Answer:
The exact value of \(\tan (\frac{5\pi}{12})\) is \(2+\sqrt{3}\)
Step-by-step explanation:
We need to calculate the exact value of \(\tan (\frac{5\pi}{12})\)
Since, \(\tan \frac{x}{2} = \frac{1- \cos x}{\sin x}\)
Put \(x= \frac{5 \pi}{6}\) in above
\(\tan (\frac{5\pi}{12}) = \frac{1- \cos \frac{5\pi}{6}}{\sin \frac{5\pi}{6}}\)
Since,
\(\cos \frac{5\pi}{6}}=\frac{-\sqrt{3}}{2}\)
\(\sin \frac{5\pi}{6}}=\frac{1}{2}\)
\(\tan (\frac{5 \pi }{12}) = \frac{1+\frac{\sqrt{3}}{2}}{\frac{1}{2}}\)
\(\tan (\frac{5 \pi }{12}) = \frac{\frac{2+\sqrt{3}}{2}}{\frac{1}{2}}\)
\(\tan (\frac{5 \pi }{12}) = 2+\sqrt{3}\)
Therefore, the exact value of \(\tan (\frac{5\pi}{12})\) is \(2+\sqrt{3}\)
