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*(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.

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