# Precalculus : Polar Coordinates and Complex Numbers

## Example Questions

### Example Question #1 : Convert Rectangular Coordinates To Polar Coordinates And Vice Versa

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 #2 : Convert Rectangular Coordinates To Polar Coordinates And Vice Versa

How could you express  in rectangular coordinates?

Explanation:

The polar coordinates given have an angle of but a negative radius, so our coordinates are located in quadrant III.

This means x and y are both negative. You can figure out these x and y coordinates using trigonometric ratios, or since the angle is , special right triangles.

The hypotenuse of this triangle is 5, but in the special right triangle it's 2, so we know we're multiplying each side by .

That makes the x-coordinate or adjacent side be

and the y-coordinate or opposite side be

.

In this case, once again, both are negative, so our answer is

.

### Example Question #3 : Convert Rectangular Coordinates To Polar Coordinates And Vice Versa

How could you express  in rectangular coordinates?

Round to the nearest hundredth.

Explanation:

In order to determine the rectangular coordinates, look at the triangle representing the polar coordinates:

We can see that both x and y are positive. We can figure out the x-coordinate by using the cosine:

multiply both sides by 10.

We can figure out the y-coordinate by using the sine:

### Example Question #131 : Polar Coordinates And Complex Numbers

Convert  into rectangular coordinates.

Explanation:

If the angle in the polar coordinates is , that means it's in the second quadrant, and away from the x-axis.

This means that the x-coordinate will be negative, and the y-coordinate will be positive:

We can find the x-coordinate using cosine:

multiply both sides by 7

, however we know that the x-coordinate is negative, so we'll use -5.66.

We can find the y-coordinate using sine:

multiply both sides by 7

the coordinates are .

### Example Question #132 : Polar Coordinates And Complex Numbers

Convert to rectangular coordinates

Explanation:

Just to double-check our answer, the point is in quadrant IV, so the x-coordinate is positive and the y-coordinate is negative.

We can convert by using the formula and

In this case, r is 3 and theta is -30

The coordinate is

### Example Question #133 : Polar Coordinates And Complex Numbers

Convert the rectangular coordinates  to polar form with an angle between  and .

Explanation:

We must first recall that the polar coordinates of a point are expressed in the form , where  is the radius (or the distance from the origin to the point) and  is the angle formed between the positive x-axis to the radius.

The radius can be calculated using the distance formula.

Our first point is the origin  and our second point is the one in question

Therefore, substituting gives us

We can find our angle  using the formula

Substituting the coordinates of our point  gives

We can use our knowledge or a chart or calculator to determine that the angle that gives this tangent value is  or .  Since we want a postive angle less than , we need to go with the latter option.

Therefore, the polar coordinates of our point are

### Example Question #134 : Polar Coordinates And Complex Numbers

Convert  to polar coordinates.

Explanation:

Write the Cartesian to polar conversion formulas.

Substitute the coordinate point to the equations to solve for .

Ensuring that  is located the first quadrant, the correct angle is zero.

### Example Question #135 : Polar Coordinates And Complex Numbers

Convert the following rectangular coordinates to polar coordinates:

Explanation:

To convert from rectangular coordinates to polar coordinates :

Using the rectangular coordinates given by the question,

The polar coordinates are

### Example Question #136 : Polar Coordinates And Complex Numbers

Convert the rectangular coordinates to polar coordinates:

Explanation:

To convert from rectangular coordinates to polar coordinates :

Using the rectangular coordinates given by the question,

The polar coordinates are

### Example Question #71 : Polar Coordinates

Convert the rectangular coordinates to polar coordinates: