Expert: Ahmed Salami - 5/3/2010

Question
Determine all solutions of the equations in radians.

Find tan x/2, given that tan x = 3 and x terminates in pi < x < pi/2.

Answer
Hi Meghan,
First up, the interval π < x < π/2 is invalid since π/2 < π

But from the double-angle formulae,
tan2A = 2tanA/(1 - tan²A)
Let x = 2A, then x/2 = A and
tanx = 2tan(x/2)/(1 - tan²(x/2))
Taking t = tan(x/2)
tanx = 2t/(1 - t²)
If tanx = 3
3 = 2t/(1 - t²)
2t = 3(1 - t²)
2t = 3 - 3t²
3t² + 2t - 3 = 0
Solve the quadratic equation for the two values of t and then
x/2 = arctan t
x = 2(arctan t)

Get back to me.

Regards