# Trig Product and Sum Help

#### Barry1234

Hi

I am struggling with the following question.

The instantaneous power, p, in an electric circuit is given by p = iv,where v is the voltage and i is the current.

Calculate the maximum value of power in the circuit if

$$\displaystyle v = 0.02\sin (100\pi t)$$ volts

$$\displaystyle i = 0.6\sin (100\pi t + \frac{\pi}{4} )$$ amps

Calculate the first time that the power reaches a maximum value

I have started with the two multiplied together which makes

$$\displaystyle p = V(max) * I(max) \sin (100\pi t) \sin (100\pi t + \frac{\pi}{4} )$$

Using the trigonometric formula 2.sin A.sin B = Cos(A-B) - Cos(A+B)

$$\displaystyle p = \frac{V(max) I(max)}{2} \cos 100\pi t - \frac{V(max) I(max)}{2} \cos (2 * 100\pi t + \frac{\pi}{4})$$

therefore $$\displaystyle p = \frac{0.02 * 0.6 }{2} \cos 100\pi t - \frac{0.02 * 0.6V}{2} \cos (2 * 100\pi t + \frac{\pi}{4})$$

I dont know what the next move is, or even if i am i going along the right lines??

#### romsek

MHF Helper

$A=100\pi t$

$B=100 \pi t + \dfrac \pi 4$

$A-B =-\dfrac \pi 4$

$\cos(-\pi/4)=\cos(\pi/4)$

so one of your trig terms ends up being a constant.

You can then easily find the maximum of the other trig term.

#### Barry1234

i have re-calculated this and come up with

$$\displaystyle p = \frac{0.02 * 0.6}{2} [ \cos ( 2 * 100 \pi t + \frac{\pi}{4}) + \cos ( - \frac{\pi}{4})$$

hence

$$\displaystyle p = \frac{0.02 * 0.6}{2} [ \cos ( 2 * 100 \pi t + \frac{\pi}{4}) + \cos ( \frac{\pi}{4})$$

I am a little unsure how to progress from here??

#### Barry1234

The maxmum of the

$$\displaystyle [ \cos ( 2 * 100 \pi t + \frac{\pi}{4}) = 1$$

So this makes

$$\displaystyle p = \frac{0.02 * 0.6}{2} [ \cos ( 1 + \cos \frac{\pi}{4})$$

Is this correct?

#### skeeter

MHF Helper
$P = 0.006\bigg[\cos\left(-\dfrac{\pi}{4}\right)-\cos\left(200\pi t + \dfrac{\pi}{4}\right)\bigg]$

$P = 0.006\bigg[\dfrac{\sqrt{2}}{2}-\cos\left(200\pi t + \dfrac{\pi}{4}\right)\bigg]$

max power will occur when $\cos\left(200\pi t + \dfrac{\pi}{4}\right)=-1$

$P_{max} = 0.006\bigg[\dfrac{\sqrt{2}}{2}+1\bigg]$

Barry1234

#### Barry1234

Thanks Skeeter

So this works out to a value of about 0.01W.

How would i go about working out the first time that the power reaches a maximum value?

#### skeeter

MHF Helper
max power will occur when $\cos\left(200\pi t + \dfrac{\pi}{4}\right)=-1$ and note that $\cos{\pi} = -1$

what now?

#### Barry1234

does that make

$$\displaystyle \cos (200 \pi t + \frac{\pi}{4} ) = \cos \pi$$

And a trigonometric equation?

$$\displaystyle \cos (200 \pi t + \frac{\pi}{4}) - \cos \pi = 0$$

Is this correct?

#### skeeter

MHF Helper

$200\pi t + \dfrac{\pi}{4} = \pi$

... solve for $t$

#### Barry1234

Thanks Skeeter

$$\displaystyle t = \frac {\pi - \frac {\pi}{4}} {200 \pi}$$

t = 3.75 * 10^-3