Graphing and Converting Polar and Rectangular Coordinates Butterflies Are Among the Most Celebrated of All Insects

Using Polar Coordinates

Graphing and converting polar and rectangular coordinates Butterflies are among the most celebrated of all insects. It’s hard not to notice their beautiful colors and graceful flight. Their symmetry can be explored with trigonometric functions and a system for plotting points called the polar coordinate system. In many cases, polar coordinates are simpler and easier to use than rectangular coordinates. We are going to look at a You are familiar with new coordinate system plotting with a rectangular called the polar coordinate system. coordinate system. The center of the graph is Angles are measured from called the pole. the positive x axis.

Points are represented by a radiusradius and an angle (r, )

To plot the point    5,   4  First find the angle

Then move out along the terminal side 5 The polar coordinate system is formed by fixing a point, O, which is the pole (or origin). The polar axis is the ray constructed from O. Each point P in the plane can be assigned polar coordinates (r, ). P = (r, )

O  = directed angle Polar Pole (Origin) axis

r is the directed distance from O to P.  is the directed angle (counterclockwise) from the polar axis to OP.

The grid at the left is a polar grid. The A typical angles of 30o, 45o, 90o, … are shown on the graph along with circles of radius 1, 2, 3, 4, and 5 units.

Points in polar form are given as (r,  ) where r is the radius to the point and  is the angle of the point. On one of your polar graphs, plot the point (3, 90o)?

The point on the graph labeled A is correct. A negative angle would be measured clockwise like usual.  3  3,   4  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.  2   4,   3  Polar coordinates can also be given with the angle in degrees.

90 (8, 210°) 120 60 135 45

150 30 (6, -120°) 180 0

330 210 (-5, 300°) 315 225 300 240 270 (-3, 540°) Graphing Polar Coordinates

  3  Now, try graphing  2 ,  .  4  A C Did you get point B? Polar points have a new aspect. A radius can be negative! A negative radius means to go in the exact opposite B direction of the angle. To graph (-4, 240o), find 240o and move 4 units in the opposite direction. The opposite direction is always a 180o difference. Point C is at (-4, 240o). This point could also be labeled as (4, 60o). Graphing Polar Coordinates

How would you write point A with a negative radius? A C A correct answer would be (-3, 270o) or (-3, -90o). In fact, there are an infinite number of ways to label a single polar point. B Is (3, 450o) the same point?

Don’t forget, you can also use radian angles as well as angles in degrees. On your own, find at least 4 different polar coordinates for point B. Plotting Points The point ( r ,  )  2,  lies two units from the pole on the  3  terminal side of the angle    .  3 2    3 2,   3 

 1 2 3 0

3,  3  4   3 4 3 units from 3 the pole 2

 2,  5 2  3  2,  2,  5  2, 24  2,   3   3   33   2,   3  additional ways  0 to represent the 1 2 3 point 2,   3  (r , ) r ,  2 n   (r , )  r ,   (2 n  1)   3 2

Notice unlike in the rectangular coordinate system, there are many ways to list the same point. Converting from Rectangular to Polar

Find the polar form for the rectangular point (4, 3).

To find the polar coordinate, we must calculate the radius and angle to the (4, 3) given point. r We can use our knowledge of right 3  triangle trigonometry to find the radius and angle. 4 r2 = 32 + 42 tan  = ¾ r2 = 25  = tan-1(¾) r = 5  = 36.87o or 0.64 rad

The polar form of the rectangular point (4, 3) is (5, 36.87o) Converting from Rectangular to Polar

In general, the rectangular point (x, y) is converted to polar form (r, θ) by:

1. Finding the radius r2 = x2 + y2 (x, y) r 2. Finding the angle y  tan  = y/x or  = tan-1(y/x) Recall that some angles require x the angle to be converted to the appropriate quadrant. Converting from Rectangular to Polar

On your own, find polar form for the point (-2, 3). 3 tan  (-2, 3) r2 = (-2)2 + 32  2 r2 = 4 + 9 1  3  2   tan   r = 13  2  r =   56.31o However, the angle must be in the second quadrant, so we add 180o to the answer and get an angle of 123.70o. The polar form is ( 13 , 123.70o) Converting from Polar to Rectanglar

Convert the polar point (4, 30o) to rectangular coordinates.

We are given the radius of 4 and angle of 30o. Find the values of x and y. Using trig to find the values of x and y, we know 4 that cos  = x/r or x = r cos  . Also, sin  = y/r or y y = r sin  . 30o x  r cos y  rsin x  o  o y  4sin30  x  4cos30    y  41  2 3 2 x  4  2 3 2 The point in rectangular form is: 2 3, 2 Converting from Polar to Rectanglar

On your own, convert (3, 5π/3) to rectangular coordinates.

We are given the radius of 3 and angle of 5π/3 or 300o. Find the values of x and y.

o x  r cos -60 y  rsin o  o  x  3cos300  y 3sin300    1 3 3 3 x  3  y 3 3  2 2 2 2

 3  3 3    The point in rectangular form is:  ,   2 2  The relationship between rectangular and polar coordinates is as follows. y The point (x, y) lies on a circle of radius r, therefore, r2 = x2 + y2.

(x, y) Definitions of (r, ) trigonometric functions r y sin  y r cos  x Pole  r x y tan  (Origin) x x

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 19 Coordinate Conversion cos  x r xr cos y sin  yr sin r y 2 2 2 tan  (Pythagorean Identity) x r x y Example: Convert the point 4,   into rectangular coordinates.  3  xrcos 4 cos4  1 2  3   2  3 yrsin 4 sin  4    2 3  3  2 xy,  2, 2 3

r x2  y 2 11 2  2  2 One set of polar coordinates is ( r , ) 2, .  4  Another set is (r , ) 2,5 .  4 

(3, 4) Based on the trig you know can you see r how to find r and ? 4  32  42  r 2 3 r = 5 4 tan  3 We'll find  in radians 1 4  polar coordinates are: (5, 0.93)   tan    0.93  3  Let's generalize this to find formulas for converting from rectangular to polar coordinates. (x, y) x2  y2  r 2 r  y r  x2  y2 x y tan  x  y    tan1   x  Now let's go the other way, from polar to rectangular coordinates. Based on the trig you know can you see    4,  how to find x and y?  4   x 4  y cos  4 x 4 4  2    x  4   2 2  2   y sin  rectangular coordinates are: 4 4  2    2 2, 2 2 y  4   2 2  2  Let's generalize the conversion from polar to rectangular coordinates.

x cos  r r,  r x  r cos  y x y sin  r

y  r sin Convert the rectangular coordinate system equation to a polar coordinate system equation. From conversions, how 2 2  r  x2  y2 r  3 x  y  9 was r related to x2 and y2 ? Here each r unit is 1/2 and we went out 3 and did all angles.

r must be  3 but there is no Before we do the conversion restriction on  so consider let's look at the graph. all values. Convert the rectangular coordinate system equation to a polar coordinate system equation. x2  4y What are the polar conversions substitute in for we found for x and y? x and y

x  r cos r cos 2  4r sin  y  r sin r 2 cos2   4r sin

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.

When trying to figure out the graphs of polar equations we can convert them to rectangular equations particularly if we recognize the graph in rectangular coordinates. r  7 We could square both sides r 2  49 Now use our conversion: r 2  x2  y2 x2  y2  49 We recognize this as a circle with center at (0, 0) and a radius of 7.

On polar graph paper it will centered at the origin and out 7 Let's try another:    Take the tangent of both sides 3 To graph on a polar plot   tan  tan   3 we'd go to where  3  and make a line. Now use our conversion: y tan  x y  3 Multiply both x sides by x

y  3x

We recognize this as a line with slope square root of 3. Let's try another: rsin  5

Now use our conversion: y  r sin

y  5

We recognize this as a horizontal line 5 units below the origin (or on a polar plot below the pole) Example: Convert the polar equation r  4sin  into a rectangular equation.

Polar form rr2  4 sin Multiply each side by r. x22 y4 y Substitute rectangular coordinates. x22 y 40 y 

xy2  242  Equation of a circle with center (0, 2) and radius of 2

Equations in rectangular form use variables (x, y), while equations in polar form use variables (r,  ) where  is an angle.

Converting from one form to another involves changing the variables from one form to the other.

We have already used all of the conversions which are necessary.

Converting Polar to Rectangular Converting Rectanglar to Polar

cos  = x/r x = r cos  sin  = y/r y = r sin  tan  = y/x x2 + y2 = r2 r2 = x2 + y2 Convert Rectangular Equations to Polar Equations

The goal is to change all x’s and y’s to r’s and  ’s. When possible, solve for r.

Example 1: Convert x2 + y2 = 16 to polar form. Since x2 + y2 = r2, substitute into the equation. r2 = 16 Simplify.

r = 4

r = 4 is the equivalent polar equation to x2 + y2 = 16 Convert Rectangular Equations to Polar Equations

Example 2: Convert y = 3 to polar form. Since y = r sin  , substitute into the equation. r sin  = 3 Solve for r when possible.

r = 3 / sin 

r = 3 csc  is the equivalent polar equation. Convert Rectangular Equations to Polar Equations

Example 3: Convert (x - 3)2 + (y + 3)2 = 18 to polar form. Square each binomial. x2 – 6x + 9 + y2 + 6y + 9 = 18 Since x2 + y2 = r2, re-write and simplify by combining like terms. x2 + y2 – 6x + 6y = 0 Substitute r2 for x2 + y2, r cos  for x and r sin  for y. r2 – 6rcos  + 6rsin  = 0 Factor r as a common factor. r(r – 6cos  + 6sin  ) = 0 r = 0 or r – 6cos  + 6sin  = 0 Solve for r: r = 0 or r = 6cos  – 6sin  Convert Polar Equations to Rectangular Equations

The goal is to change all r’s and  ’s to x’s and y’s.

Example 1: Convert r = 4 to rectangular form. Since r2 = x2 + y2, square both sides to get r2. r2 = 16 Substitute.

x2 + y2 = 16

x2 + y2 = 16 is the equivalent polar equation to r = 4 Convert Polar Equations to Rectangular Equations

Example 2: Convert r = 5 cos  to rectangular form.

Multiply both sides by r

r2 = 5 r cos 

Substitute x for r cos 

r2 = 5x

Substitute for r2.

x2 + y2 = 5x is rectangular form. Convert Polar Equations to Rectangular Equations

Example 3: Convert r = 2 csc  to rectangular form. Since csc ß = 1/sin, substitute for csc  .

1 r  2* sin Multiply both sides by 1/sin. 1 sin r sin  2* * sin 1 Simplify

y = 2 is rectangular form.  You will notice that polar equations have graphs like the following:  Hit the MODE key.  Now, use the right  Arrow down to where arrow to choose Pol. it says Func (short  Hit ENTER. (*It's easy for "function" which to forget this step, is a bit misleading but it's crucial: until since they are all you hit ENTER you functions). have not actually selected Pol, even though it looks like you have!)  The calculator is now in polar coordinates mode. To see what that means, try this.  Hit the Y= key. Note that, instead of Y1=, Y2=, and so on, you now have r1= and so on.  In the r1= slot, type 5-5sin(θ)  Now hit the familiar X,T,θ,n key, and you get an unfamiliar result. In polar coordinates mode, this key gives you a θ instead of an X.  Finally, close off the parentheses and hit GRAPH.  If you did everything right, you just asked the calculator to graph the polar equation r=5- 5sin(θ). The result looks a bit like a valentine.  The WINDOW options are a little different in this mode too. You can still specify X and Y ranges, which define the viewing screen. But you can also specify the θ values that the calculator begins and ends with.  Graph r = 3 sin 2θ  Enter the following window values: Θmin = 0 Xmin = -6 Ymin = -4 θmax = 2π Xmax = 6 Ymax = 4 Θstep = π/24 Xscl = 1 Yscl = 1  Graph: a. r = 2 cos θ b. r = -2 cos θ c. r = 1 – 2 cos θ Each polar graph below is called a Limaçon.

r 1 2cos r 1 2sin 3 3

–5 5 –5 5

–3 –3

r22 2 sin 2 r22 3 cos2

3 3

–5 5 –5 5

–3 –3

r  2cos3 r  3sin 4 3 3 a –5 5 –5 5 a

–3 –3 The graph will have n petals if n is odd, and 2n petals if n is even.

You can graph these on your calculator. You'll need to change to polar mode and also you must be in radians.

If you are in polar function mode when you hit your button to enter a graph you should see r1 instead of y1. Your variable button should now put in  on TI-83's and it should be a menu choice in 85's & 86's.