# How do I solve a sum and difference trig equation for the smallest positive solutions?

## $\sin \left(6 x\right) \cos \left(8 x\right) - \cos \left(6 x\right) \sin \left(8 x\right) = - 0.7$. This is what I have so far: simplify $\sin \left(x\right)$. $\sin \left(6 x - 8 x\right) = - 0.7$ $\sin \left(- 2 x\right) = - 0.7$ $- \sin \left(2 x\right) = - 0.7$ solve for $x$. $2 x = \theta$ $x = \frac{1}{2} \theta$ solve for $\theta$. $\sin \left(\theta\right) = - 0.7$ sin(theta) = ???

Jun 27, 2018

22^@21; 67^@79

#### Explanation:

sin 6x.cos 8x - cos6x.sin 8x = -0.7
sin (6x - 8x) = sin (-2x) = - sin 2x = - 0.7
sin 2x = 0.7
Calculator and unit circle give 2 solutions for 2x:
$2 x = {44}^{\circ} 42$, and $2 x = 190 - 44.42 = {135}^{\circ} 58$

a. $2 x = {44}^{\circ} 42 + k {360}^{\circ}$
$x = {22}^{\circ} 21 + k {180}^{\circ}$
If k = 0 --> $x = {22}^{\circ} 21$
If k = 1 --> $x = 22.21 + 180 = {202}^{\circ} 21$

b. $2 x = {135}^{\circ} 58 + k {360}^{\circ}$
$x = {67}^{\circ} 79 + k {180}^{\circ}$
The 2 smallest positive answers are:
22^@21; 67^@79