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Chapter 4: Problem 42

Find the exact value of each trigonometric function. Do not use a calculator. $$\sec \left(-\frac{9 \pi}{4}\right)$$

Short Answer

Expert verified
The exact value of \(\sec \left(-\frac{9 \pi}{4}\right)\) is \(\sqrt{2}\).

Step by step solution

01

Simplify the Radian Measure

Bring the radian measure into the range of 0 to \(2 \pi\) by adding a multiple of \(2 \pi\). This is possible because trigonometric functions are periodic functions with period \(2 \pi\). Since the given angle is \(-\frac{9 \pi}{4}\), we add \(2 \pi= \frac{8 \pi}{4}\) to get \(-\frac{9 \pi}{4}+ \frac{8 \pi}{4}= -\frac{\pi}{4}\). So, \(\sec \left(-\frac{9 \pi}{4}\right)\) is equivalent to \(\sec(\frac{-\pi}{4})\).
02

Determine the Quadrant and Find the Reference Angle

The angle is negative, indicating a clockwise direction from the initial side, into the fourth quadrant. The reference angle in the fourth quadrant is \(\frac{\pi}{4}\). In the fourth quadrant, cosine function is positive, and since secant is reciprocally related to cosine, secant is also positive in this quadrant.
03

Find the Exact Value of the Secant Function

For an angle of \(\frac{\pi}{4}\) in the unit circle, we know the cosine value is \(\frac{\sqrt{2}}{2}\). Therefore, the secant of the angle, being the reciprocal of cosine, is \(sec \: \theta = \frac{1}{cos \: \theta} = \frac{1}{\frac{\sqrt{2}}{2}} = \sqrt{2}\). Thus, the exact value of \(\sec \left(-\frac{9 \pi}{4}\right)\) is \(\sqrt{2}\).

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Most popular questions from this chapter

Use a graphing utility to graph two periods of the function. $$y=3 \sin (2 x-\pi)+5$$

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.

Will help you prepare for the material covered in the next section. a. Graph \(y=\cos x\) for \(0 \leq x \leq \pi\) b. Based on your graph in part (a), does \(y=\cos x\) have an inverse function if the domain is restricted to \([0, \pi] ?\) Explain your answer. c. Determine the angle in the interval \([0, \pi]\) whose cosine is \(-\frac{\sqrt{3}}{2} .\) Identify this information as a point on your graph in part (a).

Have you ever noticed that we use the vocabulary of angles in everyday speech? Here is an example: My opinion about art museums took a \(180^{\circ}\) turn after visiting the San Francisco Museum of Modern Art. Explain what this means. Then give another example of the vocabulary of angles in everyday use.

Determine whether each statement makes sense or does not make sense, and explain your reasoning. Using the equation \(y=A \sin B x,\) if I replace either \(A\) or \(B\) with its opposite, the graph of the resulting equation is a reflection of the graph of the original equation about the \(x\) -axis.
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