Simple Circuit Problem? (P=IV)?

You probably have a degrees vs radians error. The functions of sine and cosine require your computation device to either be configured in degrees or radians. Google calculator is configured in radians, and when used in talking about electronic signals, these functions most commonly use radians.

The other possible error is a metric prefix error. The voltage has no metric prefix on it, it is just volts. The current has a metric prefix of milli- on its amperes, meaning that they are a thousand times less than the base unit.

Thus, the correct answers for instantaneous current and voltage are:

V(t) = -3.329 V... the negative sign means it is opposite the way that the voltage markers are labeled.

I(t) = 3.637 milliamps

Instantaneous power:

P(t) = I(t)*V(t)

P(t) = -12.11 milliwatts

If you take a look at the sign conventions of current and voltage through and across this inductor, you will see that when I and V are both positive, they are opposite what you'd expect for a resistor. The current is labeled as flowing from minus sign voltage to plus sign voltage, as it ordinarily does in a SOURCE, rather than in a load. That is what it would be doing if both V and I multiplied yielded instantaneously positive values.

But instead, they yielded a negative product for instantaneous power. This means that the instant is an instant at which it is behaving opposite of what the signs would indicate. Rather than acting as a source, it is acting as a load. -12.11 milliwatts are ABSORBED in this inductor, or rather, are being STORED in this inductor.

Inductors are interesting when alternating current flows through them. If you integrate this power over one complete cycle, you will see that net-zero energy is stored in or delivered by the inductor. All energy stored in it is later delivered to the rest of the circuit.

The hydraulic analogy for an inductor is a free spinning paddlewheel that interrupts a flow of water. This paddlewheel is "happy" staying still, and it is "happy" to rotate along with a steady flow of water. But, if the water flowrate is TRANSIENT, that is to say, changing, then the paddlewheel will exert its pushing pressure on the water in attempt to counter the change in flow rate, since the paddlewheel has inertia and doesn't like to change speeds. If the flowrate oscillates on a complete sinusoidal cycle, you notice that the paddlewheel neither speeds up nor slows down from what it was doing. Net-zero energy is stored in it.

Inductors add an "electrical inertia" to the circuit. They induce a back-voltage to oppose any changes to the current. Contrast this with a resistor which just makes a voltage drop to oppose ANY current itself.

I = 8cos1000t ma where 1000 radians per second = 2πf , where f = 1,000/(2π)= 159 Hertz

V = 4sin1000t volts

t=.002s

I = 8cos(2 radians) = 8 cos (114.6°)= -3.33 milliamps

V = 4sin( 2 radians) = 4 sin (114.6°) = 3.64 Volts

P absorbed = (3.33 milliamps)( 3.64 volts) = 12.11 milliwatts

Note: this power will be returned to the source from the inductor.

I agree with your answer. Seems the answer in red didn't take into consideration that the units of current were in milliAmperes and not straight Amps. But its quite late and I might have missed something too... :|

You need to work in RADIANS, not degrees, when you do the calculation!