Question

How do you find the exact functional value cot (113 pi/ 12) using the cosine sum or difference identity?

Answer 1

Find `cot ((113pi)/12)`

Ans: `1/(1 + sqrt2)`

Explanation:

`cot ((113pi)/12) = cot ((5pi)/12 + (108pi)/12) = cot ((5pi)/12)`
`= 1/tan ((5pi)/12)`. Call `tan ((5pi)/12) = tan t`
`tan 2t = tan ((10pi)/12) = tan ((5pi)/6) = - 1/sqrt3`

Apply the trig identity `tan 2t = (2tan t)/(1 - tan^2 t)`
`tan 2t = - 1/sqrt3 = (2tan t)/(1 - t^2)`
We get a quadratic equation in tan t
`tan^2 t - 2sqrt3t - 1 = 0`
`D = d^2 = b^2 - 4ac = 4 + 4 = 8`--> `d = +- 2sqrt2`
`tan ((113pi)/12) = tan (5pi)/12 = tan t = 1 +- sqrt2`
Because the arc (5pi)/12 is located in Quadrant I, then
`tan ((5pi)/12) = 1 + sqrt2`

Therefor:
`cot ((113pi)/12) = 1/tan ((5pi)/12) = 1/(1 + sqr2)`