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:

tan (105) = sin (105)/cos (105). Find sin 105 and cos 105.
Apply the trig identity: sin (a + b) = sin a.cos b + sin b.cos a
sin (105) = sin (60 + 45) = sin 60.cos 45 + sin 45.cos 60 =
`= (sqrt3/2)(sqrt2/2) + (sqrt2/2)(1/2) = (sqrt2/4)(1 + sqrt3)`
cos (105) = cos (60 + 45) = cos 60.cos 45 - sin 60.sin 45 =
`= (1/2)(sqrt2/2) - (sqrt3/2)(sqrt2/2) = (sqrt2/4)(1 - sqrt3)`

`tan 105 = sin/(cos) = (1 + sqrt3)/(1 - sqrt3)`

Check by calculator.
sin 105 = - 3.74
`(1 + sqrt3)/(1 - sqrt3) = 2.73/-0.73 = - 3.74`. OK