Find all the solutions of the given equation in the interval [0,2pi) sin5x + sin3x = 0
By inspection (thinking about the picture), seven solutions are k pi/4, k=1,2,3,4,5,6,7. If you need an algebraic method, consider using the formula for sin of a sum. I think you might be able to think about sin 5x = sin (2x + 3x), although I'm not sure.
I think you use the sum to product formulas but I don't know where to go from there...
I think its 2sin(5x+3x)/2 cos(5x-3x)/2
however I do not know how to find the answers after simplifying the 2sin4x cosx = 0
please help:(
Note: I overlooked k=0 as a solution. That gives eight in all.
soo what do you do know?
*now?
:/
Here is a method that works pretty well. Let 5x = 4x + x, and 3x = 4x - x. Then\[\sin(4x+x)+\sin(4x-x)=\sin 4x \cos x+\cos 4x \sin x + \sin 4x \cos x - \cos 4x \sin x =0\]\[\implies \sin 4x \cos x =0 \implies x= \frac{k \pi}{4}; k=0,1,2,3,4,5,6,7.\]
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