posted by carmen .
find general solutions for sin5x=cosx
This is pointless. There are lots of solutions, but they are nasty, and really don't provide any insights.
sin(5x) = sin^5 + 5sin*cos^4 - 10sin^3cos^2
= sin*(1-cos^2)^2 + 5sincos^4 - 10sin(1-cos^2)cos^2
= sin(12cos^4 - 12cos^2 + 1)
so, you end up trying to solve
sin(x) (12cos^4(x) - 12cos^2(x) + 1) = cos(x)
No way to get rid of the sin except to square both sides, and then you wind up with
144cos^10(x) - 432cos^8(x) + 456cos^6(x) - 192cos^4(x) + 26cos^2(x) - 1 = 0
So, you have to solve a 5th-degree polynomial in cos^2(x), then try to eliminate possible spurious solutions.
Good freakin' luck!
Visit wolframalpha.com to see the solution and the graphs.
Just type in "solve sin 5x = cos x"
Not sure what the point is to this exercise.