QuestionFind $\tan \left(-\frac{11 \pi}{3}\right)$ exactly without a calculator. Simplify your answer or state if it's undefined.

Studdy Solution

STEP 1

Assumptions1. We are asked to find the exact value of $\tan \left(-\frac{11 \pi}{3}\right)$ without using a calculator. . We know that the tangent function has a period of $\pi$ .
3. This means that $\tan \theta = \tan (\theta + n\pi)$ , where $n$ is an integer.

STEP 2

First, we need to find an equivalent angle for $-\frac{11 \pi}{}$ that lies within the interval $[-\pi/2, \pi/2]$ . This can be done by adding multiples of $\pi$ to $-\frac{11 \pi}{}$ until we get an angle in this interval.

STEP 3

We can write $-\frac{11 \pi}{3}$ as $-\frac{12 \pi}{3} + \frac{\pi}{3}$ .

STEP 4

Since $\tan \theta = \tan (\theta + n\pi)$ , we can say that $\tan \left(-\frac{11 \pi}{3}\right) = \tan \left(-\frac{12 \pi}{3} + \frac{\pi}{3}\right)$ .

STEP 5

implify $-\frac{12 \pi}{3} + \frac{\pi}{3}$ to get $-\frac{11 \pi}{3}$ .
$-\frac{12 \pi}{3} + \frac{\pi}{3} = -4\pi + \frac{\pi}{3} = -\frac{11 \pi}{3}$

STEP 6

Now, we can see that $-\frac{11 \pi}{3}$ is equivalent to $\frac{\pi}{3}$ , which lies within the interval $[-\pi/2, \pi/2]$ .

STEP 7

Now, we can find the exact value of $\tan \left(-\frac{11 \pi}{3}\right)$ by finding the value of $\tan \left(\frac{\pi}{3}\right)$ .

STEP 8

We know that $\tan \left(\frac{\pi}{3}\right)$ is equal to $\sqrt{3}$ .

STEP 9

Therefore, $\tan \left(-\frac{11 \pi}{3}\right) = \sqrt{3}$ .
The answer is A. $\tan \left(-\frac{11 \pi}{3}\right) = \sqrt{3}$ .

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