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Chapter 4: Problem 46

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

Short Answer

Expert verified
The exact value of \( \tan \left(-\frac{9 \pi}{4}\right)\) is -1.

Step by step solution

01

Simplify the Angle

Given the angle \(-\frac{9\pi}{4}\), this angle can be simplified using the periodicity property of the tangent function. The tangent function has a period of \(\pi\), which means that \(\tan(\theta) = \tan(\theta + n\pi)\) for any integer n. This property allows us to add \(\pi\) to the -9蟺/4 multiple times until we get an angle that is easier to deal with. It would be best to add \(2\pi\) which is an equivalent of a full circle, since doing so won't affect the tangent's value. Adding \(2\pi\) is the same as adding \(8\pi/4\), so \(- \frac{9\pi}{4} + \frac{8\pi}{4} = - \frac{\pi}{4}\).
02

Evaluate the Tangent Function

Now, it's left to find the tangent of the simplified angle. So, based on the unit circle, the tangent of -蟺/4 equals -1, because \(\tan(-\frac{\pi}{4}) = -1\).

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Most popular questions from this chapter

Use a graphing utility to graph each pair of functions in the same viewing rectangle. Use a viewing rectangle that shows the graphs for at least two periods. $$y=-2.5 \sin \frac{\pi}{3} x \text { and } y=-2.5 \csc \frac{\pi}{3} x$$

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

Use a graphing utility to graph each function. Use a viewing rectangle that shows the graph for at least two periods. $$y=\frac{1}{2} \tan (\pi x+1)$$

will help you prepare for the material covered in the next section. $$\text { Simplify: } \frac{-\frac{3 \pi}{4}+\frac{\pi}{4}}{2}$$

What does a phase shift indicate about the graph of a sine function? How do you determine the phase shift from the function's equation?
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