Question

How do you use a half-angle formula to simplify `tan[ (9pi) / 8 ]`?

Answer 1

Simplify `tan ((9pi)/8)`

Explanation:

`tan ((9pi)/8) = tan (pi/8 + pi)` = tan pi/8
Call tan `pi/8 = t`
`tan 2t = tan ((2pi)/8) = tan (pi/4) = 1`
Use trig identity: `tan 2t = 2t/(1 - t^2)`
`tan (pi/4) = 1 = (2t)/(1 - t^2)` -->`1 - t^2 = 2t`-->

`t^2 + 2t - 1 = 0`
`D = d^2 = b^2 - 4ac = 4 + 4 = 8`--> `d = +- 2sqrt2`

`tan ((9pi)/8) = tan (pi/8) = t = - 1 +- sqrt2`

Since the arc `(9pi)/8` is in Quadrant III, then only the positive answer is accepted.
`tan (9pi)/8 = - 1 + sqrt2` = 0.414

Check with calculator --> tan (22.5) = 0.414 OK