How could this be roughly accomplished without making an extensive table of values?

For example if I use the laplace method of analyzing an RLC circuit with a pulse excitation, I know that I will arrive at a decaying sinusoidal current represented by the following equation:

I would like to be able to quickly graph the equation incase I didn't have access to software tools at that moment.

 

For your given example

1. Draw the exponential envelope bounded by the curves

\(f1(t)=e^{-\alpha t}\)

and

\(f2(t)=-e^{-\alpha t}\)

2. Draw a sinusoid constrained in amplitude by the bounded region with damped period

\(T=\frac{2\pi}{\omega}\)

Since it is a sine function in this particular case the value at t=0 will be zero. If it were a simple cosine function the initial value would be 1. A phase displaced sine or cosine function would have an initial value determined by the value of the sine or cosine term at t=0 sec.

 

Here's an example sketch of a phase shifted exponentially decaying cosine function.

 

Oh I already know how to roughly sketch the actual shapes, but it's mostly about determining the amplitude of the first peak(for a sine wave) so I can draw it more accurately. I should have included that in my first post sorry.

 

Peaks and zero-crossings still occur in the multiples of pi/2, so these are points in time you look for.

For example, your first peak will be at the intersection of the envelope with the line x=pi/2 for a sinusoidal curve.
That is, of course, for zero time-shift. Shift your times accordingly for θ<>0.

 

Peaks and zero-crossings still occur in the multiples of pi/2, so these are points in time you look for.

For example, your first peak will be at the intersection of the envelope with the line x=pi/2 for a sinusoidal curve.
That is, of course, for zero time-shift. Shift your times accordingly for θ<>0.

Damn I feel silly for not thinking about this first haha. Thanks