maths higher - cosx cos30-sinx sin30 = 1/4 how do you do it?

Question

i know its cos(x+30)=1/4 but what is the working to get to what the 2 possibilties of x are?

Answer 1

First off, remember the identity:

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

All we have to do is observe that a = x and b = 30 in this case. Therefore,

cos(x)cos(30) - sin(x)sin(30) = 1/4
cos(x+30) = 1/4

Therefore, we ask ourselves where cos is equal to 1/4, which we don't know. In any case

x+30 = cos(inverse)(1/4)
x = cos(inverse)(1/4) - 30

Answer 2

cosx cos30-sinx sin30 = 1/4
cos(x+30)=1/4
x+30=cos-1(1/4)
x+30=75.5 (using the calc), 360-75.5
x+30=75.5, 284.5
x=75.5-30, 284.5-30
x = 45.5, 254.5