In calculus, it can be shown that e^(iθ) = cos θ + i sin θ. In...

Question

In calculus, it can be shown that e^(iθ) = cos θ + i sin θ. In Exercises 87–90, use this result to plot each complex number. -e^-πi

Accepted Answer

Welcome back, everyone. In this problem, we want to plot the complex number negative E raised to the power of five pi I using the formula E to the power of I theta equals the cosine of theta plus I multiplied by the sign of theta. And we also want to plot it on our rectangular complex grid that has the real and imaginary axis. Now, what, what do we already know? Well, the formula we mentioned, we know that that is what we call euler's formula. OK. And it's a way to relate complex exponentials to complex numbers in polar form. But the problem is our grid is in rectangular form. So what that means is that we'll have to use euler's formula to write our number in polar form, convert it to rectangular form and then we'll be able to plot. So let's see if we can figure that out. So since we know that the E raised to the power of the equals the cosine of theater plus I multiplied by the sign of theta, then we can take our expression or our complex number negative E era to the power of five pi I. And we can rewrite it as negative one multiplied by E to the power of five pi I. Now why do we do that? Because we want to apply Euller formula? Because no, using the formula where you will notice that theta equals five pi, then we can replace it as negative E is to the power of five by I equals negative one multiplied by the cosine A five K plus I multiplied by the sine of five pi. No. Where do we go from here? Well, again, now we're working with the anger five pi. What do we know about it? Recall that five pi is a reference anger to pi for both the cosine and sign. So Ricard, the sign of five pi is the same thing as the core sign of play. And the sign of five pi is the same thing as the sign of pi. OK. So if we know what these are, then we'll be able to figure out the cosines and signs of 55 respectively. So let's use our unit circle to help us. So remember that on our unit circle, OK. Hi would be right on or X axis right here. And at that, at that point, remember that the X and Y coordinates on the unit circle are the core sign of theta and the sign of theta respectively. So here at the angle pi that tells us that the core sign of pi equals negative one and the sign of pi equals zero because that's where it would be, it would be at the point negative 10. So the cosine of theta is negative one, the sign of the cosine of pi is negative one, the sign of pi is zero. So that means our expression can know our complex number can be written as negative one multiplied by negative one plus zero. I no negative one multiplied by negative one. That would just be one and negative one multiplied by zero. I, that would still just be zero. But I'm going to write zero. I, because I want to ensure I show these points on the graph. OK. So this tells us that our real part is one which would fall right here one at one on our real axis and our imaginary part is zero. So it would fall at zero on our imaginary axis. Therefore, on our graph, the point would be at 10. Thanks for listening, everyone. I hope this video helped.

In calculus, it can be shown that e^(iθ) = cos θ + i sin θ. In Exercises 87–90, use this result to plot each complex number. -e^-πi

Key Concepts

Euler's Formula

Complex Number Representation in the Complex Plane

Properties of Exponents and Negative Signs in Complex Numbers

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