In Exercises 15–18, write each complex number in rectangular form...

Question

In Exercises 15–18, write each complex number in rectangular form. If necessary, round to the nearest tenth.

6 (cos 2π/3 + i sin 2π/3)

Accepted Answer

Welcome back. Everyone. In this problem, we want to express the complex number 10 all multiplied by the cosine of 5/6 of pi plus I multiplied by the sign of 5/6 of pi in a rectangular form. And we can write or answer in one decimal place. If necessary for our answer choices, A is negative five multiplied by the spirit of three plus five. IB is five multiplied by the square of three minus five. IC is 10 multiplied by the skirt of three minus 10. I and D is negative 10 multiplied by the spirit of three plus 10. I. Now, what do we already know? Well, this number, our complex number is already in what we call polar form. OK. And to convert a number from polar form to rectangular form, recall that we just expand. So if our numbers in the form are multiplied by the cosine of theta plus are multiplied by the by sign theater, then in rectangular form, our number would now be R multiplied by the cosine of theta plus R multiplied by I sine theta. Or I can write that as R multiplied by the sign of theta, I So that's what I'm going to do to figure out my answer. So that tells us then that for our complex number, we can rewrite it as 10 multiplied by the cosine of 5/ of pi plus 10 multiplied by the sign of 5/6 of pi I. Now to go forward first, we need to figure out what is the co sign and sign of 5/6 of five. Well, from our unit circle, we can tell that the core sign of 56 of pi equals a negative square of three divided by two and the sign of 56 of pi equals a half. So we can substitute that into our formula to because no, that means our complex number is going to be 10 multiplied by the negative square of three divided by two plus 10 multiplied by a half. I let's simplify some more. We can factor out two from our expressions because two into goes five times and the same thing goes for our imaginary component. So now we can rewrite that as five negative, well, five multiplied by the negative skirt of three or negative five multiplied by the skirt of three plus five I which would have been answer choice. A thanks a lot for watching everyone. I hope this video helped.

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In Exercises 15–18, write each complex number in rectangular form. If necessary, round to the nearest tenth. 6 (cos 2π/3 + i sin 2π/3)