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Name:_______________________ Date assigned:______________ Band:________Precalculus | Packer Collegiate InstituteIntroduction to a New SpaceImagine a giant yellow flat sheet of paper. Like 10 miles by 10 miles… so gigantor that when you’re standing somewhere on it, you only see yellow paper for as far as the eye can see.That’s pretty gigantor.Now assume you wanted to describe where you are on this giant piece of paper to someone else. You have a friend somewhere else on the paper, probably miles away, and all they can see is yellow paper for as far as the eye can see. The problem is: if all of you can only see paper – and not each other – all around you… how do you find each other? Indeed, a problem. And it’s rather intractable.Nothing but vast paperSome basic information to guide you on the paperHowever with some basic markings on this vast expanse of paper, you can easily get on the phone with your friend and say: “hey, friend, I’m 2 miles east of the center and 3 miles south of the center.” And your friend can come find you at the coordinates (2,-3).We call this wonderful invention the rectangular coordinate system because it divides up the page into rectangles/squares. It helps us know where things are. The way it does this is by giving us a point of reference (the origin) and two axes with which to orient ourselves from the origin. However… it’s not the only way to describe a position on this sheet of paper. Your entire mathematical career, you’ve worked on the rectangular coordinate system. The (x,y) points! The distance left-right and the distance up-down. It’s our way of describing location in 2D space. However, there is another way. Today you’re going to be introduced to that other way.Section 1: BattleshipThere is a second way to determine where something is in 2D space, where you don’t even think in x-y coordinates. Open up the course conference and download the geogebra file titled “Battleship.” Save it to your desktop.In this file, you should see a battleship in the plane (in 2D space). You need to identify exactly where the battleship is.-2857573660You’re at the origin, using your periscope to find the battleship.Use the two controls at your command to find the battleship!You know you’ve successfully located it if you see the corny……displayed at the bottom of your screen.After you find the battleship, record its location below. Then close the file, open it up, and do it two additional times.Battleship 1Battleship 2Battleship 3DistanceAngleI said that there was a different way to identify something’s location in 2D space than giving the x-y coordinates! This is it!Look at the point below. You could identify it with the coordinates (4, 2). However, can you identify it with a distance and an angle?Point A190503175Distance:Angle:You’re going to practice this a bit more. Distance:Angle:Distance:Angle:Distance:Angle:(To be clear, those numbers are 3.41 and 2.69)Distance:Angle:To check your answers, download the geogebra file titled “Polar and Rectangular Coordinates.” Drag point A to the various locations above and check your answers!What if you have a general point on the rectangular coordinate system . What would the distance and angle be?Distance: Angle: Note: Think carefully about the formula you are deriving for your angle… Be sure to think about the quadrant!Section 2: NomenclatureThis new coordinate system is called the polar coordinate system. Instead of x and y-coordinates, we have two new coordinates: distance and angle. We designate the distance with (for radius, how far away from the origin we are!) and we designate the angle with .Plot the following coordinates on the polar graph:-24765081915Point P: Point Q: Point R: Point S: Point T*: Point U: Point V*: 17907008890*You’ve dealt with negative angles before. However, when you have a negative distance, it means go the same distance but in the opposite direction. Example: the point is plotted here: Section 3: Moving Backwards (Polar to Rectangular)Now I want you to work backwards. If I give you the distance and angle ( and ), I want you to give me the x-y coordinates.Polar coordinates: Rectangular coordinates: so… Polar coordinates: Rectangular coordinates: so… Polar coordinates: Rectangular coordinates: so… What if you have a general point on the polar coordinate system . What would the x- and y-coordinates be?x-coordinate:y-coordinate: Section 4: Some Problems(a) Are the points and different points? YES / NO(b) Are the points and different points?YES / NOWrite two different ways to represent the point .True or False: The polar coordinates of a point are unique.True or False: The rectangular coordinates of a point are unique.Home Enjoyment:Sullivan, Chapter 9.1 #31-38* (all), 39, 40, 42, 44, 45, 50, 54, 58, 60, 63, 66, 84 *Be sure to plot each point and then give your answers for (a)-(c) ................
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