq 5 \pi\) b. Consider the reciprocal function of \(y=-3 \cos \frac{x}{2}\) namely, \(y=-3 \sec \frac{x}{2} .\) What does your graph from part (a) indicate about this reciprocal function for \(x=-\pi, \pi, 3 \pi,\) and \(5 \pi ?\)
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