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Steps for Converting Equations from Rectangular to Polar form and vice versa
Four critical equivalents to keep in mind are:
ytan
x
Convert the equation: r = 2 to rectangular form
Since we know that , square both sides of the equation.
Convert the following equation from rectangular to polar form.
2 2x y x
and
Since
x r cos2r r cosr cos
Convert the following equation from rectangular to polar form.
2 22x 2y 3 2 22(x y ) 3
2 2 3x y
2
3r
2
2 3r
2
An equation whose variables are polar coordinates is called a polar equation. The graph of a polar equation consists of all points whose polar coordinates satisfy the equation.
Theorem
Let a be a nonzero real number, the graph of the equation
is a horizontal line a units above the pole if a > 0 and |a| units below the pole if a < 0.
Theorem
Let a be a nonzero real number, the graph of the equation
is a vertical line a units to the right of the pole if a > 0 and |a| units to the left of the pole if a < 0.
Theorem
Let a be a positive real number. Then,
Circle: radius ; center at ( , 0) in rectangular coordinates.
Circle: radius ; center at (- , 0) in rectangular coordinates.
r a cos
r a cos
a
2a
2
a
2a
2
Theorem
Let a be a positive real number. Then,
Circle: radius ; center at (0, ) in rectangular coordinates.
Circle: radius ; center at (0, ) in rectangular coordinates.
r a sin
r a sin
a
2a
2
a
2a
2
r 4 4cos r 4 4cos
Cardioids (heart-shaped curves) where a > 0 and passes through the origin
r a a cos r a a cos
Limacons without the inner loop
are given by equations of the form
where a > 0, b > 0, and a > b. The graph of limacon without an inner loop does not pass through the pole.
Limacons with an inner loop
are given by equations of the form
where a > 0, b > 0, and a < b. The graph of limacon with an inner loop will pass through the pole twice.
Rose curvesare given by equations of the form
and have graphs that are rose shaped. If n is even and not equal to zero, the rose has 2n petals; if n is odd not equal to +1, the rose has n petals. a represents the length of the petals.