In Exercises 27–36, write each complex number in rectangular form...

Question

In Exercises 27–36, write each complex number in rectangular form. If necessary, round to the nearest tenth.

6(cos 30° + i sin 30°)

Accepted Answer

Welcome back. I am so glad you're here. We're asked to express the given complex number in rectangular form. Write your answer in one decimal place. If necessary, our given complex number is eight multiplied by the quantity of the cosine of 45 degrees plus I sine 45 degrees. Our answer choices are answer choice. A 16 square two plus square root two. I answer choice B four square root two plus four square root two, I answer choice C four plus four I and answer choice D eight square root two plus eight square root two I, all right. So we are given our complex number in polar form, which is great because changing it to rectangular form, essentially, all we need to do is distribute this eight to the cosine of degrees and the sign of 45 degrees. Now, the cosine of 45 degrees and the sign of 45 degrees, those are on our unit circle, we can convert those to exact values and then distribute the eight. So I'm going to convert them to exact values pretty quickly if you need a refresher on how to do that. Definitely check out previous lesson videos. If we have a quick sketch of a coordinate plane, we have a vertical Y axis and a horizontal X axis. 45 degrees is in the first quadrant that is exactly halfway between the positive parts of the X axis and the Y axis. And so for 45 degrees, we recall from previous lessons that both cos cosine and sine in the first quadrant, those are going to be positive values and both the cosine and the sign of 45 degrees is the square of two divided by two. So inside the parentheses, instead of cosine 45 degrees, we can put the square root of two divided by two for the exact value and bring down plus I and then for the sine of 45 degrees that again is the square red of two divided by two, we still need to distribute that eight. So we can do that. Now, eight multiplied by square out of two divided by two is going to be four square at two. So we have four square root two plus and then we have the I multiplied by four square root two. But in our rectangular form, that's Z equals A plus B I the I comes second. So we'll put four square root two plus four square up two I and then often we put that second number that four squared of two here in parentheses. So that we know that the I is not inside of the radical with the two. All right. So we've got our answer here in rectangular form. We look at our answer choices and this matches with answer choice. B well done. We'll catch you on the next one.

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In Exercises 27–36, write each complex number in rectangular form. If necessary, round to the nearest tenth. 6(cos 30° + i sin 30°)