QuestionFind the exact value of $\tan \frac{13 \pi}{4}$ using the periodicity of trigonometric functions.

Studdy Solution

STEP 1

Assumptions1. The trigonometric functions are periodic. . The period of the tangent function is $\pi$ .
3. We need to find the exact value of $\tan \frac{13 \pi}{4}$ without using a calculator.

STEP 2

First, we need to use the periodic property of the tangent function to simplify the argument. Since the period of the tangent function is $\pi$ , we can subtract multiples of $\pi$ from the argument until we get an equivalent argument within the interval $(-\pi/2, \pi/2]$ .
$\tan \frac{13 \pi}{4} = \tan \left(\frac{13 \pi}{4} -2\pi \times n\right)$ where $n$ is an integer.

STEP 3

Now, we need to find the value of $n$ that brings the argument within the interval $(-\pi/2, \pi/2]$ . We can do this by dividing the argument by $2\pi$ and taking the integer part.
$n = \left\lfloor \frac{13 \pi}{ \times2\pi} \right\rfloor = \left\lfloor \frac{13}{8} \right\rfloor =1$

STEP 4

Substitute $n =1$ into the equation from2 to simplify the argument.
$\tan \frac{13 \pi}{4} = \tan \left(\frac{13 \pi}{4} -2\pi \times1\right) = \tan \left(\frac{13 \pi}{4} - \frac{8 \pi}{4}\right)$

STEP 5

implify the argument further.
$\tan \frac{13 \pi}{4} = \tan \left(\frac{5 \pi}{4}\right)$

STEP 6

Now, we need to find the exact value of $\tan \frac{5 \pi}{4}$ . We know that $\tan \frac{\pi}{4} =1$ , and $\tan \left(\pi + \frac{\pi}{4}\right) = \tan \frac{\pi}{4}$ because of the periodicity of the tangent function. Also, $\tan \frac{5 \pi}{4} = \tan \left(\pi + \frac{\pi}{4}\right)$ . Therefore,
$\tan \frac{5 \pi}{4} = \tan \frac{\pi}{4} =1$ So, the exact value of $\tan \frac{13 \pi}{4}$ is1.

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