how to calculate sin 75 and cos 75?

We have to calculate sin 75 and cos
75.

The values of sin 45, sin 30, cos 45 and cos 30 are
commonly known. We use these to determine the value of cos 75 and sin
75.

Use the relation cos (x + y) = (cos x)*(cos x) - (sin
x)(sin y)

cos 75 = cos (30 +
45)

=> (cos 30)(cos 45) - (sin 30)(sin
45)

cos 30 = `sqrt(3)` /2, sin 30 = 1/2, sin 45 = cos 45 =
1/`sqrt(2)`

=>`sqrt(3)` /2`sqrt(2)` -
1/2*`sqrt(2)`

=>[`sqrt(3)` -
1]/2`sqrt(2)`

Use the relation sin(x + y) = sin x * cos y +
sin y *cos x

sin (45 + 30) = (1/`sqrt 2` )(`sqrt 3` /2) +
(1/` `2 )(1/`sqrt 2` )

=> (1 + `sqrt 3` )/`sqrt
8`

The required value are cos 30 = (sqrt 30 -
1)/2*sqrt 2 and sin 75 = (1 + sqrt 3)/2*sqrt 2
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