# Calculus 2 : Polar Calculations

## Example Questions

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### Example Question #1 : Polar Calculations

Convert the polar coordinate equation  into its rectangular equivalent, and simplify.

Explanation:

The polar to rectangular transformation equations are . By substituting these into our given equation we get .

Adding  to both sides, then we get .

### Example Question #2 : Polar Calculations

Convert the following polar coordinates of the form into Cartesian coordinates of the form :

Explanation:

In order to convert the given polar coordinates into Cartesian coordinates, we must remember our formulas for x and y in terms of r and :

The problem tells us r and , so all we must do to convert these coordinates is plug them into the formulas above:

So we can see from our conversion that the given polar coordinates are expressed as    in Cartesian coordinates.

### Example Question #3 : Polar Calculations

Convert  to Cartesian coordinates.

Explanation:

The following formulas will convert polar coordinates to Cartesian coordinates.

We are given the polar coordinate, which is in  form.  Plug the coordinate into the formulas and solve for x and y.

The Cartesian coordinate form is .

### Example Question #4 : Polar Calculations

Given the Cartesian coordinate , what is  in the polar form ?

Explanation:

The formula to find theta in polar form is:

Plug in the Cartesian coordinates into the equation.

However, this angle is located in the fourth quadrant and is not in the right quadrant. Add  radians to get the correct angle since the Cartesian coordinate given is located in the second quadrant.

### Example Question #5 : Polar Calculations

Tom is scaling a mathematical mountain. The mountain's profile can be described by  between  and . Tom climbs from  to the peak of the mountain. How far did he climb? You'll need an equation solver for certain parts of the problem. Round everything to the nearest hundredth.

Explanation:

This is a two step problem. First step is to maximize the function to find the peak of the mountain. The next step is to use the arc length formula to find the distance he climbed.

To maximize, we'll take a derivative and set it equal to zero.

.

Setting this equal to zero, we get .

The derivative is positive prior to this value and negative after, so it is a max. We now must take the arc length from  to .

The formula for arc length is

.

For this case, the integral becomes

.

This will give us . No units were given in the problem, so leaving the answer unitless is acceptable.

### Example Question #6 : Polar Calculations

Find the length of the polar valued function  from  to .

Explanation:

Recall the formula for length in polar coordinates is given by

.

We were given the formula

.

In our case, this translates to

.

### Example Question #6 : Polar Calculations

Convert  to Cartesian coordinates.

Explanation:

Write the formulas to convert from polar to Cartesian.

The  and  values are known.  Substitute both into each equation and solve for  and .

The Cartesian coordinates are:

### Example Question #7 : Polar Calculations

Convert  to Cartesian coordinates and find the coordinates of the center.

Explanation:

Write the conversion formulas.

Notice the  term.  If we multiplied by  on both sides of the  equation, we will get:

Substitute this back into the first equation.

Complete the square with the  terms.

This would then become:

This is a circle centered at  with a radius of 4.

### Example Question #8 : Polar Calculations

Convert  to Cartesian coordinates.

Explanation:

When converting from polar to Cartesian coordinates, we must use the formulas

the values  and  are given, so we can calculate that

and

So the Cartesian coordinate form is

### Example Question #9 : Polar Calculations

Determine the equation in polar coordinates of

Explanation:

can be immediately transformed into polar form by:

Dividing by ,

Dividing both sides by

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