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# Examples of Polar Coordinates Help (page 2)

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## Three and Four-Leafed Roses

### Three-leafed Rose

The general form of the equation of a three-leafed rose centered at the origin in the polar plane is given by either of the following two formulas:

r = a cos 3θ

r = a sin

where a is a real-number constant greater than 0. The cosine curve is illustrated in Fig. 5-8A; the sine curve is illustrated in Fig. 5-8B.

Fig. 5-8 (A) Polar graph of a three-leafed rose with equation r = a cos 3θ. (B) Polar graph of a three-leafed rose with equation r = a sin 3θ.

### Four-leafed Rose

The general form of the equation of a four-leafed rose centered at the origin in the polar plane is given by either of the following two formulas:

r = a cos 2 θ

r = a sin 2 θ

where a is a real-number constant greater than 0. The cosine curve is illustrated in Fig. 5-9A; the sine curve is illustrated in Fig. 5-9B.

It is interesting, and a little bit mysterious, that the objects graphed in Figs. 5-8 and 5-9 are conventionally called “roses” and not “clovers.”

Fig. 5-9 (A) Polar graph of a four-leafed rose with equation r = a cos 2θ. (B) Polar graph of a four-leafed rose with equation r = a sin 2 θ .

## Spiral

The general form of the equation of a spiral centered at the origin in the polar plane is given by the following formula:

r = a θ

where a is a real-number constant greater than 0. An example of this type of spiral, called the spiral of Archimedes because of the uniform manner in which its radius increases as the angle increases, is illustrated in Fig. 5-10.

Fig. 5-10. Polar graph of a spiral; illustration for Practice 1.

## Cardioid

The general form of the equation of a cardioid centered at the origin in the polar plane is given by the following formula:

r = 2 a (1 + cos θ )

where a is a real-number constant greater than 0. An example of this type of curve is illustrated in Fig. 5-11.

Fig. 5-11. Polar graph of a cardioid; illustration for Problem 5-2.

## Examples of Polar Coordinates Practice Problems

#### Practice 1

What is the value of the constant, a, in the spiral shown in Fig. 5-10? What is the equation of this spiral? Assume that each radial division represents 1 unit.

Fig. 5-10. Polar graph of a spiral; illustration for Practice 1.

#### Solution 1

Note that if θ = π, then r = 2. Therefore, we can solve for a by substituting this number pair in the general equation for the spiral. Plug in the numbers ( θ 0 , r 0 ) = (π, 2). Proceed like this:

r 0 = a θ 0

2 = aπ

2/π = a

Therefore, a = 2/π, and the equation of the spiral is r = (2/π)θ or, in a form without parentheses, r = 2θ/π.

#### Practice 2

What is the value of the constant, a, in the cardioid shown in Fig. 5-11? What is the equation of this cardioid? Assume that each radial division represents 1 unit.

Fig. 5-11. Polar graph of a cardioid; illustration for Practice 2.

#### Solution 2

Note that if θ = 0, then r = 4. We can solve for a by substituting this number pair in the general equation for the cardioid. Plug in the numbers (θ 0 , r 0 ) = (0,4). Proceed like this:

This means that the equation of the cardioid is r = 2(1 + cos θ ) or, in a form without parentheses, r = 2 + 2 cos θ.

#### Practice 3

What is the polar equation of a straight line running through the origin and ascending at a 45° angle as you move toward the right?

#### Solution 3

The equation is θ = 45°, or if we use radians, θ = π/4. It is understood that the value of r can be any real number: positive, negative, or zero. If r is restricted to non-negative values, we get the closed-ended ray starting at the origin and pointing outward in the 45° ( π/4 rad) direction. If r is restricted to negative values, we get the open-ended ray starting at the origin and pointing outward in the 225° ( 5π/4 rad) direction. The union of these two rays forms the line running through the origin and ascending at a 45° angle as you move toward the right. In the rectangular xy -plane, this line is the graph of the equation y = x.

Practice problems for these concepts can be found at:  Polar Coordinates Practice Test

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