Question

How do you use the sum and difference identities to find the exact value of cos 75?

Answer 1

The sum and difference identities are given as follows:

sum:
`cos(alpha + beta) = cos(alpha)cos(beta) - sin(alpha)sin(beta)`

difference:
`cos(alpha - beta) = cos(alpha)cos(beta) + sin(alpha)sin(beta)`

If you have an angle whose trig values you don't know, but that can be made by two angles you do know, you will use one of these identities. `75^o` can be made of the sum of `45^o` and `30^o`, both of which are on the unit circle!

`cos(75^o) = cos(45^o + 30^o)`

Using our sum identity, where `alpha = 45^o` and `beta = 30^o`;

`=cos(45^o)cos(30^o) - sin(45^o)sin(30^o)`

Now just find these values on your unit circle and you get;

`=sqrt(2)/2 xxsqrt(3)/2 - sqrt(2)/2 xx 1/2`

and the rest is arithmetic.

To use the difference identity, I would recommend using the angles;

`cos(75^o) = cos(120^o - 45^o)`

Then, using difference identity;

`cos(120^o)cos(45^o) + sin(120^o)sin(45^o)`

Which gives;

`-1/2 xx sqrt(2)/2 + sqrt(3)/2 xx sqrt(2)/2`

Both expressions simplify to;

`(sqrt(2)(sqrt(3)-1))/4`