Question

How do you find the exact functional value tan -105 using the cosine sum or difference identity?

Answer 1

Find tan (-105)

Ans: `- (1 + sqrt3)/(1 - sqrt3)`

Explanation:

First find sin (-105) and cos (-105).
sin (-105) = - sin (105) = sin (45 + 60) = sin 45.cos 60 + sin 60.cos 45 =
`= sqrt2/4 + (sqrt2/4)(sqrt3) = - (sqrt2/4)(1 + sqrt3)`
cos (-105) = cos (105) = cos (45 + 60) = cos 45.cos 60 - sin 45.sin 60.
`= (sqrt2/4)(1 - sqrt3)`

`tan (-105) = sin/(cos) = - (1 + sqrt3)/(1 - sqrt3)`
Check by calculator:
tan (-105) = 3.73
`-(1 + sqrt3)/(1 - sqrt3) = - (2.73)/(-0.73) = 3.73.`OK