Question

How do you use the half angle identity to find exact value of tan 67.5?

Answer 1

Find tan (67.5)

Ans: (1 + sqrt2)

Explanation:

Call tan (67.5) = tan x --> tan 2x = (tan 135^@) = -1
Apply the trig identity: `tan 2x = (2tan x)/(1 - tan^2 x)`
We get:
`-1 = (2tan x)/(1 - tan^2 x)`.
`tan^2 x - 1 = 2tan x`

Solve the quadratic equation in tan x:
`tan^2 x - 2tan x - 1 = 0`
`D = d^2 = b^2 - 4ac = 4 + 4 = 8` --> `d = +- 2sqrt2`
`tan x = 2/2 +- (2sqrt2)/2 = 1 +- sqrt2`
Since the arc (67.5) is in Quadrant I, then its tan is positive.
`tan x = 1 + sqrt2`
Check by calculator.
tan (67.5) = 2.41
`(1 + sqrt2) = 1 + 1,41 = 2.41`. OK