Polar Coordinates and other cool stuff. but mostly just polar coordinates. project by Camila, Hannah, and Kristen O (the pole) (directed angle) r (directed distance) Polar axis P . (r, ø) Converting y x y } } Polar axis (x-axis) r } ø (x,y) (r,ø) The polar axis coincides with the x-axis.

r = x + y .

tan(ø)=y/x

cos(ø)=x/r

sin(ø)=y/r 2 2 2 Therefore: x = r cos(ø)

y = r sin(ø)

tan(ø) = y/x

r = sqrt(x + y ) 2 2 Learning to Plot and Read Sources because we'd be lost without them There are two items that make reading polar graphs easier.

1. Concentric circles

2. Radial lines pi/2 0 pi 3pi/2 pi/4 3pi/4 5pi/4 7pi/4 . . . . radial line concentric circles (3,pi/4) (-2,3pi/4) (1,5pi/4) (-4,7pi/4) (-2,-5pi/4) (3,-7pi/4) (-4,-pi/4) (1,-3pi/4) Practice because given our history in this class, we need it. Polar to Cartesian: (r,ø)=(2,pi/6) x = r cos(ø)

y = r sin(ø) x = 2 cos(pi/6)

y = 2 sin (pi/6) x = 2(sqrt(3)/2)

y = 2(1/2) x = sqrt(3)

y = 1 Cartesian to Polar: Cartesian coordinates: (sqrt(3),1) (y,x)=(1,-1) tan(ø) = y/x ø=3pi/4 r = (x +y ) 2 1/2 r = (2) 1/2 Polar coordinates: (sqrt(2),3pi/4) we want to get to (x,y) we use these two formulas to get the x and y values insert r and theta into the equations remember remember the fifth of-- nah, just remember your unit circle! Fin. We want to get these points in terms of (r,theta) use this formula to find theta Again, unit circle. Stress on the unit. Stress on the circle. use good old Pythagorus' brain child to find r Voila! Polar to Cartesian { { Cartesian to Polar Yep. STEP TWO: STEP ONE: Grab a graphing calculator and turn it on. Prepare yourself. This is serious business. STEP FOUR: STEP THREE: Go to "MODE," make sure you're in "Radian" and switch to "Pol" from "Func" Go to WINDOW make these changes:

ømin/max/step= 0/360/.1308996...

Xmin/max/scl = -3/3/1

Ymin/max/scl = -2/2/1 STEP FIVE: This is more of a personal preference, but 2nd ZOOM and go to "AxesOff" Symmetry to the polar axis pi pi/2 3pi/2 0 polar axis 1 2 3 1 2 3 4 (3,pi/4) (3,-pi/4) or

(-3,(pi-pi/4)) r,ø r,-ø or -r,pi-ø Symmetric about ø=pi/2 1 2 3 4 1 2 3 4 1 2 3 4 pi/2 pi 3pi/4 0 (3,pi/4) (3,pi-(pi/4)) or

(-3,-pi/4) r,ø r,pi-ø or

-r,-ø Symmetry about the pole pi pi/2 0 3pi/2 1 2 3 4 1 2 3 4 (3,pi/4) (-3,pi/4) or

(3,pi+(pi/4)) r,ø -r,ø or

r,pi+ø Special graphs The three of us dedicated a large chunk of our time in class fooling usefully experimenting with our calculators making fun graphs using polar coordinates. We have made a list of some of our favorites and even written down a step by step process of how we make them together. I will call it... "Polar-curve-bonding-time-because-the-end-is-near." r=cos(1/7ø) theta is the same button you use to get x. r=cos(4/3ø) When making these graphs, they should be entered into the calculus like so: r1=cos(n/dø) For example: except this is actually a sin curve. Liars who produced this picture. from polar to cartesian and cartesian to polar Can I like, make my calculator do all that junk? Polar-curve-bonding-time-because-the-end-is-near 2 Polar coordinates are simply another way to determine a point in 2d space. We have been mighty good chums with the Cartesian System which involves our two friends, the x and y axes. In this method, we use a point O as our origin and call it the pole. From O, we construct a ray called the polar axis. Then, point P can be assigned as a polar cordinate via: 1. r = the directed distance from O to P, and

2. ø (theta) = the directed angle found by moving counterclockwise from the polar axis to the segment OP What are polar coordinates? Start. http://mathworld.wolfram.com/PolarCoordinates.html(general knowledge)

http://tutorial.math.lamar.edu/Classes/CalcII/PolarCoordinates.aspx (general knowledge)

http://www.math.tamu.edu/~fulling/coalweb/polar.htm (general knowledge)

http://www.docstoc.com/docs/6351757/Polar_coordinates (Navigation)

Finney, Swokowski, Best, and my dad's college calculs book

http://mapnavigation.net/polar-coordinates (navigation)

http://www-istp.gsfc.nasa.gov/stargaze/Skepl1st.htm

http://mathworld.wolfram.com/SemilatusRectum.html

http://www.mathwarehouse.com/ellipse/eccentricity-of-ellipse.php

http://www.math.uh.edu/~jiwenhe/Math1432/lectures/lecture12_handout.pdf

http://www.squarecirclez.com/blog/polar-coordinates-and-cardioid-microphones/2496 Polar navigation Applications φ φ ϑ ϑ ϴ Θ θ ø Oribts? Elliptical. Polar coordinates? Circular. Close enough, in my opinion. Close enough in Kepler's opinion, too, which is why he used them. Kepler's first law says the orbit of every planet is an ellipse with the sun at one of the foci. The orbit of the planet can be found using polar coordinates via this equation: r,ø are the heliocentric polar coordinates for the equation, p is the semi-latus rectum which pretty much means nothing to us because it has to do with a chord through a focus parallel to a conic section directrix. Yeah, that means nothing to us. And e is the eccentricity which is the measure of how ciruclar the ellipse is. (Eccentricity=0 means it is a circle) Ellipses, planets, n' stuff So, for all you future planetary orbit plotters out there... here you go! Limaçons: Spiral of Archimedes: r=ø fun stuff: change østep to 2 or 3, or 4. aka: snails r=1/2 cos(ø) Lemniscates r=sqrt(cos2ø) NOTE: SWITCH SIN FOR COS AND THE GRAPH WILL TILT BY PI/4 aka: ribbons Cardioids: aka: uzumaki aka: hearts r=1+cosø, r=1-cosø,

r=1-sinø, r=1+sinø, MUST MAKE WINDOW BIGGER. Keep 3:2 ratio. However, since most polar graphs are symmetrical in some way, shape, or form, symmetry is an easy way to get out of doing a lot of work. For instance here, we have symmetry about the polar axis. The only way to make a polar graph other than using a calculator (which we will get to later) is by plotting each individual point. Can you hear me now? Your microphone sure can. 'Cause it uses polar coordinates! Oh snap. A microphone's pick up patterns can be represented in polar coordinates. Omnidirectional microphone the lower the frequency, the more circular the pattern. Cardioid microphone r=sin(ø) this is a cardioid graph that we will look at later. It is uni-directional. r=1+sin(ø) r=ø Shotgun microphone "super-directional" will only pick up things in front of it with little spill over Bi-directional microphone Used for interviews. The reason is obvious. Polar navigation isn't 100% polar coordinates. Instead of using radians they use degrees. Instead of going in a counterclockwise manner, they go clockwise. However, the same idea of finding a point on a map applies. Figure out where you are, find the angular measurement from you to your destination, measure the distance, and you can set out. With a compass, preferably. Cami Cami Cami Kristen Kristen Kristen Hannah Hannah Hannah Kristen Kristen

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