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
Nghi N.
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#

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