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Chapter 4: Problem 72

In Exercises \(61-86,\) use reference angles to find the exact value of each expression. Do not use a calculator. $$\tan \frac{9 \pi}{2}$$

Short Answer

Expert verified
\(\tan(\frac{9\pi}{2})\) is undefined

Step by step solution

01

Identifying the quadrant and reference angle

In order to evaluate the expression, the first step is to find where the angle \(\frac{9\pi}{2}\) lies on the unit circle. Remember that \(\frac{2\pi}{2} = \pi\) represents a complete revolution around the unit circle. Therefore, \(\frac{9\pi}{2} = 4\pi + \frac{\pi}{2}\). This is the same as making 4 complete revolutions around the circle and then moving an additional \(\frac{\pi}{2}\) radians, which ends in the positive y-axis.
02

Evaluating Tangent Function

Once we know where the angle ends up, we know that the reference angle is 0, because we end up at the positive y-axis, moving counter-clockwise. The point on the unit circle corresponding to this angle is \((0,1)\). The tangent of an angle in the unit circle is the ratio of the y-coordinate to the x-coordinate. In this case, the x-coordinate is 0 and the y-coordinate is 1. Therefore, \(\tan(\frac{9\pi}{2})\) is undefined because we cannot divide by zero.

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