# Precalculus : Polar Coordinates and Complex Numbers

## Example Questions

### Example Question #6 : Express Complex Numbers In Polar Form

Express this complex number in polar form.

None of these answers are correct.

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:

Explanation:

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:

Explanation:

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

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:

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?

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:

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?

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

Determine the equation of the following polar graph:

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 .

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

Which of the following polar equations would produce this graph?