Question

Given `tantheta=-3/4` and `90<theta<180`, how do you find `tan(theta/2)`?

Answer 1

`tan (t/2) = 3`

Explanation:

Call tan (t/2) = x and use trig identity:
`tan 2t = (2tan t)/(1 - tan^2 t)`
`tan 2x = (-3/4) = (2x)/(1 - x^2)`
Cross multiply:
`3x^2 - 3 = 8x`
Solve the quadratic equation for x.
`3x^2 - 8x - 3 = 0`
`D = d^2 = b^2 - 4ac = 64 + 36 = 100` --> `d = +- 10`
There are 2 real roots:
`x = - b/(2a) +- d/(2a) = 8/6 +- 10/6 = (4 +- 5)/3`
x1 = 3, and `x2 = - 1/3`
`tan (t/2) = x1 = 3`
and `tan (t/2) = x2 = - 1/3`
Because t is in Quadrant II, then `(t/2)` is in Quadrant I, and `tan (t/2)` is positive.
There for: `tan (t/2) = 3`