Solve for x : $\sqrt{2}\sin(x)+\sqrt{6}\cos(x) = \sqrt{3} +1$

Question

Solve $\sqrt{2}\sin(x)+\sqrt{6}\cos(x) = \sqrt{3} +1$ for $x$

I started by multiplying both sides of the equation by $\frac{1}{2\sqrt{2}}$ to obtain

$$\displaystyle\frac{\sin(x)}{2}+\frac{\sqrt{3}\cos(x)}{2} = \frac{\sqrt{3} +1}{2\sqrt{2}}$$ $$\iff \sin(60+x) = \frac{\sqrt{3} +1}{2\sqrt{2}}$$

I am stuck here. Any hints on solving the R.H.S will be appreciated.

Answer 1

$\sin 75^\text o= \dfrac{\sqrt{3}+1}{2\sqrt2}$

Can you find now ?

PROOF :

$\sin (x+y) = \sin x\cos y +\cos x \sin y $
$\sin(75^\text o) = \sin(30^\text o+45^\text o)\\\qquad\quad= \sin 30^\text o\cos 45^\text o +\cos 30^\text o \sin 45^\text o\\\qquad\quad = \dfrac{1}{2\sqrt2}+\dfrac{\sqrt3}{2\sqrt2} = \dfrac{\sqrt3+1}{2\sqrt2} $

Answer 2

Try to find what is $$\sin\frac{5\pi}{12}$$ If you get it $$\frac{\sqrt3+1}{2\sqrt2}$$ then you did correctly and you know what to do next using the general definition of $$\sin x=\sin y $$ yields what you know it !

you can find $$\sin\frac{5\pi}{12}$$ using the formula for $$\sin \frac{x}{2}$$ by taking $x=\frac{5\pi}{6}$ in degrees which is equivalent to 75 degrees

Answer 3

square both sides, we get $$2 \sin^2(x) + 6 \cos^2(x) + 2\sqrt{12}\cos(x)\sin(x) = (1 + \sqrt{3})^2$$ that is $$2 + 4 \cos^2(x) + 2\sqrt{12}\cos(x)\sin(x) = (1 + \sqrt{3})^2$$ that is $$2 + 4 \cos^2(x) + \sqrt{12}\sin(2x) = (1 + \sqrt{3})^2$$ that is $$2 + 4 \frac{1}{1+\tan^2(x)} + \sqrt{12}\frac{2\tan x}{1+\tan^2(x)} = (1 + \sqrt{3})^2$$ Multiply everything by $1 + \tan^2(x)$ $$2(1+\tan^2(x)) + 4 +2 \sqrt{12}\tan x = (1 + \sqrt{3})^2(1+\tan^2(x)) $$ This is quadratic in $y = \tan(x)$

$$ay^2 + by + c = 0$$ where \begin{align} a &= 2- (1 + \sqrt{3})^2\\ b &=2\sqrt{12}\\ c &= 2 + 4 - (1 + \sqrt{3})^2 \end{align} Solving you get something like \begin{align} y_1 &= 1.0000 \\ y_2 &= 0.2679 \end{align}

Now taking the inverse tangent of that you get \begin{align} x_1 &= \tan^{-1} y_1 =\tan^{-1} 1.0000 = 45^\circ\\ x_2 &= \tan^{-1} y_2 =\tan^{-1} 0.2679= 15^\circ \end{align}

Answer 4

$$\frac{\sqrt{3} +1}{2\sqrt{2}}=\frac{\sqrt{3}\sqrt{2} +\sqrt{2}}{4}=\frac{\sqrt{2}}{2}\cdot\frac{\sqrt{3}}{2}+\frac{\sqrt{2}}{2}\cdot\frac{1}{2}=\sin{45^\circ}\cos{30^\circ}+\cos{45^\circ}\sin{30^\circ}=\sin{(45^\circ+30^\circ)}=\sin{75^\circ}$$