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

sin(6x)cos(8x)-cos(6x)sin(8x) = -0.7.

This is what I have so far:
simplify sin(x).
sin(6x-8x) = -0.7
sin(-2x) = -0.7
-sin(2x) = -0.7

solve for x.
2x = theta
x = 1/2 theta

solve for theta.
sin(theta) = -0.7
sin(theta) = ???

1 Answer
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:
2x = 44^@42, and 2x = 190 - 44.42 = 135^@58

a. 2x = 44^@42 + k360^@
x = 22^@21 + k180^@
If k = 0 --> x = 22^@21
If k = 1 --> x = 22.21 + 180 = 202^@21

b. 2x = 135^@58 + k360^@
x = 67^@79 + k180^@
The 2 smallest positive answers are:
22^@21; 67^@79