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Chapter 6: Problem 29

Write each complex number in rectangular form. If necessary, round to the nearest tenth. $$4\left(\cos 240^{\circ}+i \sin 240^{\circ}\right)$$

Short Answer

Expert verified
-2 - 3.46i

Step by step solution

01

Identity of the complex number

Firstly, recognize that the complex number is in the form \(r(\cos \theta + i \sin \theta)\). Here, \(r = 4\), \(\theta = 240^{\circ}\).
02

Conversion to rectangular form

Next, use the conversion formulas to find \(a = r\cos \theta\) and \(b = r\sin \theta\). Therefore, calculate \(a = 4\cos 240^{\circ}\) and \(b = 4\sin 240^{\circ}\) using a calculator.
03

Calculation

After applying the conversion formulas, you find that \(a = 4\cos 240^{\circ} = -2\) and \(b = 4\sin 240^{\circ} = -3.46\) rounded to the nearest tenth.
04

Putting it all together

Finally, put \(a\) and \(b\) together in the form \(a + bi\) to give the rectangular form of the complex number. Thus, \(4(\cos 240^{\circ}+i\sin240^{\circ}) = -2 - 3.46i \) rounded to the nearest tenth.

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Most popular questions from this chapter

A force of 80 pounds on a rope is used to pull a box up a ramp inclined at \(10^{\circ}\) from the horizontal. The rope forms an angle of \(33^{\circ}\) with the horizontal. How much work is done pulling the box 25 feet along the ramp?

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Will help you prepare for the material covered in the next section. Refer to Section 2.1 if you need to review the basics of complex numbers. In each exercise, perform the indicated operation and write the result in the standard form \(a+b i\). $$(1+i)(2+2 i)$$
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