POLAR COORDINATES MIT – Polar Coordinates click PatrickJMT Polar coordinates – the Basics Graphing Polar Curve – Part 1 Graphing Polar Curve – Part 2 Areas and Polar Coordinates Student recommended videos

Cartesian coordinates One way to give someone directions is to tell them to go three blocks East and five blocks South ⇒ Cartesian coordinates Polar coordinates Another way to give directions is to point and say “Go a half mile in that direction ⇒ Polar coordinates Polar graphing is like the second method of giving directions. Each point is determined by a distance and an angle.

Polar coordinate system Polar coordinate system: a pole (fixed point) and a polar axis (directed ray with endpoint at pole).

Circle centered at the origin Line through the origin Some curves/areas are easier to describe with polar coordinates: Area ① ② ③

If the angle is measured in a clockwise direction, the angle is negative. The directed distance, r, is measured from the pole to point P. If point P is on the terminal side of angle θ, then the value of r is positive. If point P is on the opposite side of the pole, then the value of r is negative. More than one coordinate pair can refer to the same point. The angle, θ, is measured from the polar axis to a line that passes through the point and the pole. If the angle is measured in a counterclockwise direction, the angle is positive.

Example All of the polar coordinates of this point are:

Problem : P (x, y) = (1,  3). Express it in polar coordinates (r, θ) two different ways such that 0≤ θ < 2  (r, θ) = (2,  /3), (- 2, 4  /3). Problem : P(x, y) = (-4, 0). Express it in polar coordinates (r, θ) two different ways such that 0≤ θ < 2 . (r, θ) = (4,  ),(- 4, 0). Problem : P (x, y) = (-7, -7), express it in polar coordinates (r, θ) two different ways such that 0≤ θ < 2 . (r, θ) = (  98, 5  /4),(-  98,  /4). Problem : Given a point in polar coordinates (r, θ) = (3,  /4), express it in rectangular coordinates (x, y). (x, y) = (3√2/2, 3√2/2)

Problem : Transform the equation x 2 + y 2 + 5x = 0 to polar coordinate form. x 2 + y 2 + 5x = 0 r 2 + 5(r cos θ) = 0 r ( r + 5 cos θ) = 0 The equation r = 0 is the pole. Thus, keep only the other equation: r + 5 cos θ = 0 Problem : Transform the equation r = 4sin θ to Cartesian coordinate form. What is the graph? Describe it fully!!!

Problem : What is the maximum value of | r| for the following polar equations: a) r = cos(2 θ) b) r = 3 + sin(θ) c) r = 2 cos(θ) - 1 θ = n  /2 where n is an integer and | r| = 1 θ =  /2+2n  where n is an integer and | r| = 4 θ = (2n + 1)  where n is an integer and | r| = 3 Problem : Find the intercepts and zeroes of the following polar equations: a) r = cos(θ) + 1 b) r = 4 sin(θ) Polar axis intercepts: (r, θ) = (2, 2n  ),(0, (2n + 1)  ), where n is an integer. Line θ =  /2 intercepts: (r, θ) = (1,  /2 + n  ), where n is an integer. r = cos(θ) + 1 = 0 for θ = (2n + 1) , where n is an integer Polar axis intercepts: (r, θ) = (0, n  ) where n is an integer. Line θ =  /2 intercepts: (r, θ) = (4,  /2 +2n  ) where n is an integer. r = 4 sin(θ) = 0 for θ = n , where n is an integer.

TI – 84 plus Graphing polar equations example: graph the polar equation r = 1- sinθ 1.Hit the MODE key. 2.Arrow down to where it says Func and then use the right arrow to choose Pol. 3. Hit ENTER The calculator is now in parametric equations mode. 4. Hit the Y= key. 5. In the r 1 slot, type r = 1- sin(θ) Hit X,T,θ,n key for typing θ Press [WINDOW] and enter the following settings: θmin = 0 θmax = 2π θstep = π /24 Xmin = -3 Xmax = 3 Xscl = 1 Ymin = -3 Ymax = 1 Yscl = 1 With these settings the calculator will evaluate the function from θ = 0 to θ = 2 π in increments of π /24. Press [GRAPH].

ExampleSpiral of Archimedes: r = θ, θ ≥ 0 The curve is a nonending spiral Here it is shown in detail from θ = 0 to θ = 2π

Example θ0π/4π/3π/22 π/33 π/4π5 π/44 π/33 π/25 π/37 π/42 π r– 1– –0.41–1

convex limacon carotid limacon limacon with a dimple with an inner loop

Cardioids (Heart-Shaped): r = 1 ± cosθ, r = 1 ± sinθ

Flowers Petal Curve: r = cos 2 θ

Petal Curves: r = a cos n θ, r = a sin n θ r = sin 3θ r = cos 4 θ If n is odd, there are n petals. If n is even, there are 2n petals.

Tests for Symmetry: x-axis: If (r,  ) is on the graph,so is (r, -  ).

Tests for Symmetry: y-axis: If (r,  ) is on the graph,so is (r,  -  )or (-r, -  ).

Tests for Symmetry: origin: If (r,  ) is on the graph,so is (-r,  )or (r,  +  ).

Tests for Symmetry: If a graph has two symmetries, then it has all three: 

Try graphing this on the TI-89.

and now good luck Note that rather than trying to remember this formula it would probably be easier to remember how we derived it. First and second derivative of r = r(  ):

Example: Find the slope of a polar curve:

Area Inside a Polar Graph: For a very small , the curve could be approximated by a straight line and the area could be found using the triangle formula:  1 ≤  ≤  2

Example: Find the area enclosed by:

example: Find the area of the inner loop of r = cos θ

Length of a Polar Curve: