Question

How do you find the exact value of arctan(2-√3)

Answer 1

so we need an angle θ , so that
tan θ = 2 - √3

( we could be sneaky and use our calculator, sure enough it says 15°)

so we will make an "intelligent guess", then prove my guess is correct
we know tan 30° = 1/√3

and I also know that tan 30° = 2 tan15°/(1 - tan^2 15°)
let tan15° = x
so tan30° = 2 tan15/(1 - tan^2 15°) becomes

1/√3 = 2x/(1-x^2)
2√3 x = 1 - x^2
x^2 + 2√3 x - 1 = 0
x = ( -2√3 ± √(12+4) )/2
= -√3 ± 2 <----- well, who would have guessed!
but we know x = tan 15
so 2 - √3 = tan 15° , and

arctan(2-√3) = 15° or π/12 radians

Answer 2

OR

Multiply and divide 2 - √3 by 1/ 2

2 - √3 = [ ( 2 - √3 ) / 2 ] / ( 1 / 2 ) =

( 1 - √3 / 2 ) / ( 1 / 2 )

Since:

sin 30°= 1 / 2

cos 30° =√3 / 2

and half-angle formula for tangent is:

tan ( θ / 2 ) = ( 1 - cos θ ) / sin ( θ )

tan ( 30° / 2 ) =

( 1 - cos 30° ) / sin 30 °=

( 1 - √3 / 2) / ( 1 / 2 ) = tan 15°

arctan ( 2 - √3 ) =

arctan ( 1 - √3 / 2) / ( 1 / 2 ) =

15° = π / 12 rad