Examples of Polar Coordinates—Circles
To see how the polar coordinate system works, let’s look at the graphs of some familiar objects. Circles, ellipses, spirals, and other figures whose equations are complicated in Cartesian coordinates can often be expressed much more simply in polar coordinates. In general, the polar direction θ is expressed in radians. In the examples that follow, the “rad” abbreviation is eliminated, because it is understood that all angles are in radians.
Circle Centered At Origin
The equation of a circle centered at the origin in the polar plane is given by the following formula:
r = a
where a is a real-number constant greater than 0. This is illustrated in Fig. 5-3.
Fig. 5-3. Polar graph of a circle centered at the origin, with radius a.
Circle Passing Through Origin
The general form for the equation of a circle passing through the origin and centered at the point ( θ _{0} , r _{0} ) in the polar plane (Fig. 5-4) is as follows:
r = 2r _{0} cos ( θ – θ _{0} )
Fig. 5-4. Polar graph of a circle passing through the origin, with center at ( θ _{0} , r _{0} ) and radius r _{0} .
Ellipse Centered At Origin
The equation of an ellipse centered at the origin in the polar plane is given by the following formula:
r = ab /( a ^{2} sin ^{2} θ + b ^{2} cos ^{2} θ ) ^{1/2}
where a and b are real-number constants greater than 0.
In the ellipse, a represents the distance from the origin to the curve as measured along the “horizontal” ray θ = 0 , and b represents the distance from the origin to the curve as measured along the “vertical” ray θ = π/2. This is illustrated in Fig. 5-5. The values a and b represent the lengths of the semi-axes of the ellipse. The greater value is the length of the major semi-axis, and the lesser value is the length of the minor semi-axis.
Fig. 5-5. Polar graph of an ellipse centered at the origin, with semi-axes a and b.
Hyperbola Centered At Origin
The general form of the equation of a hyperbola centered at the origin in the polar plane is given by the following formula:
r = ab /( a ^{2} sin ^{2} θ – b ^{2} cos ^{2} θ ) ^{1/2}
where a and b are real-number constants greater than 0.
Let D represent a rectangle whose center is at the origin, whose vertical edges are tangent to the hyperbola, and whose vertices (corners) lie on the asymptotes of the hyperbola (Fig. 5-6). Let a represent the distance from the origin to D as measured along the “horizontal” ray θ = 0 , and let b represent the distance from the origin to D as measured along the “vertical” ray θ = π/2. The values a and b represent the lengths of the semi-axes of the hyperbola. The greater value is the length of the major semi-axis, and the lesser value is the length of the minor semi-axis.
Fig. 5-6. Polar graph of a hyperbola centered at the origin, with semi-axes a and b.
Lemniscate
The general form of the equation of a lemniscate centered at the origin in the polar plane is given by the following formula:
r = a(cos 2θ ) ^{1/2}
where a is a real-number constant greater than 0, representing the maximum radius. This is illustrated in Fig. 5-7.
Fig. 5-7. Polar graph of a lemniscate centered at the origin, with radius a.
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 3θ
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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From Trigonometry Demystified: A Self-Teaching Guide. Copyright © 2003 by The McGraw-Hill Companies, Inc. All Rights Reserved.