SOLUTION: Solve: sin 3x*cos x + cos 3x*sin x=-1/2

SOLUTION: Solve: sin 3xcos x + cos 3xsin x=-1/2

Question 639149: Solve:
sin 3x*cos x + cos 3x*sin x=-1/2
Answer by jsmallt9(3758) (Show Source): You can put this solution on YOUR website!
The easiest way to solove this is by recognizing that the left side of the equation fits the pattern for sin(A+B):
sin(A+B) = sin(A)cos(B) + cos(A)sin(B)
with the "A" being "3x" and the "B" being "x". So by sin((A+B) the left side is equal to:
sin(3x+x) = -1/2
or simply
sin(4x) = -1/2

This equation we can solve. We should recognize that 1/2 is a special angle value for sin and that the reference angle will be . Since the sin is negative, we know that the angle terminates in the 3rd or 4th quadrants. Putting the reference angle and the quadrants together we get:
(for the 3rd quadrant)
which simplifies to
(for the 3rd quadrant)
and
(for the 4th quadrant)
(Instead of we could also have used or )

Now we just divide both sides by 4 (or multiply by 1/4):

and

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