How to find the general solution 5 sin(x)+2 cos(x)=3?

May 10, 2018

$\rightarrow x = n \pi + {\left(- 1\right)}^{n} \cdot \left({\sin}^{- 1} \left(\frac{3}{\sqrt{29}}\right)\right) - {\sin}^{- 1} \left(\frac{2}{\sqrt{29}}\right)$ $n \in \mathbb{Z}$

Explanation:

$\rightarrow 5 \sin x + 2 \cos x = 3$

rarr(5sinx+2cosx)/(sqrt(5^2+2^2))=3/(sqrt(5^2+2^2)

$\rightarrow \sin x \cdot \left(\frac{5}{\sqrt{29}}\right) + \cos x \cdot \left(\frac{2}{\sqrt{29}}\right) = \frac{3}{\sqrt{29}}$

Let $\cos \alpha = \frac{5}{\sqrt{29}}$ then $\sin \alpha = \sqrt{1 - {\cos}^{2} \alpha} = \sqrt{1 - {\left(\frac{5}{\sqrt{29}}\right)}^{2}} = \frac{2}{\sqrt{29}}$

Also, $\alpha = {\cos}^{- 1} \left(\frac{5}{\sqrt{29}}\right) = {\sin}^{- 1} \left(\frac{2}{\sqrt{29}}\right)$

Now, given equation transforms to

$\rightarrow \sin x \cdot \cos \alpha + \cos x \cdot \sin \alpha = \frac{3}{\sqrt{29}}$

$\rightarrow \sin \left(x + \alpha\right) = \sin \left({\sin}^{- 1} \left(\frac{3}{\sqrt{29}}\right)\right)$

$\rightarrow x + {\sin}^{- 1} \left(\frac{2}{\sqrt{29}}\right) = n \pi + {\left(- 1\right)}^{n} \cdot \left({\sin}^{- 1} \left(\frac{3}{\sqrt{29}}\right)\right)$

$\rightarrow x = n \pi + {\left(- 1\right)}^{n} \cdot \left({\sin}^{- 1} \left(\frac{3}{\sqrt{29}}\right)\right) - {\sin}^{- 1} \left(\frac{2}{\sqrt{29}}\right)$ $n \in \mathbb{Z}$

May 11, 2018

$x = {12}^{\circ} 12 + k {360}^{\circ}$
$x = {124}^{\circ} 28 + k {360}^{\circ}$

Explanation:

5sin x + 2cos x = 3.
Divide both sides by 5.
$\sin x + \frac{2}{5} \cos x = \frac{3}{5} = 0.6$ (1)
Call $\tan t = \sin \frac{t}{\cos t} = \frac{2}{5}$ --> $t = {21}^{\circ} 80$ --> cos t = 0.93.
The equation (1) becomes:
$\sin x . \cos t + \sin t . \cos x = 0.6 \left(0.93\right)$
$\sin \left(x + t\right) = \sin \left(x + 21.80\right) = 0.56$
Calculator and unit circle give 2 solutions for (x + t) -->
a. x + 21.80 = 33.92
$x = 33.92 - 21.80 = {12}^{\circ} 12$
b. x + 21.80 = 180 - 33.92 = 146.08
$x = 146.08 - 21.80 = {124}^{\circ} 28$
$x = {12}^{\circ} 12 + k {360}^{\circ}$
$x = {124}^{\circ} 28 + k {360}^{\circ}$
$x = {12}^{\circ} 12$ --> 5sin x = 1.05 --> 2cos x = 1.95
$x = {124}^{\circ} 28$ --> 5sin x = 4.13 --> 2cos x = -1.13