Question

Find the exact value for tan7/8 using half angle identities

Answer 1

Given the function:

`\text{ Tan }(7\pi)/(8)`

Let's determine the Cosine Equivalent:

`\text{Tan}(7\pi)/(8)\text{ =}(Opposite)/(Adjacent)`
`\text{Opposite = 7}\pi`
`\text{Hypotenuse = }\sqrt[]{Opposite^2+Adjacent^2}`
`\text{ = }\sqrt[]{(7\pi)^2\text{ + }8^2}`
`\text{Hypotenuse = }\sqrt[]{64\text{ + 49}\pi^2}`
`\text{ Cosine = }(Adjacent)/(Hypotenuse)\text{ = }\frac{8}{\text{ }\sqrt[]{64\text{ + 49}\pi^2}}`

Let's determine the Sine Equivalent:

`\text{ Sine = }(Opposite)/(Hypotenuse)`
`\text{ Sine = }\frac{7\pi}{\text{ }\sqrt[]{64\text{ + 49}\pi^2}}`

Let's now determine the exact value using half-angle identities:

`\text{ Tan }(u)/(2)\text{ = }\frac{1\text{ - Cosu}}{\text{ Sinu}}`
`=\text{ }\frac{1\text{ - }\frac{8}{\text{ }\sqrt[]{64\text{ + 49}\pi^2}}\text{ }}{\frac{7\pi}{\text{ }\sqrt[]{64\text{ + 49}\pi^2}}}`
`=\text{ }\frac{-8\text{ + }\sqrt[]{64+49\pi^2}}{7\pi}`
`\text{ = 0.70033092876 }\approx\text{ 0.70}`