Precalculus : Polar Coordinates and Complex Numbers

Study concepts, example questions & explanations for Precalculus

varsity tutors app store varsity tutors android store

Example Questions

Example Question #6 : Express Complex Numbers In Polar Form

Express this complex number in polar form.

Possible Answers:

None of these answers are correct.

Correct answer:

Explanation:

 

Given these identities, first solve for  and . The polar form of a complex number is: 

 

 at  (because the original point, (1,1) is in Quadrant 1)

Therefore...

 

Example Question #1 : Express Complex Numbers In Polar Form

Convert to polar form:

Possible Answers:

Correct answer:

Explanation:

First, find the radius :

Then find the angle, thinking of the imaginary part as the height and the radius as the hypotenuse of a right triangle:

according to the calculator.

We can get the positive coterminal angle by adding :

The polar form is

Example Question #8 : Express Complex Numbers In Polar Form

Convert to polar form:

Possible Answers:

Correct answer:

Explanation:

First find the radius, :

Now find the angle, thinking of the imaginary part as the height and the radius as the hypotenuse of a right triangle:

according to the calculator.

This is an appropriate angle to stay with since this number should be in quadrant I.

The complex number in polar form is

Example Question #9 : Express Complex Numbers In Polar Form

Convert the complex number to polar form

Possible Answers:

Correct answer:

Explanation:

First find :

Now find the angle. Consider the imaginary part to be the height of a right triangle with hypotenuse .

according to the calculator.

What the calculator does not know is that this angle is actually located in quadrant II, since the real part is negative and the imaginary part is positive.

To find the angle in quadrant II whose sine is also , subtract from :

The complex number in polar form is

Example Question #1 : Convert Polar Coordinates To Rectangular Coordinates

Convert the polar coordinates to rectangular coordinates:

Possible Answers:

Correct answer:

Explanation:

To convert polar coordinates  to rectangular coordinates ,

Using the information given in the question, 

The rectangular coordinates are 

Example Question #1 : Determine The Polar Equation Of A Graph

Which polar equation would produce this graph?

Polar ppt 3

Possible Answers:

Correct answer:

Explanation:

This is the graph of a cardiod. Based on its orientation where the cusp [pointy part] is on the y-axis, it is a sine and not cosine function. The x-intercepts are at , so the first number must be 2. Since vertically the graph goes from 0 to 4, the second number must be 2, because and .

Example Question #1 : Graphs Of Polar Equations

Give the polar equation for this graph:

Polar 7sin5x

Possible Answers:

Correct answer:

Explanation:

This graph shows a rose curve with an odd number of petals.

This means that the equation will be in the form , where represents the length of the "petals" and represents the number of petals.

There are 5 petals of length 7.

Example Question #3 : Determine The Polar Equation Of A Graph

Which polar equation would produce this graph?

Polar ppt 5

Possible Answers:

Correct answer:

Explanation:

This graph shows a rose curve with an even number of petals. The first petal also intersects with the x-axis. This means that the equation will be in the form where is the length of each petal, and is half the number of petals. (Note that for an odd number of petals, the rose curve will have exactly petals). In this case, the petals have length 5, and there are 8 of them [half of 8 is 4].

Example Question #4 : Determine The Polar Equation Of A Graph

Please choose the best answer from the following choices.

Determine the equation of the following polar graph:

 

Msp6661d6g1fca347a4f9a00002g2fc12725bd10gh

Possible Answers:

Correct answer:

Explanation:

Each loop is on the line of each axis, which means the equation will have a cosine.

The loops stretch out to the third unit on each axis, which means there will be a three in front of the cosine.

Lastly, there are 4 loops. Whenever there is an even number of loops, you divide that number by 2, and that is the number which goes in front of theta in the parentheses. In this case, it is  because .

Thus our answer is,

.

Example Question #1 : Determine The Polar Equation Of A Graph

Which of the following polar equations would produce this graph?

 

Polar

Possible Answers:

Correct answer:

Explanation:

You can tell from the direction of the cardioid, extending horizontally, that you are dealing with a cosine graph. Then you should note that the y-intercepts are at 1 and -1, telling you that the first number must be 1. And since the graph extends to x = 2, you should note that the two numeric terms in  should add to 2, making the correct choice .

Learning Tools by Varsity Tutors