# How do I solve for the two smallest positive solutions for: sin(2x)cos(6x)-cos(2x)sin(6x) = -0.35 ?

## I understand that I am using a sine sum and difference identity (sin(A+B)=sin(A)cos(B)+cos(A)sin(B)) but I have no clue what to do with the negative decimal number at the end of the equation. This is what I have so far: $\sin \left(2 x\right) \cos \left(6 x\right) - \cos \left(2 x\right) \sin \left(6 x\right) = - 0.35$ Then I take sine and simplify. $\sin \left(5 x - 10 x\right) = - 0.35$ $\sin \left(- 5 x\right) = - 0.35$ $- \sin \left(5 x\right) = - 0.35$ Then I solve for $x$. $5 x = \theta$ $x = \frac{1}{5} \theta$ Then I have to solve for $\theta$. $\sin \left(\theta\right) = - 0.35$ sin(theta) = ??? Once I figure out how to fine $\theta$, then I'll be able to find the solutions.

Jun 27, 2018

5^@12; 39^@88

#### Explanation:

This equation comes from the trig identity:
sin (a - b) = sin a.cos b - sin b.cos a.
In this case:
sin (2x - 6x) = sin 2x.cos 6x - sin 6x.cos 2x
sin (2x - 6x) = sin (-4x) = - sin 4x = -0.35
sin 4x = 0.35
Calculator and unit circle give 2 solutions for 4x:
a. $4 x = {20}^{\circ} 49 + k {360}^{\circ}$ -->
$x = {5}^{\circ} 12 + k {90}^{\circ}$, and
b. $4 x = 180 - \left(20.49\right) = {159}^{\circ} 51 + k {360}^{\circ}$ -->
$x = {39}^{\circ} 88 + k {90}^{\circ}$
The 2 smallest positive answers are (k = 0):
$x = {5}^{\circ} 12$, and $x = {39}^{\circ} 88$

Jun 27, 2018

$x = {5.12}^{\circ}$ or $x = {39.88}^{\circ}$

and in radians $x = 0.0894$ or $x = 0.696$

#### Explanation:

As $\sin \left(2 x\right) \cos \left(6 x\right) - \cos \left(2 x\right) \sin \left(6 x\right) = - 0.35$, we have

$\cos \left(2 x\right) \sin \left(6 x\right) - \sin \left(2 x\right) \cos \left(6 x\right) = 0.35$

or $\sin \left(6 x\right) \cos \left(2 x\right) - \cos \left(6 x\right) \sin \left(2 x\right) = 0.35$ (here we have used commutative property)

As $\sin \left(A - B\right) = \sin A \cos B - \cos A \sin B$, we can write the above as

$\sin \left(6 x - 2 x\right) = 0.35 = \sin {20.49}^{\circ}$

(We have used scientific calculator to find your $\theta$ here.)

and hence either $4 x = {20.49}^{\circ}$ i.e. $x = {5.12}^{\circ}$

or $4 x = {180}^{\circ} - {20.49}^{\circ} = {159.51}^{\circ}$ i.e. $x = {39.88}^{\circ}$

If you need to find in radians $\sin \left(0.35757\right) = 0.35$

and then $4 x = 0.35757$ or $x = 0.0894$

we can also have $4 x = \pi - 0.35757 = 2.78402$ and $x = 0.696$

If scientific calculator is not available, one can use tables too.