Question

How do you find the exact value of the half angle of `tan 157.5`?

Answer 1

Use the trig identity: `tan 2x. (1 - tan^2 x) = 2.tan x`
Call tan x = t. Calculator gives: `tan 2x = tan 157.5 deg = -0.41`
`-0.41.(1 - t^2) - 2t = 0`
`0.41 t^2 - 2t - 0.41 = 0`. This is a quadratic equation. Use the formula to solve it.

d^2 = b^2 - 4ac = 4 + 0.67 = 4.67 -> d = 2.16
x1 = 2/0.82 + 2.16/0.82 = 5.07
x2 = 2/0.82 - 2.16/0.82 = -0.20 (rejected since tan x = tan (157.5/2) must be positive).
Answer: t = tan x = 5.07 --> x = 78.84