Question

find the exact value of the following expression.

\tan left( \frac { 11 pi } { 6 } + \frac { 3 pi } { 4 }
ight)

find the exact value of the expression.

\tan left( \frac { 11 pi } { 6 } + \frac { 3 pi } { 4 }
ight) = \frac { - ( sqrt { 3 } + 3 ) } { 3 - sqrt { 3 } }

(simplify your answer. type an exact answer, using radicals as needed. use integers or fractions for a expression. rationalize all denominators.)

Explanation:

Step1: Use the tangent addition formula

The tangent addition formula is \(\tan(A + B)=\frac{\tan A+\tan B}{1-\tan A\tan B}\). Let \(A = \frac{11\pi}{6}\) and \(B=\frac{3\pi}{4}\).
First, find \(\tan\frac{11\pi}{6}\) and \(\tan\frac{3\pi}{4}\).
\(\tan\frac{11\pi}{6}=\tan(2\pi-\frac{\pi}{6})=-\tan\frac{\pi}{6}=-\frac{\sqrt{3}}{3}\)
\(\tan\frac{3\pi}{4}=- 1\)

Step2: Substitute into the formula

Substitute \(\tan A=-\frac{\sqrt{3}}{3}\) and \(\tan B = - 1\) into \(\tan(A + B)=\frac{\tan A+\tan B}{1-\tan A\tan B}\)

$$
LATEXBLOCK0
$$

Step3: Rationalize the denominator

Multiply the numerator and denominator by \(3 + \sqrt{3}\)

$$
LATEXBLOCK1
$$

Answer:

\(-(\sqrt{3}+2)\)