# Calculus 2 : Polar Form

## Example Questions

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### Example Question #1 : Polar Form

What would be the equation of the parabola  in polar form?

Explanation:

We know  and .

Subbing that in to the equation  will give us .

Multiplying both sides by  gives us

.

### Example Question #2 : Polar Form

A point in polar form is given as .

Find its corresponding  coordinate.

Explanation:

To go from polar form to cartesion coordinates, use the following two relations.

In this case, our  is  and our  is .

Plugging those into our relations we get

which gives us our  coordinate.

### Example Question #3 : Polar Form

What is the magnitude and angle (in radians) of the following cartesian coordinate?

Give the answer in the format below.

Explanation:

Although not explicitly stated, the problem is asking for the polar coordinates of the point . To calculate the magnitude, , calculate the following:

To calculate , do the following:

### Example Question #2 : Polar Form

What is the following coordinate in polar form?

Provide the angle in degrees.

Explanation:

To calculate the polar coordinate, use

However, keep track of the angle here. 68 degree is the mathematical equivalent of the expression, but we know the point (-2,-5) is in the 3rd quadrant, so we have to add 180 to it to get 248.

.

### Example Question #3 : Polar Form

What is the equation  in polar form?

Explanation:

We can convert from rectangular form to polar form by using the following identities:  and . Given , then .

. Dividing both sides by ,

### Example Question #4 : Polar Form

What is the equation  in polar form?

None of the above

Explanation:

We can convert from rectangular form to polar form by using the following identities:  and . Given , then . Multiplying both sides by ,

### Example Question #4 : Polar Form

What is the equation  in polar form?

None of the above

Explanation:

We can convert from rectangular form to polar form by using the following identities:  and . Given , then . Simplifying accordingly,

### Example Question #5 : Polar Form

Given  and , what is  in terms of  (rectangular form)?

Explanation:

Knowing that  and , we can isolate  in both equations as follows:

Since both of these equations equal , we can set them equal to each other:

### Example Question #5 : Polar Form

Convert the following function into polar form:

Explanation:

The following formulas were used to convert the function from polar to Cartestian coordinates:

Note that the last formula is a manipulation of a trignometric identity.

Simply replace these with x and y in the original function.

### Example Question #6 : Polar Form

Convert from rectangular to polar form:

Explanation:

To convert from rectangular to polar form, we must use the following formulas:

It is easier to find our angle  first, which is done by plugging in our x and y into the second formula:

Find the angle by taking the inverse of the function:

Now find r by plugging in our angle and x and y into the first formula, and solving for r:

Our final answer is reported in polar coordinate form :

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