Graphing in Polar Coordinates
Author: June Patton

This applet shows the connection between the graphs of functions in rectangular and polar coordinates

Use the sliders to change parameters of these rectangular and polar functions.

x

f(x) = a cos(b (x - c)) + d

0

 \frac{ π }{2 } 

π

 \frac{ 3π }{2 } 

2π

5

4

3

2

1

-1

-2

-3

-4

-5

Show/Hide θ

Show/Hide f(x)

Show/Hide f(x)

Show/Hide r(θ)

Show/Hide r(θ)

r(θ) = a cos(b (θ - c)) + d

f(x) = a sin(b (x - c)) + d

r(θ) = a sin(b (θ - c)) + d

f(x) = a x + d

r(θ) = a θ + d

linear

cosine

sine

Function

0

 \frac{π }{2 } 

π

 \frac{3π }{2} 

 \frac{ π }{ 6} 

 \frac{ π }{3 } 

 \frac{ 2π }{3 } 

 \frac{ 5π }{6 } 

 \frac{ 7π }{6 } 

 \frac{ 4π }{3 } 

 \frac{ 5π }{3 } 

 \frac{ 11π }{6 } 

n = 1 (slider from 1 to 3, step 1)
B = (6.3, 0)
d = 0 (slider from -5 to 5, step 0.1)
A = Point(yAxis)
a_1 = Segment(A, B)
C = Point(a_1)
a = 1 (slider from -5 to 5, step 0.1)
b = 1 (slider from -5 to 5, step 0.1)
c = 0 (slider from -5 to 5, step 0.1)
f_3 = Function(a * sin(b * (x - c)) + d, 0.0, 6.283185307179586)
f_1 = Function(a * b * (x - c) + d, 0.0, 6.283185307179586)
f_2 = Function(a * cos(b * (x - c)) + d, 0.0, 6.283185307179586)
flist: {f_1, f_2, f_3}
f = Function(Element[flist, n], 0.0, 6.283185307179586)
polarCircles = Sequence(Circle[(12.56637, 0.0), k / 2.0], k, 1.0, 10.0)
polarLines = Sequence(Segment[(12.56637, 0.0), (4.0 * 3.141592653589793 + 5.0 * cos(a), 5.0 * sin(a))], a, 0.0, 6.021385919380437, 0.2617993877991494)
xLines = Sequence(Segment[(0.0, -(5.0) + k), (2.0 * 3.141592653589793, -(5.0) + k)], k, 0.0, 10.0, 0.5)
yLines = Sequence(Segment[(m, -(5.0)), (m, 5.0)], m, 0.0, 6.283185307179586, 0.7853981633974483)
aa = Segment((0.0, 0.0), (6.283185307179586, 0.0))
bb = Segment((0.0, -5.0), (0.0, 5.0))
e = Segment(A, C)
H: (4.0 * 3.141592653589793 + 5.0 * cos(x(C)), 5.0 * sin(x(C)))
J: (4.0 * 3.141592653589793 - 5.0 * cos(x(C)), -(5.0) * sin(x(C)))
h = Segment(J, H)
D: (x(C), f(x(C)))
text4: "θ = " + (x(C)) + " rad"
K = (12.566370614359172, 0)
L = (17.566370614359172, 0)
θ = Angle(L, K, H)
i = false
text21: "(" + (x(D)) + ", " + (y(D)) + ")"
k = false
l = false
j = false
m = false
text23: "r(θ) = " + a + " cos(" + b + " (θ - " + c + ")) + " + d
ff = Function(f(x), 0.0, x(C))
g = CurveCartesian(f(t) * cos(t) + 4.0 * 3.141592653589793, f(t) * sin(t), t, 0.0, 6.28319)
gg = CurveCartesian(f(t) * cos(t) + 4.0 * 3.141592653589793, f(t) * sin(t), t, 0.0, x(C))
E: (f(x(C)) * cos(x(C)) + 4.0 * 3.141592653589793, f(x(C)) * sin(x(C)))
text1: "f(x) = " + f
text43: "r(θ) = " + a + " sin(" + b + " (θ - " + c + ")) + " + d
text44: "r(θ) = " + a + "  θ  + " + d
F = Point(bb)
G = (6.270619999999998, 5.00964)
I = (6.270619999999998, -5.026099999999998)
M = Point(bb)
poly1, f_4, g_1, i_1, m_1 = Polygon(F, G, I, M)
N = (12.12702, -0.39422)
O = (17.13158, -0.28774)
text3: "(" + (y(D)) + " , " + (x(D)) + ")"
P = (12.55294, -0.02154)
Q = (17.58412, -0.15464)
p = Circle(P, Q)
R = (12.552939999999996, 9.401939999999998)
S = (21.976419999999994, 9.481799999999998)
T = (22.029659999999996, 6.819799999999999)
U = (12.552939999999996, 6.7931799999999996)
poly2, r, s, t, u = Polygon(R, S, T, U)
V = (-1.2894600000000003, 9.455179999999999)
W = (-1.2628400000000004, 6.979519999999999)
Z = (7.894439999999998, 6.979519999999999)
A_1 = (7.867819999999997, 9.481799999999998)
poly3, v, w, z, a_2 = Polygon(V, W, Z, A_1)
B_1 = (8.187259999999998, 9.934339999999999)
C_1 = (8.187259999999998, 6.04782)
D_1 = (12.339979999999997, 6.021199999999999)
E_1 = (12.339979999999997, 9.934339999999999)
poly4, b_1, c_1, d_1, e_1 = Polygon(B_1, C_1, D_1, E_1)
q = Vector(K, H)