(r,  )

You are familiar with plotting with a rectangular coordinate system. We are going to look at a new coordinate system called the polar coordinate system.

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,  ) radiusangle 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.

Let's plot the following points: Notice unlike in the rectangular coordinate system, there are many ways to list the same point.

Do Now: look at the graph’s xy axis Plot the point: (3,4) on a polar graph even though it is a “rectangular point” Converting from Rectangular to Polar coordinates

Let's take a point in the rectangular coordinate system and convert it to the polar coordinate system. (3, 4) r  Based on the trig you know can you see how to find r and  ? 4 3

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

When converting from Rectangular to Polar, always know what quadrant you are in! Plot the point, (-5,-12) Quadrant III Find the angle to the nearest degree

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

If we use the formula, we get r=13 and But, in the third quadrant, the angle would be Approximately 247 degrees So the conversion to polar is…(13,247)

Now plot the rectangular point (-3,0) Just by inspection, find the radius and The angle! Quadrantal points don’t need the conversions.

rectangular point (-3,0) is Just by inspection, find the radius and The angle!

Do Now: Graph the point:.

Converting from polar to rectangular points Make a guess as to the approximate rectangular point (x,y), that would convert from the do now.

Let's generalize the conversion from polar to rectangular coordinates. Then find the exact rectangular coordinates For the point you just plotted. r y x

Now let's go the other way, from polar to rectangular coordinates. Based on the trig you know can you see how to find x and y? rectangular coordinates are: 4 y x

330  315  300  270  240  225  210  180  150  135  120  00 90  60  30  45  **Be sure the calculate the cos and sin before multiplying by the radius in order to get exact values for these: (8, 210°) (6, -120°) (-5, 300°) (-3, 540°)

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

CONVERTING POLAR EQUATIONS TO RECTANGULAR: Do Now: Graph the equation to see what it looks like. Copy these steps: 1.Eliminate all cos and sin 2.Eliminate r 3.Simplify to recognize

CONVERTING POLAR EQUATIONS TO RECTANGULAR:

multiply both sides by r to get all “r” together

CONVERTING POLAR EQUATIONS TO RECTANGULAR: Now Simplify: Remember “Complete the square?”

We need algebra to recognize what we have: This is a circle, radius =2, center: (0,2) X 2 + (y 2 -4y + ___)=0 + ____

Example 2 use the conversions:

Circle: Center: (4,-2) radius =

Ex. 3:

Convert the rectangular coordinate system equation to a polar coordinate system equation. r must be  3 but there is no restriction on  so consider all values. Here each r unit is 1/2 and we went out 3 and did all angles. Before we do the conversion let's look at the graph.

Convert the rectangular coordinate system equation to a polar coordinate system equation.  r must be  3 but there is no restriction on  so consider all values. Here each r unit is 1/2 and we went out 3 and did all angles. Before we do the conversion let's look at the graph.

Convert the rectangular coordinate system equation to a polar coordinate system equation. substitute in for x and y What are the polar conversions we found for x and y?

Convert the rectangular coordinate system equation to a polar coordinate system equation. 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. What are the polar conversions we found for x and y?

Convert the rectangular coordinate system equation to a polar coordinate system equation. substitute in for x and y What are the polar conversions we found for x and y?

Convert to polar form: Replace x and y with the conversions above Solve for r when possible.

Convert to polar form:

This is one point Graph the result…..

Convert the rectangular coordinate system equation to a polar coordinate system equation. substitute in for x and y What are the polar conversions we found for x and y?

Convert the rectangular coordinate system equation to a polar coordinate system equation. substitute in for x and y What are the polar conversions we found for x and y?

Turn to page 560 for a library of polar functions Lets graph a circle first

. Try:

Let's let each unit be 1/4. Let's plot the symmetric points This type of graph is called a cardioid.

Let's let each unit be 1. Let's plot the symmetric points This type of graph is called a limacon without an inner loop.

Equations of limacons without inner loops would look like one of the following: r = a +b cos  r = a +b sin  r = a - b cos  r = a - b sin  where a > 0, b > 0, and a > b These graphs DO NOT pass through the pole.

Let's let each unit be 1/2. Let's plot the symmetric points This type of graph is called a limacon with an inner loop.

Equations of limacons with inner loops would look like one of the following: r = a +b cos  r = a +b sin  r = a - b cos  r = a - b sin  where a > 0, b > 0, and a < b These graphs will pass through the pole twice.

Let's let each unit be 1/2. Let's plot the symmetric points This type of graph is called a rose with 4 petals.

Equations of rose curves would look like one of the following: r = a cos(n  ) r = a sin(n  ) Where n even has 2n petals and n odd has n petals (n  0 or  1) Complete table and graph #1 on pg. 548

15 degrees 30 degrees 15 30

Limacon With Inner Loop made with TI Calculator Rose with 7 petals made with graphing program on computer

Examples of Polar Curves