# How do you solve 2\sin x = 3\cos ( x + 45^ { \circ } )?

Jul 27, 2018

color(brown)(x = arctan (3 / (3 + 2 sqrt2) ) ~~ 27.2357^@

#### Explanation:

$2 \sin x = 3 \cos \left(x + {45}^{\circ}\right)$

color(crimson)("Using identity " cos (A + B) = cos A cos B - sin A sin B,

$\frac{2}{3} \sin x = \cos x \cdot \cos 45 - \sin x \cdot \sin 45$

$\frac{2}{3} \sin x = \cos \frac{x}{\sqrt{2}} - \sin \frac{x}{\sqrt{2}}$

$\left(\frac{2 \sqrt{2}}{3}\right) \sin x = \cos x - \sin x$

$\cos \frac{x}{\sin} x - {\left(\cancel{\sin} \frac{x}{\cancel{\sin}} x\right)}^{\textcolor{red}{1}} = \frac{2 \sqrt{2}}{3}$

$\cot x = 1 + \frac{2 \sqrt{2}}{3}$

$\tan x = \frac{1}{1 + \left(\frac{2 \sqrt{2}}{3}\right)}$

$\tan x = \frac{3}{3 + 2 \sqrt{2}}$

$x = \arctan \left(\frac{3}{3 + 2 \sqrt{2}}\right)$

color(brown)(x ~~ 27.2357^@

Jul 27, 2018

x = ( kpi + 0.47535)) rad = (180 k + 27.2357)^o,
$k = 0 , \pm 1 , \pm 2 , \pm 3 , \ldots$

#### Explanation:

$= 2 \sin x - 3 \cos \left(x + \frac{\pi}{4}\right)$

$= 2 \sin x - 3 \left(\cos x \cos \left(\frac{\pi}{4}\right) - \sin x \sin \left(\frac{\pi}{4}\right)\right)$

$= \left(\left(2 + \frac{3}{\sqrt{2}}\right) \sin x - \frac{3}{\sqrt{2}} \cos x\right)$

$= a \left(\sin x \sin \alpha - \cos x \cos \alpha\right)$

$= a \sin \left(x - \alpha\right)$,

where

$a = \sqrt{{\left(2 + \frac{3}{\sqrt{2}}\right)}^{2} + {\left(\frac{3}{\sqrt{2}}\right)}^{2}}$

$= \sqrt{13 + 6 \sqrt{2}} = 4.63522$, nearly.

$\cos \alpha = \frac{2 - \frac{1}{\sqrt{2}}}{a} \mathmr{and} \sin \alpha = \frac{\frac{3}{\sqrt{2}}}{a}$. Now,

sin ( x - alpha ) = 0, and so,

$x - \alpha = k \pi , k = 0 , \pm 1 , \pm 2 , \pm 3 , \ldots$, giving

$x = k \pi + \alpha = {\left(180 k + \arcsin \left(0.45765\right)\right)}^{o}$

= (180 k + 27.2357))^o = ( kpi + 0.47535)) rad.

GRAPH CHECK, 6-sd x = 0.474353 rad :
graph{y-2 sin x + 3 cos ( x + pi/4 ) = 0[0.47535.475356 -0.00001 0.00001]}{