SOLUTION: Please help me with this problem, express sin(3x)cos(5x) as a sum or difference of trig functions. Thank You

Algebra ->  Trigonometry-basics -> SOLUTION: Please help me with this problem, express sin(3x)cos(5x) as a sum or difference of trig functions. Thank You      Log On


   

Question 515587: Please help me with this problem,
express sin(3x)cos(5x) as a sum or difference of trig functions.
Thank You
Answer by drcole(72) About Me  (Show Source):
You can put this solution on YOUR website!
Let's recall the sum and difference formulae for the sine function:
sin%28alpha+%2B+beta%29+=+sin%28alpha%29+cos%28beta%29+%2B+cos%28alpha%29+sin%28beta%29+
sin%28alpha+-+beta%29+=+sin%28alpha%29+cos%28beta%29+-+cos%28alpha%29+sin%28beta%29+
If we sum these two equations (in other words, add together the left hand sides and the right hand sides), the term with cos%28alpha%29+sin%28beta%29 will cancel out, leaving:
sin%28alpha+%2B+beta%29+%2B+sin%28alpha+-+beta%29+=+2+sin%28alpha%29+cos%28beta%29+
(dividing both sides by 2)
So . In your problem, alpha+=+3x and beta+=+5x, so we get:
sin%283x%29+cos%285x%29+=+%281%2F2%29+%2A+%28sin%283x+%2B+5x%29+%2B+sin%283x+-+5x%29%29+
+sin%283x%29+cos%285x%29+=+%281%2F2%29+%2A+%28sin%288x%29+%2B+sin%28-2x%29%29+ (simplifying)
+sin%283x%29+cos%285x%29+=+%281%2F2%29+%2A+%28sin%288x%29+-+sin%282x%29%29+ (recalling that for any theta, sin%28-theta%29+=+-sin%28theta%29)
+sin%283x%29+cos%285x%29+=+%281%2F2%29+%2A+sin%288x%29+-+%281%2F2%29+%2A+sin%282x%29+ (distributing the 1%2F2)