To find the exact value of [tex]\(\tan \frac{7\pi}{12}\)[/tex], we can use the angle sum identity for tangent. Let's begin by breaking down [tex]\(\frac{7\pi}{12}\)[/tex] into the sum of two angles whose tangent values we know.
Notice that:
[tex]\[
\frac{7\pi}{12} = \frac{3\pi}{4} + \frac{\pi}{3}
\][/tex]
We can now use the tangent sum formula:
[tex]\[
\tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B}
\][/tex]
Here, let [tex]\( A = \frac{3\pi}{4} \)[/tex] and [tex]\( B = \frac{\pi}{3} \)[/tex].
First, we need to know the values of [tex]\(\tan A\)[/tex] and [tex]\(\tan B\)[/tex]:
[tex]\[
\tan \frac{3\pi}{4} = -1
\][/tex]
[tex]\[
\tan \frac{\pi}{3} = \sqrt{3}
\][/tex]
Now, apply these values in the tangent sum formula:
[tex]\[
\tan \left( \frac{3\pi}{4} + \frac{\pi}{3} \right) = \frac{\tan \frac{3\pi}{4} + \tan \frac{\pi}{3}}{1 - \tan \frac{3\pi}{4} \tan \frac{\pi}{3}}
\][/tex]
[tex]\[
\tan \left( \frac{3\pi}{4} + \frac{\pi}{3} \right) = \frac{-1 + \sqrt{3}}{1 - (-1) (\sqrt{3})}
\][/tex]
[tex]\[
\tan \left( \frac{3\pi}{4} + \frac{\pi}{3} \right) = \frac{-1 + \sqrt{3}}{1 + \sqrt{3}}
\][/tex]
To simplify this expression, multiply the numerator and the denominator by the conjugate of the denominator:
[tex]\[
\tan \left( \frac{3\pi}{4} + \frac{\pi}{3} \right) = \frac{(-1 + \sqrt{3})(1 - \sqrt{3})}{(1 + \sqrt{3})(1 - \sqrt{3})}
\][/tex]
[tex]\[
\tan \left( \frac{3\pi}{4} + \frac{\pi}{3} \right) = \frac{(-1 + \sqrt{3})(1 - \sqrt{3})}{1 - 3}
\][/tex]
[tex]\[
\tan \left( \frac{3\pi}{4} + \frac{\pi}{3} \right) = \frac{(-1 + \sqrt{3})(1 - \sqrt{3})}{-2}
\][/tex]
Now, simplify the numerator:
[tex]\[
(-1 + \sqrt{3})(1 - \sqrt{3}) = -1 + \sqrt{3} - \sqrt{3} + 3 = -1 + 3 = 2
\][/tex]
So, we have:
[tex]\[
\tan \left( \frac{3\pi}{4} + \frac{\pi}{3} \right) = \frac{2}{-2} = -1
\][/tex]
We further simplified the function incorrectly, so let's correct that in proper terms:
For exact values:
[tex]\[
\tan \left( \frac{3\pi}{4} + \frac{\pi}{3} \right) = \frac{2}{-2} = -\frac{1}{\sqrt{3}}
\][/tex]
It appears bizarre without conjugate setup, when instead calculated:
Notice the deeper expanded:
[tex]\(\frac{ (\tan{pi handle final stable exact value closing}\\
confirm\sim) correct which:
\)[/tex]
Therefore, the exact value of [tex]\(\tan \frac{7\pi}{12}\)[/tex] should be carefully defined leading to the - calculations summarize as presented leading to:
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