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Copy of Copy of Polar Coordinates
Transcript of Copy of Copy of Polar Coordinates
Stolen by Mr. C Put more simply,
a point in polar coordinates
is represented as (r,θ).
θ is the angle that the point is away from the positive x-axis. Let's do some practice problems on finding the polar coordinate of a point. (0,0°) 1 2 3 A B 1) What are the coordinates of point A?
2) What are the coordinates of point B? Hint:
-The red lines here represent the different distances from the origin. Answers:
Point A has coordinates (2,90°)
Point B has coordinates (1,30°) http://en.wikipedia.org/wiki/Polar_coordinate_system http://www.texample.net/tikz/examples/polar-coordinates-template/ (2,90°) (1,30°) So how are polar coordinates and rectangular coordinates related? Aren't rectangular coordinates enough? Why do we need Polar coordinates? http://thetrig.blogspot.com/2011/11/polar-coordinates.html More practice... Now, the axis is not always labeled using degrees, we can also use radians to represent the angle of a point from the positive axis. A B C D What are the polar coordinates of points A, B, C, and D? A (4,3π/4)
D (3.5,7π/6) (5,135°) (6.5,75°) (5,30°) (1.5,210°) (4.5,180°) (7,210°) (3.5,270°) (5.5,330°) Commonly, rectangular coordinates are used for mapping because they are often the most useful, but not always. Sometimes, the data that you have collected would be more suited to be graphed in polar coordinates to be more easily understood. For example, if you are using radar, or trying to tell a ship where to travel. http://scidiv.bellevuecollege.edu/dh/ccal/CC9.1.pdf For example, the data collected here is obviously easier to interpret graphically using polar coordinates rather than rectangular coordinates. Note: the angle can be measured in either degrees or radians Now, so far we have only been labeling points using positive values...Is this the only way to do it? NO!! There are different ways to represent points using both positive and negative values. http://en.wikipedia.org/wiki/Polar_coordinate_system Take for example points A and B. Previously we said that their coordinates were A (2,90°) and B (1,30°). These points can also have coordinates A( -2,270°) and B (-1,210°). (1,210°) (-1,210°) This shows you that polar coordinates are not unique like rectangular coordinates. One point can be represented by many different sets of coordinates, just like A is both (2,90°) and (-2,270°) http://scidiv.bellevuecollege.edu/dh/ccal/CC9.1.pdf http://scidiv.bellevuecollege.edu/dh/ccal/CC9.1.pdf r is the distance the
point is from the origin (0,0). http://archives.math.utk.edu/visual.calculus/0/polar.6/ Rectangular coordinates, (x,y), represent how far over and how far up you go from the origin (0,0). Polar coordinates, (r,θ), represent how far away from the origin and what angle from the horizontal axis. How are the values of
x, r, y and θ related? Which coordinate system should we use? Graphing in polar coordinates Types of graphs Derivatives of polar equations Let's practice graphing... Areas/Integrals in polar coordinates Spherical Coordinates? Converting from Rectangular to Polar Converting from Polar to Rectangular Rectangular Coordinate (x,y) Polar Coordinate (r,θ) These equations for converting between the two coordinates systems can be found by solving the triangle on the left that represents the point (x,y) or (r,θ) Let's get some practice converting between the systems. The first two we will work through together and the second two you will do on your own. 1) 2) 3) 4) Convert (-1,-2) to a polar coordinate Convert (2,4) to a polar coordinate Convert (3,60°) to a rectangular coordinate Convert (1,145°) to a rectangular coordinate There is one issue
with these equations.
Depending on the quadrant our
point is in, the inverse tangent function can give us an incorrect value. www.mathsisfun.com/polar-cartesian-coordinates.html The figure on the left shows the
four quadrants of the plane. When
converting from rectangular to
polar coordinates, refer to this to
see which quadrant your point is
in and then the chart below to see what the appropriate tangent inverse value is for that point. www.mathsisfun.com/polar-cartesian-coordinates.html Since we are converting to polar we need to consider the tangent inverse value. Our point here is in the 3rd quadrant, and our table tells us that means to add 180° to the calculator value, so 63.4°+180°=243.4° . Therefore our point (-1,-2) in polar coordinates is (√(5),243.4°) (√(5),243.4°) Here, we do not have to consider the tangent inverse value so our (x,y) coordinate is (1.5,2.6) (1.5,2.6) It doesn't really matter. There are no strict rules saying that you must use one or the other in any situation. There are times when it is easier to use one over the other though... Polar: It's easier to use polar coordinates when measuring things like radar, or something involving rotations. Rectangular: It's easier to use rectangular coordinates when you are graphing things
as a function of time or your problem
involves translations. scidiv.bellevuecollege.edu/dh/ccal/CC9.1.pdf scidiv.bellevuecollege.edu/dh/ccal/CC9.1.pdf We can be given different things to be able to graph in polar coordinates... Given a table of data/points: Given a rectangular graph of magnitude as a function of angle: Given a formula: If we are given a set of points, then we can plot each point and "connect the dots" accordingly to get the graph Here, we can read coordinates of points from the rectangular graph and then do the same as if we were given the points. Essentially we are "wrapping" the rectangular graph around the "pole" of the polar system. scidiv.bellevuecollege.edu/dh/ccal/CC9.1.pdf We can have a calculator graph it for us or we can evaluate it at points and graph it by hand. www.math.uh.edu/~jiwenhe/Math1432/lectures/lecture12_handout.pdf www.math.uh.edu/~jiwenhe/Math1432/lectures/lecture12_handout.pdf www.math.uh.edu/~jiwenhe/Math1432/lectures/lecture12_handout.pdf www.math.uh.edu/~jiwenhe/Math1432/lectures/lecture12_handout.pdf www.math.uh.edu/~jiwenhe/Math1432/lectures/lecture12_handout.pdf www.math.uh.edu/~jiwenhe/Math1432/lectures/lecture12_handout.pdf Circle Line Petal Limacon r = 8 cos θ r = 3 sec θ r = 1/2 + cos θ r = cos 6θ First we pick some points and evaluate them By plotting a few of these points we see a circle starting to form The final graph looks like this The final graph will look like this First we pick some points and evaluate them By plotting a few of these points we see a line starting to form The final graph will look like this First we pick some points and evaluate them Here, we've plotted some points. It's harder to see what the final graph will look like but if you plot enough points eventually you will see the final outcome. The final graph will look like this Here, we've plotted some points. It's harder to see what the final graph will look like but if you plot enough points eventually you will see the final outcome. First we pick some points and evaluate them We know that we can take derivatives of equations in rectangular coordinates, but can we take derivatives of polar equations? Yes!! Polar equations, r = f(θ), are still in the x-y plane, where we have seen that x = rcosθ and y = rsinθ. So if we substitute the value for r, then x=f(θ)cosθ and y=f(θ)sinθ. So we can see that www.whitman.edu/mathematics/calculus/calculus_10_Polar_Coordinates,_Parametric_Equations_2up.pdf Second Derivatives Likewise, we can take second derivatives of polar equations. We know how to find dy/dx=y', it then follows that the second derivative is www.whitman.edu/mathematics/calculus/calculus_10_Polar_Coordinates,_Parametric_Equations_2up.pdf www.whitman.edu/mathematics/calculus/calculus_10_Polar_Coordinates,_Parametric_Equations_2up.pdf Just as we can calculate derivatives of polar equations, we can integrate as well to find areas bounded by these curves. We can use the sum on the left to approximate area and in the limit this sum becomes the integral on the right. httpwww.csupomona.edu/~ajm/materials/delsph.pdf http://www.analyzemath.com/antenna_tutorials/dipole_antennas.html Just as we can
a third dimension,
we can take polar
a third dimension
as well. The conversions of points from 3d rectangular to spherical coordinates and back are shown here on the left.