Question

How do you use the half angle identity to find exact value of tan 112.5?

Answer 1

Find exact value of `tan (112.5)`

Explanation:

Call tan (112.5) = tan x.
tan 2x = tan (225) = tan (45 + 180) = tan 45 = 1
Apply the trig identity: `tan 2x = (2tan x)/(1 - tan^2 x)`, we get:
`1 = (2tan x)/(1 - tan^2 x)`
`1 - tan^2 x = 2tan x`.
Solve the quadratic equation:
`tan^2 x + 2tan x - 1 = 0`
`D = d^2 = b^2 - 4ac = 4 + 4 = 8` --> `d = +- 2sqrt2`
Ther3 are 2 real roots:
`tan x = -2/2 +- 2sqrt2/2 = -1 +- sqrt2.`
Since (tan 112.5) is negative (Quadrant II), then
`tan x = -1 - sqrt2`
Check by calculator.
sin (112.5) = -2.14
`(-1 - sqrt2) = -1 - 1,41 = - 2.41`. OK