Posted by carmen on Friday, August 24, 2012 at 8:49am.
find general solutions for sin5x=cosx

maths  Steve, Friday, August 24, 2012 at 10:39am
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*(1cos^2)^2 + 5sincos^4  10sin(1cos^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 5thdegree 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.
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