Find the real positive constants C and F for all real t such that:
2.5cos(3t) - 1.5sin(3t+pi/3) = Ccos(3t+F)
Your post title was Any help getting me started solving these constants would be appreciated
You've been given that start. Now it's your turn to show some working. It's assumed you're familar with the compound angle formula for expanding sin (x + y) .... (if not, then I don't see why you'd be given a question like this to solve in the first place).
Yes, you'd think after four years of engineering school this sort of thing would be old hat for me but I've never really cared much for doing math for the sake of doing math.
Thanks for the tip on expanding with the sum/difference formulas.
Next question: I can see in the previous that solving for coefficients "a" and "b" would net me coefficient "C", but I'm not exactly sure where that comes from. I mean, I know that sin^2(3t) + cos^2(3t) = 1, but how does asin(3t) + bcos(3t) = Ccos(3t+F) ?? and wouldn't the fact that "F" is in the parenthesis mess it up?
BTW, you misspelled familar(sic).
Expand the left hand side of the given equation using the compound angle formula. Expand the right hand side using the compound formula. Equate the coefficients of cos(3t) and sin(3t) on each side. Do all this and you will get the following two equations:
$\displaystyle \frac{10 - 3 \sqrt{3}}{4} = C \cos F$ .... (1)
$\displaystyle \frac{3}{4} = C \sin F$ .... (2)
After 4 years of engineering school (which I'll assume included the odd maths course along the way) you should be familir with how to solve these two equations simultaneously (which you should carefully check).
Edit: I only just saw your above post but don't have time to review it right now.
Since your purpose is to find the constants $\displaystyle C $ and $\displaystyle F $ for all real $\displaystyle t $. Then you can pick any $\displaystyle t $ and put it into your equation.
For example, let $\displaystyle 3t=0 $, you get:
$\displaystyle
\frac{5}{2} - \frac{3}{2}\times \frac{\sqrt{3}}{2} = C\cos F
$
- Let $\displaystyle 3t=-\frac{\pi}{2} $, you get:
$\displaystyle
\frac{3}{2} \times \frac{1}{2} = C\sin F
$