# Math Help - help for a trigonometric problem

1. ## help for a trigonometric problem

Can anyone please give me some idea for this problem?
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 v=0.02 sin(100 pi t)volts and i=0.6 sin(100pi t+ pi/4 ) amp
Calculate the first time that the power reaches a maximum value.

2. Hi

$p = iv=0.02 \sin(100 \pi t) \:0.6 \sin\left(100 \pi t+ \frac{\pi}{4}\right)$

Using
$\sin a \:\sin b = \frac12\\cos(a-b)-\cos(a+b))" alt="\sin a \:\sin b = \frac12\\cos(a-b)-\cos(a+b))" />

Spoiler:

$p = 0.006 \:\left(\cos\left(\frac{\pi}{4}\right) - \cos\left(200 \pi t+ \frac{\pi}{4}\right)\right)$

$p = 0.003 \:\sqrt{2} - 0.006\:\cos\left(200 \pi t+ \frac{\pi}{4}\right)$

The maximum is obtained when cos = -1

$p_{max} = 0.003 \:\sqrt{2} + 0.006 = 0.010\: W$

It is obtained for $200 \pi t+ \frac{\pi}{4} = \pi \implies t = \frac{3}{800} = 0.00375\:s$