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Chapter 2: Q4P (page 69)

Find each of the following in rectangular form x+iyand check your results by computer. Remember to save time by doing as much as you can in your head.

role="math" localid="1658739754131" e3ln-iπ.

Short Answer

Expert verified

The rectangular form of the given question is e3ln2-iπ=-8.

Step by step solution

01

Given Information.

The given expression ise3ln2-iπ.

02

Meaning of rectangular form.

Representing the complex number in rectangular form means writing the given complex number in the form of x+iy, in which x is the real part and y is the imaginary part.

03

Step 3: Separate the exponential.

The given question ise3ln2-iπ

Break the exponential part in the given question.

e3ln2-iπ=e3ln2e-iπe3ln2-iπ=eln23e-iπe3ln2-iπ=eln8e-iπe3ln2-iπ=8e-iπ

04

Step 4: Convert it into rectangular form.

Use the complex number formulae-iθ=cosθ-isinθ to rewrite the above expression.

8e-iπ=8cosπ-isinπ=8(-1-i(0))=-8

Therefore, the rectangular form of e3ln2-iπis -8.

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Most popular questions from this chapter

For each of the following numbers, first visualize where it is in the complex plane. With a little practice you can quickly findx,y,r,θin your head for these simple problems. Then plot the number and label it in five ways as in Figure 3.3. Also plot the complex conjugate of the number.

(cosπ-isinπ).

Express the following complex numbers in the form. Try to visualize each complex number, using sketches as in the examples if necessary. The first twelve problems you should be able to do in your head (and maybe some of the others—try it!) Doing a problem quickly in your head saves time over using a computer. Remember that the point in doing problems like this is to gain skill in manipulating complex expressions, so a good study method is to do the problems by hand and use a computer to check your answers.

3.9e3iπ12

For each of the following numbers, first visualize where it is in the complex plane. With a little practice you can quickly find x,y,r,θin your head for these simple problems. Then plot the number and label it in five ways as in Figure 3.3. Also plot the complex conjugate of the number.

role="math" localid="1658476746206" (cos3Ï€2+isin3Ï€2)

Solve for all possible values of the real numbers x and y in the following equations.

(x+iy)3=-1

For each of the following numbers, first visualize where it is in the complex plane. With a little practice, you can quickly find x,y,r,θin your head for these simple problems. Then plot the number and label it in five ways as in Figure 3.3. Also, plot the complex conjugate of the number.

7(cos110°-isin110°).

See all solutions

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