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

1 Answer
Dec 6, 2014

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#