Solve V = 5 + 12*j in Form v(t) = A*cos(w*t) + B*sin(w*t)

[anengstudent]

Homework Statement

If V = 5 + 12*j then write v(t) in the form v(t) = A*cos(w*t) + B*sin(w*t)

w = 10000 rad/s

Homework Equations

N/A

The Attempt at a Solution

I can easily put it into the form v(t) = C*cos(w*t + phi) and I can probably get the above form using a trig identity (haven't tried yet).

However, apparently it should be easily solvable by looking at it graphically.

I look at the case when t=0, then v(t) = C*cos(phi), the projection on the x-axis = 5.

Since x = cos(t) and y = sin(t) on the unit circle, then I figured v(t) = 5*cos(w*t) + 12*sin(w*t)

That works for t=0, but the actual answer is 5*cos(w*t) - 12*sin(w*t) (I found this by plotting it and 13*cos(w*t + arctan(12/5), they matched)

I seem to be on the right track but I don't understand the actual answer. The only resource I've found on the net puts +cos(t) on the positive x-axis but +sin(t) on the negative y axis... ?

I just don't understand and no one can help because it's an assignment question. I've got the mark now though because I know the answer, but I don't understand how to get it exactly.

 

[gneill]

Consider the trig identity for the cosine of the sum of two angles:
$$cos(\alpha + \beta) = cos(\alpha) cos(\beta) - sin(\alpha) sin(\beta)$$
and work from your A*cos(ωt + φ) form.