Harmonic oscillation question

[Krupski69]

(6.10)

 

[Krupski69]

The angular frequency omega=4.

I am having difficulties finding the complex numbers C and D. Any help given is greatly appreciated.

 

[MarkFL]

I would begin by writing:

[MATH]3\cos(4t)=\frac{3}{2}\left(e^{i4t}+e^{-i4t}\right)[/MATH]
[MATH]5\sin(4t)=\frac{5}{2i}\left(e^{i4t}-e^{-i4t}\right)[/MATH]
What do you get when you add the two equations above?

 

[Krupski69]

I would begin by writing:

[MATH]3\cos(4t)=\frac{3}{2}\left(e^{i4t}+e^{-i4t}\right)[/MATH]
[MATH]5\sin(4t)=\frac{5}{2i}\left(e^{i4t}-e^{-i4t}\right)[/MATH]
What do you get when you add the two equations above?

My values of C and D:

C = (3/2)-(5/2)i

D = (3/2)+(5/2)i

 

[MarkFL]

My values of C and D:

C = (3/2)-(5/2)i

D = (3/2)+(5/2)i

Adding, we get:

[MATH]3\cos(4t)+5\sin(4t)=\frac{3}{2}\left(e^{i4t}+e^{-i4t}\right)+\frac{5}{2i}\left(e^{i4t}-e^{-i4t}\right)[/MATH]
[MATH]3\cos(4t)+5\sin(4t)=\left(\frac{3}{2}+\frac{5}{2i}\right)e^{i4t}+\left(\frac{3}{2}-\frac{5}{2i}\right)e^{-i4t}[/MATH]
[MATH]3\cos(4t)+5\sin(4t)=\frac{3i+5}{2i}e^{i4t}+\frac{3i-5}{2i}e^{-i4t}[/MATH]
[MATH]3\cos(4t)+5\sin(4t)=\frac{3i^2+5i}{2i^2}e^{i4t}+\frac{3i^2-5i}{2i^2}e^{-i4t}[/MATH]
[MATH]3\cos(4t)+5\sin(4t)=\frac{-3+5i}{-2}e^{i4t}+\frac{-3-5i}{-2}e^{-i4t}[/MATH]
[MATH]3\cos(4t)+5\sin(4t)=\frac{3-5i}{2}e^{i4t}+\frac{3+5i}{2}e^{-i4t}[/MATH]
Yes, I agree with your result.