Question

How do you find Tan 22.5 using the half angle formula?

Answer 1

Find tan (22.5)

Answer: `-1 + sqrt2`

Explanation:

Call tan (22.5) = tan t --> tan 2t = tan 45 = 1

Use trig identity: `tan 2t = (2tan t)/(1 - tan^2 t)` (1)

`tan 2t = 1 = (2tan t)/(1 - tan^2 t)` -->
--> `tan^2 t + 2(tan t) - 1 = 0`
Solve this quadratic equation for tan t.

`D = d^2 = b^2 - 4ac = 4 + 4 = 8` --> `d = +- 2sqrt2`
There are 2 real roots:
tan t = -b/2a +- d/2a = -2/1 + 2sqrt2/2 = - 1 +- sqrt2
Answer:
`tan t = tan (22.5) = - 1 +- sqrt2`
Since tan 22.5 is positive, then take the positive answer:
tan (22.5) = - 1 + sqrt2