Attention Required!

     
Find the number of solutions to lớn $sin x+sin2x+sin3x=cos x+cos2x+cos3x$ with $xin<0,2pi)$

I tried lớn solve it by doing this:-

$$2sin(2x)cos x + sin(2x) = 2cos(2x)cos x + cos(2x)$$

$$sin(2x) (2cos x+1) = cos(2x) (2cos x + 1)$$

$$ sin(2x) = cos(2x)$$

$$ 2cos xsin x = 1 - 2sin^2 x$$

$$ 2cos xsin x + 2sin^2 x - 1 = 0$$

$$ 2sin x(cos x + sin x) -1=0$$

$$ 2sin x(2sin x) -1=0$$

$$ 4sin^2 x = 0$$

$$ sin^2 x = (frac12)^2$$

$$ sin^2 x = sin^2 (fracpi6)$$

Now by using $x = nπ -fracpi6$ I got 4 solutions between $0,2pi$. But the answer is 6. Where I did wrong. Please don"t mind for mistakes và answer ASAP. THANKS IN ADVANCE.

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proof-verification trigonometry
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edited May 28, 2019 at 14:44
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Michael Rozenberg
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asked May 28, 2019 at 14:09
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Diya SarkarDiya Sarkar
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6
$egingroup$
You have also $$2cosx+1=0,$$ which gives another two roots.

Because $$sin2x(2cosx+1)=cos2x(2cosx+1)$$ it"s$$sin2x(2cosx+1)-cos2x(2cosx+1)=0$$ or$$(2cosx+1)(sin2x-cos2x)=0$$ & we obtain:$$2cosx+1=0$$ or$$sin2x-cos2x=0.$$


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edited May 28, 2019 at 14:23
answered May 28, 2019 at 14:17
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Michael RozenbergMichael Rozenberg
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