# how to calculate sin 75 and cos 75?

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### 2 Answers

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**``` `

We'll replace 75 by the sum 30+45 = 75 and we'll use the following trigonometric identities:

sin(a+b)=sin a*cos b + sin b*cos a

sin(30 + 45) = sin 30*cos 45 + sin 45*cos 30

sin 30 = 1/2; cos 30 = sqrt3/2

sin 45 = cos 45 = sqrt2/2

sin(30 + 45) = sqrt2/4 + sqrt6/4

sin(30 + 45) = (sqrt 2 + sqrt6)/4

cos (a+b) = cos a*cos b - sin a*sin b

cos (30 + 45) = cos 30*cosĀ 45 - sin 30*sin 45

cos (30 + 45) = (sqrt6 - sqrt2)/4

**Therefore, the requested values for sin 75 and cos 75 are: sin 75 = (sqrt 2 + sqrt6)/4 and cos 75 = (sqrt6 - sqrt2)/4.**