Precalculus : Polar Coordinates and Complex Numbers

Study concepts, example questions & explanations for Precalculus

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Example Questions

Example Question #1 : Evaluate Powers Of Complex Numbers Using De Moivre's Theorem

Evaluate

Possible Answers:

Correct answer:

Explanation:

First convert the complex number into polar form:

Since the real part is negative but the imaginary part is positive, the angle should be in quadrant II, so it is

We are evaluating

Using DeMoivre's Theorem:

DeMoivre's Theorem is

 

We apply it to our situation to get.

simplify and take the exponent

is coterminal with since it is an odd multiple of pi

Example Question #8 : Powers And Roots Of Complex Numbers

Use DeMoivre's Theorem to evaluate the expression .

Possible Answers:

Correct answer:

Explanation:

First convert this complex number to polar form:

so

Since this number has positive real and imaginary parts, it is in quadrant I, so the angle is

So we are evaluating

Using DeMoivre's Theorem:

DeMoivre's Theorem is

We apply it to our situation to get.

Example Question #1 : Powers And Roots Of Complex Numbers

Evaluate:

Possible Answers:

Correct answer:

Explanation:

First, convert this complex number to polar form.

 

Since the point has a positive real part and a negative imaginary part, it is located in quadrant IV, so the angle is .

This gives us

To evaluate, use DeMoivre's Theorem:

DeMoivre's Theorem is

 

We apply it to our situation to get.

 

simplifying

is coterminal with  since it is an even multiple of 

Example Question #10 : Powers And Roots Of Complex Numbers

Possible Answers:

Correct answer:

Explanation:

First, convert the complex number to polar form:

Since both the real and the imaginary parts are positive, the angle is in quadrant I, so it is

This means we're evaluating

Using DeMoivre's Theorem:

DeMoivre's Theorem is

 

We apply it to our situation to get.

First, evaluate . We can split this into which is equivalent to

[We can re-write the middle exponent since is equivalent to ]

This comes to

Evaluating sine and cosine at is equivalent to evaluating them at since

This means our expression can be written as: 

Example Question #1 : Find The Roots Of Complex Numbers

Evaluate , where  is a natural number and  is the complex number .

Possible Answers:

Correct answer:

Explanation:

Note that,

 

Example Question #2 : Find The Roots Of Complex Numbers

What is the  length of 

?

Possible Answers:

Correct answer:

Explanation:

We have

.

Hence,

.

Example Question #11 : Powers And Roots Of Complex Numbers

Solve for  (there may be more than one solution).

Possible Answers:

Correct answer:

Explanation:

Solving that equation is equivalent to solving the roots of the polynomial .

Clearly, one of roots is 1.

Thus, we can factor the polynomial as 

so that we solve for the roots of .

Using the quadratic equation, we solve for roots, which are .

 

This means the solutions to  are

 

Example Question #1 : Find The Roots Of Complex Numbers

Recall that  is just shorthand for  when dealing with complex numbers in polar form. 

Express   in polar form.

Possible Answers:

Correct answer:

Explanation:

First we recognize that we are trying to solve  where .

Then we want to convert  into polar form using,

  and .

Then since De Moivre's theorem states,

  if  is an integer, we can say

.

Example Question #5 : Find The Roots Of Complex Numbers

Solve for  (there may be more than one solution).

Possible Answers:

Correct answer:

Explanation:

To solve for the roots, just set equal to zero and solve for z using the quadratic formula () :  and now setting both  and  equal to zero we end up with the answers  and  

 

Example Question #6 : Find The Roots Of Complex Numbers

Compute

 

Possible Answers:

Correct answer:

Explanation:

To solve this question, you must first derive a few values and convert the equation into exponential form: :  

 

Now plug back into the original equation and solve:   

 

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