A quick Trig Identity Question.

  1. Hopefully this will make sense...

    We have the trig. identities shown below:
    sin(u)cos(v) = 0.5[sin(u+v) + sin(u-v)]
    cos(u)sin(v) = 0.5[sin(u+v) - sin(u-v)]

    How are these different? I realize u and v switched between the sine and cosine functions, but what is the difference between u and v? I recognize that there is a difference between taking sine of a number u and sine of a different number v, and same with taking the cosine of a those numbers, I just don't see how we differentiate between u and v. Like say we have...

    x(t) = sin(2πt)cos(2π10t) and we choose u = 2πt and v = 2π10t
    so that x1(t) = 0.5[sin(2π11t) + sin(2π9t)]
    but what if we choose u = 2π10t and v = 2πt
    then x2(t) = 0.5[sin(2π11t) - sin(2π9t)], which is different than the original x(t) even though we simply chose u and v to be different?
  2. jcsd
  3. jbunniii

    jbunniii 3,378
    Science Advisor
    Homework Helper
    Gold Member

    Your mistake is here. The formula is ##0.5[\sin(u+v) + \sin(u-v)]##, so the argument to the second ##\sin## should be ##-2\pi 9 t##, not ##2\pi 9 t##. Then use the fact that ##\sin(-x) = -\sin(x)## to see that the two answers are in fact the same.
    1 person likes this.
  4. Oh I see now. Thank you.
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