10.4 Polar Coordinates and Polar Graphs Defining Polar Coordinates and Creating Graphs We can plot points using polar coordinates OR rectangular coordinates. Use the following relationships to convert back and forth between the two forms. tan θ =
y
y
sin θ =
x
r
cos θ =
x
r
Polar to Rectangular: Use x = r cos θ
y = r sin θ ⎛ π⎞
Convert the point ⎜ 3, ⎟ to rectangular coordinates. ⎝ 6⎠
Rectangular to Polar: Use tan θ =
y
r = x 2 + y 2 x
⎛ −1 −1 ⎞
Convert the point ⎜
to polar coordinates. ,
⎝ 2 2 ⎟⎠
Note: In rectangular coordinates there is one way to name any given point. In polar coordinates there are infinitely any ways to name a given point. Graphing polar equations Use your graphing calculator to create and compare graphs of these interesting equations. Use Radian mode with θ min = 0 , θ max = 2π , and θ step = .125π . Use a variety of values for the constants a and b. r = a cos θ
r = a sin θ
r = aθ
r = a + a cos θ
r = a + a sin θ
r = a + b cos θ
r = a + b sin θ
r = a sin(nθ ) for n an even integer
r = a sin(nθ ) for n an odd integer
r = a cos(nθ ) for n an even integer
r = a cos(nθ ) for n an odd integer
Note the direction in which the graph develops. Use the trace feature on your calculator to slowly observe the development of the graph. Note whether or not a particular curve is “traced” once or twice, etc. as you graph from 0 to 2π. Calculus with Polar Graphs Slope in Polar Form If f is a differentiable function of θ then the slope of the tangent line to the graph of r = f (θ ) at the point ( r, θ ) is given by dy dy / dθ
f (θ )cos θ + f '(θ )sin θ
=
=
dx dx / dθ − f (θ )sin θ + f '(θ )cos θ
Solutions to Solutions to If dy
dx
= 0 yield horizontal tangents, provided ≠ 0. dθ
dθ
dx
dy
= 0 yield vertical tangents, provided ≠ 0. dθ
dθ
dy
dx
and both equal 0, no conclusion can be drawn about tangent lines. dθ
dθ
Homework Examples: 1. Plot the point in polar coordinates and find the corresponding rectangular 5π ⎞
⎛
coordinates for the point. ⎜ −3, ⎟ ⎝
3⎠
2. Plot the point in rectangular coordinates and find two sets of polar coordinates for the point for 0 ≤ θ ≤ 2π . −6 3, 6 (
)
3. Convert the rectangular equation to polar form and sketch its graph. 2
2
x = 3 x + y = 25 6x − x 2 = y 2 4. Convert the polar equation to rectangular form and sketch its graph. r = 3sec θ r = −2 sin θ r 2 sin 2θ = 2 5. Use a graphing utility to graph the polar equation. Find an interval for θ over which the graph is traced only once. 5θ
r = sin
r = 4 + 3cos θ 2
6. Evaluate the derivative of r = 3 − 2 cos θ at θ = 0 and at θ = π / 2 . Check the graph to see if your answer makes sense. dy dy / dθ
f (θ )cos θ + f '(θ )sin θ =
=
dx dx / dθ − f (θ )sin θ + f '(θ )cos θ
7. Find any points on the graph having horizontal or vertical tangents. r = 3sin θ dy dy / dθ
f (θ )cos θ + f '(θ )sin θ
=
=
dx dx / dθ − f (θ )sin θ + f '(θ )cos θ
8. Determine any tangents at the pole for the polar equation: r = − sin 5θ Tangents at the poll occur when f (θ ) = 0 and f '(θ ) ≠ 0 . Graph the equation in your graphing calculator and see if your solution(s) make sense.