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The sum of all solution of the equation

$\sin x+2\cos x=1+\sqrt{3}\cos x$ in $[0,2\pi]$

My Try:

$$(\sin x+\cos x)+(\sin x-\sqrt{3}\cos x)=1$$

$$\sqrt{2}\sin \bigg(x+\frac{\pi}{4}\bigg)+2\sin \bigg(x-\frac{\pi}{3}\bigg)=1$$

Could some Help me to solve it. Thanks in Advance

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  • $\begingroup$ Your working shows an equation that is not the same as the original equation. $\endgroup$ – Peter Foreman Mar 31 at 14:52
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Remeber that we can write $$f(x)= a\sin x +b\cos x $$ like this : $$f(x)=A \sin (x+\phi)$$

where $A= \sqrt{a^2+b^2}$ and $\tan \phi = b/a$.

So $$\sin x+(2-\sqrt{3})\cos x =1$$

$A = \sqrt{8-4\sqrt{3}}$ and $\phi = \pi/12$

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Hint: Substitute $$\sin(x)=\frac{2t}{1+t^2}$$ $$\cos(x)=\frac{1-t^2}{1+t^2}$$ the so-called Weierstrass substitution

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Set $X=\cos x$ and $Y=\sin x$; then $Y=1+(\sqrt{3}-2)X$. Substitute into $X^2+Y^2=1$.

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Hint:

$$2-\sqrt3=\csc30^\circ-\cot30^\circ=\tan15^\circ=\cot75^\circ$$

If $$\sin x+\cot A\cos x=1$$

$$\cos(x-A)=\sin A=\cot(90^\circ-A)$$

$x=?$

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