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Chapter 4: Problem 71

In Exercises \(61-86,\) use reference angles to find the exact value of each expression. Do not use a calculator. $$\tan \frac{9 \pi}{4}$$

Short Answer

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

Step by step solution

01

Understanding the periodicity of the tan function

Since the tangent function has a periodicity of \(\pi\), every integer multiple of \(\pi\) would effectively bring us back to the same point. Hence, we can subtract any multiple of \(\pi\) from our given angle until we get an equivalent angle that is within our standard unit circle. Thus, \(\tan \frac{9 \pi}{4} = \tan \frac{8 \pi}{4} + \frac{\pi}{4} = \tan \frac{\pi}{4}\).
02

Calculate the tangent of the simplified angle

The angle \( \frac{\pi}{4} \) represents a 45-degree angle, and it is known that \( \tan(45掳) = 1 \). So, \( \tan \frac{\pi}{4} = 1 \).

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Most popular questions from this chapter

Use a graphing utility to graph $$ y=\sin x-\frac{\sin 3 x}{9}+\frac{\sin 5 x}{25} $$ in a \(\left[-2 \pi, 2 \pi, \frac{\pi}{2}\right]\) by [-2,2,1] viewing rectangle. How do these waves compare to the smooth rolling waves of the basic sine curve?

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