Solution of Trig equation $\sin x+2\cos x=1+\sqrt{3}\cos x$

Question

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

Answer 1

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

Answer 2

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 $\phi = \pi/12$

Answer 3

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

Answer 4

Hint:

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

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

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

$x=?$