Question

How do you evaluate cos 75?

Answer 1

`cos(75^@)=((sqrt(3)-1)sqrt(2))/4 ~~0.258819`

Explanation:

General Formula:
`color(white)("XXX")cos(A+B)=cos(A) * cos(B) - sin(A) * sin(B)`

Note: `75^@ = 30^@ +45^@`

`30^@` and `45^@` are two of the standard angles with:
`color(white)("XXX")cos(30^@)=sqrt(3)/2color(white)("XXX")sin(30^@)=1/2`

`color(white)("XXX")cos(45^@)=sqrt(2)/2color(white)("XXX")sin(45^@)=sqrt(2)/2`

Therefore
`cos(75^@) =cos(30^@+45^@)`

`color(white)("XXX")=cos(30^@) * cos(45^@) -sin(30^@) * sin(45^@)`

`color(white)("XXX")=sqrt(3)/2 * sqrt(2)/2 - 1/2 * sqrt(2)/2`

`color(white)("XXX")=((sqrt(3)-1)sqrt(2))/4`

(using a calculator)
`color(white)("XXX")~~0.258819`