Question

sin 3x cos x - cos 3x sin x=.5

Answer 1

do you know that
sin(A-B)= sin(A)cos(B) - cos(A)sin(B)

if you let A= 3x and B= x, can you finish?

Answer 2

that's were i got stuck i dont know what to do from there

Answer 3

OK. start with
sin(A)cos(B) - cos(A)sin(B) and match it to your problem
sin 3x cos x - cos 3x sin x

what is A and what is B?

Answer 4

3x is A ad x is B

Answer 5

so we replace sin(A)cos(B) - cos(A)sin(B)
with
sin(A-B), but replace A with 3x and B with x

Answer 6

Ok. but what do you do now

Answer 7

what did you get?

Answer 8

i got sin 2x

Answer 9

OK, so you have simplified the problem from
sin 3x cos x - cos 3x sin x=.5

to
sin(2x)= 0.5

Do you know the sin of what angle is 1/2 ?

Answer 10

You could use a calculator, but generally people memorize a few key angles.

Answer 11

pi/6 and 5pi/6

Answer 12

so that means if we start with
sin(2x)= 1/2
then
2x = pi/6 or
2x= 5pi/6

can you solve for x?

Answer 13

pi/3 and 5pi/3

Answer 14

to solve for x you multiply by 1/2 on both sides:
\[\frac{1}{\cancel{2}}\cancel{2}x=\frac{1}{2}\frac{\pi}{6}\]
when you multiply the fractions on the right hand side, multiply top*top and bottom*bottom

Answer 15

you can check your answer by copy and paste this into the google search window
sin(pi/4)*cos(pi/12)- cos( pi/4 )*sin(pi/12) =