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.

8(cos 7π/4 + i sin 7π/4)

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 12 multiplied by the quantity of the cosine of five pi divided by three plus I sign of five pi divided by three. Our answer choices are answer choice. A six plus six square at three. I answer choice B six minus six square at three. I answer choice C 24 minus 24 square at three I and answer choice D 12 minus 12 square root three. I, all right. We are given a complex number in polar form which is great. In order to get it to rectangular form, essentially, all we need to do is distribute this 12 that's in front of our parentheses to both the cosine of five pi divided by three and the sign of five pi divided by three. Now, because five pi divided by three is on our unit circle, it's a lot easier to get an exact value for this one. You could plug it into a calculator, but we can do exact values. I'm going to get these exact values pretty quickly. But if you need a refresher, definitely check out previous lesson videos. We know that five pi divided by three. If we have a coordinate plane with a vertical Y axis, a horizontal X axis is down in the fourth quadrant, closer to the negative part of the Y axis than the positive part of the X axis. And our five pi divided by three reference angle in quadrant, one would be pi divided by three or 60 degrees. So knowing that we're in quadrant four, looking at pi divided by three, the cosine of pi divided by three or the cosine of degrees is a positive one half. But since we're down in quadrant four, we're talking about cosine and quadrant four, that's also positive. So we have a positive one half for the exact value of the cosine of five pi divided by three plus. And we can bring down this, I. Now for the sign of five pi divided by three, the sign of pi divided by three or 60 degrees is going to be the square root of three divided by two. However, down in the fourth quadrant for sign that's going to be negative. So we can change the sign of this one to negative. So for our exact values inside of the parentheses, we have one half minus I square at three divided by two, that's all being multiplied by the 12. Now we can distribute that 12 to the one half and the negative square at three divided by two, 12 multiplied by one half is a positive six. And then we can bring down minus. And now for the standard form, we'll put for our rectangular form Z equals A plus B I. So we want the regular numbers first and then the I, so 12 multiplied by the square root of three, divided by two is going to be six square at three. And then don't forget the I, sometimes we put parentheses around six square at three. So we know that the I is not inside of the radical, but there is our complex number 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.

In Exercises 27–36, write each complex number in rectangular form. If necessary, round to the nearest tenth. 8(cos 7π/4 + i sin 7π/4)

Key Concepts

Complex Numbers

Polar to Rectangular Conversion

Trigonometric Values of Common Angles

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