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