A quick Trig Identity Question.

[logan233]

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?

 

[jbunniii]

x(t) = sin(2πt)cos(2π10t) and we choose u = 2πt and v = 2π10t
so that x₁(t) = 0.5[sin(2π11t) + sin(2π9t)]

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.

 

[logan233]

Oh I see now. Thank you.