Three and FourLeafed Roses
Threeleafed Rose
The general form of the equation of a threeleafed 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 3θ
where a is a realnumber constant greater than 0. The cosine curve is illustrated in Fig. 58A; the sine curve is illustrated in Fig. 58B.
Fourleafed Rose
The general form of the equation of a fourleafed 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 realnumber constant greater than 0. The cosine curve is illustrated in Fig. 59A; the sine curve is illustrated in Fig. 59B.
It is interesting, and a little bit mysterious, that the objects graphed in Figs. 58 and 59 are conventionally called “roses” and not “clovers.”
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 realnumber 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. 510.
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 realnumber constant greater than 0. An example of this type of curve is illustrated in Fig. 511.
Examples of Polar Coordinates Practice Problems
Practice 1
What is the value of the constant, a, in the spiral shown in Fig. 510? What is the equation of this spiral? Assume that each radial division represents 1 unit.
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. 511? What is the equation of this cardioid? Assume that each radial division represents 1 unit.
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 nonnegative values, we get the closedended 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 openended 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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