Circuit: Total Energy into a component over interval

[Saladsamurai]

Homework Statement

The voltage at terminal a relative to terminal b of an electric component is given by $v(t) = 20\cos(120\pi t)$ Volts and the current into a is $i(t) = -4\sin(120\pi t)$ amps. Find 1) the total energy that flows into component from time t₁ to t₂ and 2) in particular find the energy absorbed when t₂ = t₁ + 1/15.

Homework Equations

Power = i(t)*v(t)

The Attempt at a Solution

So I set Power = dE/dt = i(t)*v(t) and integrated to yield the final expression

$$\DeltaE_{t_1\rightarrow t_2} = \frac{1}{6\pi}\left[\cos(240\pi t)\right]_{t_1}^{t_2}\qquad(1)$$

I believe that this expression takes care of part 1). However, for part 2), I am unclear on how to evaluate the expression from t1 to t1 + 1/15. This becomes:

$$\DeltaE_{t_1\rightarrow t_2} = \frac{1}{6\pi}\left[\cos(240\pi t_1+1/15) - \cos(240\pi t_1) \right]$$

Is there some sort of trig trick I a can use to evaluate this? Or somehow use the fact the a cosine function is periodic?

Just need a hint here

 

[collinsmark]

$$\DeltaE_{t_1\rightarrow t_2} = \frac{1}{6\pi}\left[\cos({\color{red}240\pi t_1+1/15}) - \cos(240\pi t_1) \right]$$

Maybe it's just a typo within LaTeX, but you didn't multiply through the 240π quite correctly.

Is there some sort of trig trick I a can use to evaluate this? Or somehow use the fact the a cosine function is periodic?

Just need a hint here

Here's a hint: Note that

$$\cos (\omega t + \theta) = \frac{e^{j(\omega t + \theta)} + e^{-j( \omega t + \theta)}}{2}$$

Try to express your equation exponentially. You'll be able to pull out a complex exponential (that's not a function of t₁), and functions as a complex constant. Recalling that [itex] e^{j \theta} = \cos \theta + j \sin \theta [/tex], the whole thing can be reduced in this particular problem.