11.1 Polar Coordinates and Graphs Objective To graph polar equations. To convert polar to rectangular To convert rectangular to polar

A polar coordinate pair One way to give someone directions is to tell them to go three blocks East and five blocks South. This is like x-y Cartesian graphing. Another way to give directions is to point and say “Go a half mile in that direction.” Polar graphing is like the second method of giving directions. Each point is determined by a distance and an angle. A polar coordinate pair O Initial ray determines the location of a point.

(r, ) The center of the graph is called the pole. Angles are measured from the positive x-axis. Points are represented by a radius and an angle (r, ) To plot the point First find the angle Then move out along the terminal side 5

A negative angle would be measured clockwise like usual. To plot a point with a negative radius, find the terminal side of the angle but then measure from the pole in the negative direction of the terminal side.

Therefore unlike in the rectangular coordinate system, there are many ways to express the same point.

r r = 5 Convert Cartesian Coordinates to Polar Coordinates (5, 0.93) Let's take a point in the rectangular coordinate system and convert it to the polar coordinate system. Based on the trig you know can you see how to find r and ? (3, 4) r 4  3 r = 5 We'll find  in radians polar coordinates are: (5, 0.93)

Let's generalize this to find formulas for converting from rectangular to polar coordinates. (x, y) x = r cos, y = r sin r y  x

Giving Alternative Forms for Coordinates of a Point (–1, 1) lies in quadrant II. Since one possible value for  is 135º. Also, Therefore, two possible pairs of polar coordinates are

Convert Polar Coordinates to Cartesian Coordinates Now let's go the other way, from polar to rectangular coordinates. 4 y x rectangular coordinates are:

Convert Polar Coordinates to Cartesian Coordinates Let's generalize the conversion from polar to rectangular coordinates. r y x

Graphs of Polar Equations Equations such as r = 3 sin , r = 2 + cos , or r = , are examples of polar equations where r and  are the variables. The simplest equation for many types of curves turns out to be a polar equation. Evaluate r in terms of  until a pattern appears.

Find a rectangular equation for r = 4 cos θ

substitute in for x and y Converting a Cartesian Equation to a Polar Equation What are the polar conversions we found for x and y? substitute in for x and y We wouldn't recognize what this equation looked like in polar coordinates but looking at the rectangular equation we'd know it was a parabola.

Solve the rectangular equation for y to get

Convert a Cartesian Equation to a Polar Equation 3x + 2y = 4 Let x = r cos  and y = r sin  to get Cartesian Equation Polar Equation

Convert r = 5 cos to rectangular equation. Since cos = x/r, substitute for cos. Multiply both sides by r, we have r2 = 5x x² + y² = 5x Substitute for r2 by x2 + y2, then This represents a circle centered at (5/2, 0) and of radius 5/2 in the Cartesian system.

Now you try: Convert r = 2 csc to rectangular form. Since csc = r/y, substitute for csc. Multiply both sides by y/r. Simplify, we have (a horizontal line) is the rectangular form. y = 2

For the polar equation convert to a rectangular equation, use a graphing calculator to graph the polar equation for 0    2, and use a graphing calculator to graph the rectangular equation. (a) Multiply both sides by the denominator.

Convert to a rectangular equation: Multiply both sides by the denominator.

The figure shows (c) Solving x2 = –8(y – 2) Square both sides. rectangular equation It is a parabola vertex at (0, 2) opening down and p = –2, focusing at (0, 0), and with diretrix at y = 4. The figure shows (c) Solving x2 = –8(y – 2) a graph with polar for y, we obtain coordinates.

Theorem Tests for Symmetry Symmetry with Respect to the Polar Axis (x-axis):

Theorem Tests for Symmetry

Theorem Tests for Symmetry Symmetry with Respect to the Pole (Origin):

The tests for symmetry just presented are sufficient conditions for symmetry, but not necessary. In class, an instructor might say a student will pass provided he/she has perfect attendance. Thus, perfect attendance is sufficient for passing, but not necessary.

Identify points on the graph:

Check Symmetry of: Polar axis: Symmetric with respect to the polar axis.

The test fails so the graph may or may not be symmetric with respect to the above line.

The pole: The test fails, so the graph may or may not be symmetric with respect to the pole.

Cardioids (a heart-shaped curves) are given by an equation of the form where a > 0. The graph of cardioid passes through the pole.

Graphing a Polar Equation (Cardioid) Example 3 Graph r = 1 + cos . Analytic Solution Find some ordered pairs until a pattern is found.  r = 1 + cos  0º 2 135º .3 30º 1.9 150º .1 45º 1.7 180º 60º 1.5 270º 1 90º 315º 120º .5 360º The curve has been graphed on a polar grid.

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.

This type of graph is called a limacon without an inner loop. Graph r = 3 + 2cos Let's let each unit be 1. Since r is an even function of , let's plot the symmetric points.

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. Ex: r = 1 – 2cosθ

Lemniscates are given by equations of the form and have graphs that are propeller shaped. Ex: r =

Graphing a Polar Equation (Lemniscate) Graph r2 = cos 2. Solution Complete a table of ordered pairs.  0º ±1 30º ±.7 45º 135º 150º 180º Values of  for 45º <  < 135º are not included because corresponding values of cos 2 are negative and do not have real square roots.

Rose curves are 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. Ex: r = 2sin(3θ) and r = 2sin(4θ)

Assignment P. 400 #1 – 11 odd ( a and b is enough but can do all if want more practice)

Polar coordinates can also be given with the angle in degrees. (8, 210°) 330 315 300 270 240 225 210 180 150 135 120 0 90 60 30 45 (6, -120°) (-5, 300°) (-3, 540°)

Give three other pairs of polar coordinates for the point P(3, 140º). (3, –220º) (–3, 320º) (–3, –40º)

Plot each point by hand in the polar coordinate system. Then determine the rectangular coordinates of each point. Since r is –4, Q is 4 units in the negative direction from the pole on an extension of the ray. The rectangular coordinates:

Graphing a polar Equation Using a Graphing Utility Solve the equation for r in terms of θ. Select a viewing window in POLar mode. The polar mode requires setting max and min and step values for θ. Use a square window. Enter the expression from Step1. Graph.