Question

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) = ???`

Answer 1

`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`