If 3*sin(x) + 5*cos(x) = 5, then find all possible values of 5*sin(x) - 3*cos(x).
3sin(x) + 5cos(x) = 5
Using the identity: sin2(x) + cos2(x) = 1:
3sin(x) + 5cos(x) = 5
3sin(x) + 5·sqrt( 1 - sin2(x) ) = 5
5·sqrt( 1 - sin2(x) ) = 5 - 3sin(x)
[ 5·sqrt( 1 - sin2(x) ) ]2 = [ 5 - 3sin(x) ]2
25( 1 - sin2(x) ) ) = 25 - 30sin(x) + 9sin2(x)
25 - 25sin2(x) = 25 - 30sin(x) + 9sin2(x)
- 25sin2(x) = - 30sin(x) + 9sin2(x)
0 = -30sin(x) + 34sin2(x)
34sin2(x) - 30sin(x) = 0
sin(x)[34sin(x) - 30 ] = 0
Either six(x) = 0 or 34sin(x) - 30 = 0
x = 0 34sin(x) = 30
sin(x) = 30/34
x = sin-1(x)
For all the solutions, you will need to add 2·n·pi to each of the above solutions.