# AP Calculus BC : Polar Form

## Example Questions

### Example Question #11 : Polar Form

What is the polar form of ?

None of the above

Explanation:

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

### Example Question #21 : Functions, Graphs, And Limits

What is the polar form of ?

Explanation:

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

### Example Question #22 : Parametric, Polar, And Vector Functions

What is the polar form of ?

Explanation:

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

Dividing both sides by , we get:

### Example Question #23 : Parametric, Polar, And Vector Functions

Convert the following cartesian coordinates into polar form:

Explanation:

Cartesian coordinates have x and y, represented as (x,y). Polar coordinates have

is the hypotenuse, and  is the angle.

Solution:

### Example Question #24 : Parametric, Polar, And Vector Functions

Convert the following cartesian coordinates into polar form:

Explanation:

Cartesian coordinates have x and y, represented as (x,y). Polar coordinates have

is the hypotenuse, and  is the angle.

Solution:

### Example Question #25 : Parametric, Polar, And Vector Functions

Calculate the polar form hypotenuse of the following cartesian equation:

Explanation:

In a cartesian form, the primary parameters are  and . In polar form, they are  and

is the hypotenuse, and  is the angle created by .

2 things to know when converting from Cartesian to polar.

You want to calculate the hypotenuse,

Solution:

### Example Question #26 : Parametric, Polar, And Vector Functions

Graph the equation  where .

Explanation:

At angle  the graph as a radius of . As it approaches , the radius approaches .

As the graph approaches , the radius approaches .

Because this is a negative radius, the curve is drawn in the opposite quadrant between  and .

Between  and , the radius approaches  from  and redraws the curve in the first quadrant.

Between  and , the graph redraws the curve in the fourth quadrant as the radius approaches  from .

### Example Question #27 : Parametric, Polar, And Vector Functions

Draw the graph of  from .

Explanation:

Because this function has a period of , the x-intercepts of the graph   happen at a reference angle of  (angles halfway between the angles of the axes).

Between  and  the radius approaches  from .

Between  and , the radius approaches  from  and is drawn in the opposite quadrant, the third quadrant because it has a negative radius.

From  to  the radius approaches  from  , and is drawn in the fourth quadrant, the opposite quadrant.

Between  and , the radius approaches  from .

From  and , the radius approaches  from .

Between  and , the radius approaches  from . Because it is a negative radius, it is drawn in the opposite quadrant, the first quadrant.

Then between  and  the radius approaches  from  and is draw in the second quadrant.

Finally between  and , the radius approaches  from .

### Example Question #7 : Derivatives Of Polar Form

Find the derivative of the following function:

Explanation:

The derivative of a polar function is given by the following:

First, we must find

We found the derivative using the following rules:

Finally, we plug in the above derivative and the original function into the above formula:

### Example Question #31 : Functions, Graphs, And Limits

What is the derivative of ?