find the exact value of the trigonometric expression without the use of a calculator.\n(\tanleft(\frac{11pi}{…

find the exact value of the trigonometric expression without the use of a calculator.\n(\tanleft(\frac{11pi}{6}-\frac{7pi}{4}\right))\nfind the exact value of the expression.\n(\tanleft(\frac{11pi}{6}-\frac{7pi}{4}\right)=square)\n(simplify your answer. type an exact answer, using radicals as needed)

find the exact value of the trigonometric expression without the use of a calculator.\n(\tanleft(\frac{11pi}{6}-\frac{7pi}{4}\right))\nfind the exact value of the expression.\n(\tanleft(\frac{11pi}{6}-\frac{7pi}{4}\right)=square)\n(simplify your answer. type an exact answer, using radicals as needed)

Answer

Explanation:

Step1: Calculate the difference of angles

First, find the common denominator of (6) and (4), which is (12). [ \begin{align*} \frac{11\pi}{6}-\frac{7\pi}{4}&=\frac{11\pi\times2}{6\times2}-\frac{7\pi\times3}{4\times3}\ &=\frac{22\pi}{12}-\frac{21\pi}{12}\ &=\frac{\pi}{12} \end{align*} ]

Step2: Use the tangent formula for (\frac{\pi}{12})

We know that (\tan\frac{\pi}{12}=\tan\left(\frac{\pi}{3}-\frac{\pi}{4}\right)) By the tangent subtraction formula (\tan(A - B)=\frac{\tan A-\tan B}{1 + \tan A\tan B}), where (A=\frac{\pi}{3}), (\tan A=\sqrt{3}) and (B = \frac{\pi}{4}), (\tan B=1) [ \begin{align*} \tan\left(\frac{\pi}{3}-\frac{\pi}{4}\right)&=\frac{\sqrt{3}-1}{1+\sqrt{3}\times1}\ &=\frac{(\sqrt{3}-1)(\sqrt{3}-1)}{(1 + \sqrt{3})(\sqrt{3}-1)}\ &=\frac{3-2\sqrt{3}+1}{3-1}\ &=\frac{4 - 2\sqrt{3}}{2}\ &=2-\sqrt{3} \end{align*} ]

Answer:

(2-\sqrt{3})