Student recommended videos POLAR COORDINATES MIT Polar Coordinates

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

One way to give someone directions is to tell them to go three blocks East and five blocks South ⇒ Cartesian 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: a pole (fixed point) and a polar system axis (directed ray with endpoint at pole).

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

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. 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.

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) = (3√ 2/2, 3√ 2/2)

Problem : Transform the equation x 2 + y 2 + 5 x = 0 to polar coordinate form. x 2 + y 2 + 5 x = 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 = 4 sin 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) θ = n /2 where n is an integer and | r| = 1 b) r = 3 + sin(θ) θ = /2+2 n where n is an integer and | r| = 4 c) r = 2 cos(θ) - 1 θ = (2 n + 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 • Polar axis intercepts: (r, θ) = (2, 2 n ), (0, (2 n + 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 = (2 n + 1) , where n is an integer b) r = 4 sin(θ) • Polar axis intercepts: (r, θ) = (0, n ) where n is an integer. • Line θ = /2 intercepts: (r, θ) = (4, /2 +2 n ) 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 With these settings the calculator will evaluate the θstep = π /24 function from θ = 0 to θ = 2 π in increments of π /24. Xmin = -3 • Press [GRAPH]. Xmax = 3 Xscl = 1 Ymin = -3 Ymax = 1 Yscl = 1

Example Spiral 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 θ r 0 – 1 π/4 – 0. 41 π/3 0 π/2 1 2 π/3 3 π/4 2 2. 41 π 3 5 π/4 4 π/3 3 π/2 5 π/3 7 π/4 2. 41 2 1 0 – 0. 41 2π – 1

convex limacon carotid 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 2 n 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.

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

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 = 2 + 4 cos θ

Length of a Polar Curve: