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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#

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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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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) = ???#

#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) = ???#