Question

How do you use the half angle identity to find exact value of tan(75) degrees?

Answer 1

Find exact value of tan (75)

Ans: 3 + sqrt3

Explanation:

Call tan 75 = tan x
Trig Table --> tan (150) = - sqrt3/3 = - 1/sqrt3
Apply the trig identity: tan 2x = (2tan x)/(1 - tan^2 x)
- 1/(sqrt3) = (2tan x)/(1 - tan^2 x)
-1 + tan^2 x = 2sqrt3.tan x
`tan^2 x - 2sqrt3.tan c - 1 = 0`
`D = d^2 = b^2 - 4ac = 12 + 4 = 16.` --> `d = +- 4`
`tan 75 = tan (x) = (2sqrt3)/2 +- 4/2 = sqrt3 +- 2`.
Since tan 75 is positive, then tan 75 = sqrt3 + 2 = 3.73
Check by calculator.
tan 75 = 3.73
`2 + Sqrt3 = 3.73`OK