Polar equations are math functions given in the form of R= f (θ). To express these functions you use the polar coordinate system. The graph of a polar function R is a curve that consists of points in the form of ( R, θ). Due to the circular aspect of this system, it's easier to graph polar equations using this method.
Understand Polar Equations
Understand that in the polar coordinate system you denote a point by (R, θ) where R is the polar distance and θ is the polar angle in degrees.
Use radian or degrees to measure θ. To convert radians to degrees, multiply the value by 180/π. For example, π/2 X 180/π = 90 degrees.
Know that there are many curve shapes given by polar equations. Some of these are circles, limacons, cardioids and rose-shaped curves. Limacon curves are in the form R= A ± B sin(θ) and R= A ± B cos(θ) where A and B are constants. Cardioid (heart-shaped) curves are special curves in the limacon family. Rose petalled curves have polar equations in the form of R= A sin(nθ) or R= A cos(nθ). When n is an odd number, the curve has n petals but when n is even the curve has 2n petals.
Simplify the Graphing of Polar Equations
Look for symmetry when graphing these functions. As an example use the polar equation R=4 sin(θ).You only need to find values for θ between π (Pi) because after π the values repeat since the sine function is symmetrical.
Choose the values of θ that makes R maximum, minimum or zero in the equation. In the example given above R= 4 sin (θ), when θ equals 0 the value for R is 0. So (R, θ) is (0, 0). This is a point of intercept.
Find other intercept points in a similar manner.
Graph Polar Equations
Consider R= 4 sin(θ) as an example to learn how to graph polar coordinates.
Evaluate the equation for values of (θ) between the interval of 0 and π. Let (θ) equal 0, π /6 , π /4, π /3, π /2, 2π /3, 3π /4, 5π /6 and π. Calculate values for R by substituting these values into the equation.
Use a graphing calculator to determine the values for R. As an example, let (θ) = π /6. Enter into the calculator 4 sin(π /6). The value for R is 2 and the point (R, θ) is (2, π /6). Find R for all the (θ) values in Step 2.
Plot the resulting (R, θ ) points from Step 3 which are (0,0), (2, π /6), (2.8, π /4), (3.46,π /3), (4,π /2), (3.46, 2π /3), (2.8, 3π /4), (2, 5π /6), (0, π) on graph paper and connect these points. The graph is a circle with a radius of 2 and center at (0, 2). For better precision in graphing, use polar graph paper.
Graph the equations for limacons, cardioids or any other curve given by a polar equation by following the procedure outlined above.