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/4)

Accepted Answer

Welcome back, everyone. In this problem, we want to plot the complex number E raise to the power of 1/ of pi I using the formula E race to the power of I theta equals the cosine of theta plus I sine theta. And we also want to plot it on a rectangular graph for complex numbers where we have our real and imaginary axis. Now, what do we already know? Well, we know that this formula is what we call euler's formula. OK? And Euler's formula is just a way to link a complex exponential to a complex number in polar form. But the problem is we have a rectangular good. So we'll need to convert our number using euler's formula to polar form and then rewrite it in rectangular form in order for us to plot the number. So let's go ahead and do that. So using our formula, OK, using U formula since E to the I of theta, E equals the core sign of theta plus I multiplied by the sign of theater, then if I have E to the core of 1/12 of pie, I, it just means that wherever I see the I'm going to replace it with 1/12 of pie. So that could be rewritten as the co sign of 1/12 of pie plus oops. Yeah, plus I multiplied by the sign of 1/12 of pi no. If I want to move further, I need to figure out what these things are. Now, if we recall, if we require some common trick identities, OK. The common trigonometric identities, then you'll probably remember that the cosine of 1/12 of pi equals the square root of six plus the square root of two all divided by four. Likewise, you recall that the sign of 1/12 of pi equals the square of six minus the square of two all divided by four. So with both of these, that means we can rewrite our expression. That is e to the power of 1/12 of pi I as the square root of six plus the squared of two all divided by four plus the square root of six minus the squared of two, all divided by four. I now that is what it would look like in a rectangular form. Know that we have it in rectangular form. All we need to do is to plot these on our grid because now you'll notice that here we have a real part and we have an imaginary part. So let's do that. So our real part is a square of six plus a square of two, all divided by four. And that is, it's approximately one right. So we are going to put it about somewhere right here. So this is where the X value of our cord in it would be likewise the square of six minus the square of two all divided by four is approximate. It's about halfway between zero and 0.5. So it's approximately 0.25. So we're going to put it there on our grid. That means our point would fall right at this point where it would be a skirt of six plus a square of two all divided by four. So let me write that out nicely, a skirt of six plus the square of two divided by four and the square of six minus the square of two all divided by four. And that's what our complex number would be on our rectangular grid. Thanks a lot for watching 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/4)

Key Concepts

Euler's Formula

Complex Numbers

Polar Coordinates

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