Respuesta :
Given the function:
[tex]\text{ Tan }\frac{7\pi}{8}[/tex]Let's determine the Cosine Equivalent:
[tex]\text{Tan}\frac{7\pi}{8}\text{ =}\frac{Opposite}{Adjacent}[/tex][tex]\text{Opposite = 7}\pi[/tex][tex]\text{Hypotenuse = }\sqrt[]{Opposite^2+Adjacent^2}[/tex][tex]\text{ = }\sqrt[]{(7\pi)^2\text{ + }8^2}[/tex][tex]\text{Hypotenuse = }\sqrt[]{64\text{ + 49}\pi^2}[/tex][tex]\text{ Cosine = }\frac{Adjacent}{Hypotenuse}\text{ = }\frac{8}{\text{ }\sqrt[]{64\text{ + 49}\pi^2}}[/tex]Let's determine the Sine Equivalent:
[tex]\text{ Sine = }\frac{Opposite}{Hypotenuse}[/tex][tex]\text{ Sine = }\frac{7\pi}{\text{ }\sqrt[]{64\text{ + 49}\pi^2}}[/tex]Let's now determine the exact value using half-angle identities:
[tex]\text{ Tan }\frac{u}{2}\text{ = }\frac{1\text{ - Cosu}}{\text{ Sinu}}[/tex][tex]=\text{ }\frac{1\text{ - }\frac{8}{\text{ }\sqrt[]{64\text{ + 49}\pi^2}}\text{ }}{\frac{7\pi}{\text{ }\sqrt[]{64\text{ + 49}\pi^2}}}[/tex][tex]=\text{ }\frac{-8\text{ + }\sqrt[]{64+49\pi^2}}{7\pi}[/tex][tex]\text{ = 0.70033092876 }\approx\text{ 0.70}[/tex]