Question

Given sin 30degree=1/2 and sin 45degree=1/squareroot2. Find the exact value of sin 75degree and cos 75degree?

Answer 1

sin 75 = (sqrt2/4)(1 + sqrt3)
cos 75 = (sqrt2/2)(sqrt3 - 1)

Explanation:

Use trig identity:
sin (a + b) = sin a.cos b + sin b.cos a
In this case:
sin (75) = sin (30 + 45) = sin 30.cos 45 + sin 45.cos 30 (1)
We have from trig table:
`sin 30 = 1/2`, and `cos 30 = sqrt3/2`
`sin 45 = sqrt2/2`, and `cos 45 = sqrt2/2`
Replace these values into equation (1):
`sin 75 = (1/2)(sqrt2/2) + (sqrt2/2)(sqrt3/2) = (sqrt2/4)(1 + sqrt3)`
The same method:
cos (a + b) = cos a.cos b - sin a.sin b
cos (75) = cos (30 + 45) = cos 30.cos 45 - sin 45.sin 30
`cos 75 = (sqrt3/2)(sqrt2/2) - (sqrt2/2)(1/2) = (sqrt2/4)(sqrt3 - 1)`