# Solve 4sinx-cosx+3=0 ?

May 4, 2018

$x = {240}^{\circ} 93 + k {360}^{\circ}$
$x = {327}^{\circ} 12 + k {360}^{\circ}$

#### Explanation:

4sin x - cos x = -3
Divide both sides by 4 -->
$\sin x - \left(\frac{1}{4}\right) \cos x = - \frac{3}{4}$ (1)
Call $\tan t = \frac{\sin}{\cos t} = \frac{1}{4}$ --> $t = {14}^{\circ} 04$ -->
$\cos t = 0.97$.
The equation (1) becomes:
$\sin x . \cos t - \sin t . \cos x = - \left(\frac{3}{4}\right) \cos t = - \left(\frac{3}{4}\right) \left(0.97\right) = - 0.73$
$\sin \left(x - t\right) = \sin \left(x - 14.04\right) = - 0.73$
Calculator and unit circle give 2 solutions for (x - 14.04) -->
a. $x - 14.04 = - 46.89$
x = - 46.89 + 14.04 = - 32.85, or
$x = {327}^{\circ} 15$ (co-terminal).
b. $x - 14.04 = 180 - \left(- 46.89\right) = {226}^{\circ} 89$
$x = 226.89 + 14.04 = {240}^{\circ} 93$
For general answers, add $k {360}^{\circ}$
Check by calculator.
x = 240.93 --> 4sin x = - 3.50 --> cos x = - 0.50
4sin x - cos x + 3 = - 3.50 + 0.50 + 3 = 0. Proved.