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Transcript of Polar Equations
Polar EquationsProject by Brenna Nelson,Stewart Foster, Kathy Huynh
Converting From Polar to Rectangular CoordinatesA point P in a polar coordinate system is represented by an ordered pair of numbers (r, )(r, ): polar coordinatesr: radius: angleA point with the polar coordinates (r, ) can also be represented by either of the following:(r, measure(in radians) 2k) or (-r, + + 2k) where k is any integerPolar coordinates of the pole are (0, ) where can be any angle
Polar to Rectangular CoordinatesIf P is a point with polar coordinates (x,y) of P are given byx = rcoscos = x/ry = rsinsin = y/rtan = y/x
r = x + y where r is the hypotenuse and x and y are the corresponding sides to the trianglePlug the values of r and into the x and y equations to find the values of the rectangular coordinates
Converting from Polar to RectangularPolar Coordinates (r, )Given: (6, /6)r = 6 = /6Use the equations x=rcos and y=rsin to find the values for x and y by plugging in the given values of r and x=rcosy=rsin x=(6)cos(/6)y=(6)sin(/6)x=(6) ( )y=(6) (1/2)x=3y=3
(x, y) = (3 , 3) Insert the values for r and into the equations Find the numerical values from solving the found equations The found values for x and y are the rectangular coordinates
Rectangular to Polar Coordinatesr = tan = y/x so =Plug the values of the x and y coordinates into the equations to find the values of the polar coordinatesSteps for conversion:Step 1) Always plot the point (x,y) firstStep 2) If x=0 or y=0, use your illustration to find (r, ) polar coordinatesStep 3) If x does not equal zero and y does not equal zero, then r=Step 4) To find , first determine the quadrant that the point lies in
Converting from Rectangular to PolarRectangular Coordinates (x, y)Given (2, -2)x = 2y = -2By plugging the values of x and y into the polar coordinate equations r = and , you can thus find the values of r and .r =
Polar coordinates (r, ) = ( , -/4)
Try these on your own:Convert r = 4sin from the polar equation the rectangular equation.
Convert 4xy=9 from the rectangular equation to the polar equation.
Solutions: Example 1Convert r = 4sin from the polar equation the rectangular equation.
r = 4sinGiven equationr = 4rsinMultiply each side by rr = 4yy = rsinx + y = 4yr = x + y Equation of a circlex + (y - 4y) = 0 Subtract 4y from each sidex + (y - 4y + 4) = 4 Complete the square in yx + (y 2) = 4Factor y
This is the standard form of the equation of a circle with center (0,2) and radius 2.
Solutions: Example 2Convert 4xy = 9 from the rectangular equation to the polar equation.Use x =rcos and y = rsin to substitute into the equation4(rcos)(rsin) = 9x = rcos, y =rsin4rcossin = 92r(2sincos) = 92rsin(2) = 9
This is the standard polar equation for the rectangular equation 4xy = 9 Double Angle Formula2sincos = sin(2)
Polar EquationsLimaons:Gen. equation:r = a bcos(0 < a, 0 < b)r = a bsin
Rose Curves:Gen. equation:r = a acos(n)r = a asin(n)(n petals if n is odd,2n petals if n is even)
Polar EquationsCircles:Gen. equation:r = ar = cos()Lemniscates:Gen. equation:r2 = a a2cos(2)r2 = a a2sin(2)
How to Sketch Polar EquationsSketch the graph of the polar equation:r = 2 + 3cosThe function is a graph of a limaon because it matches the general formula:r = a bcos
Method 1r = 2 + 3cosConvert the equation from polar to rectangular~ x = rcos ~ y = rsinx = (2 + 3cos)cosy = (2 + 3cos)sinSubstitute different valuesfor to find the remainingcoordinates
Method 2r = 2 + 3cosSubstitute values of and use radial lines to plot pointsUse a number of radial lines to ensure that the entire graph of the polar function is sketchedRadial line: the lines that extends from the origin, forming an angle equivalent to the radian valueEx. Because = 90 , the radial line for isREMEMBER: draw arrows to show inwhich direction the polar function isbeing sketched
Method 2 (cont.)r = 2 + 3cosMethod 2 is used to sketch the polar equationThe work is shown below:Because you know that the equation is a limaon, you can roughly sketch the rest of the graph.NOTE: this method is only an approximation; it should not be used for calculations.
Method 3r = 2 + 3cosUsing a calculatorThe easiest way to graph a polar equation is to just put the equation into the calculator
The method for graphing the polar equations with the calculator are explained in a later slide.
Try these on your own:Graph the polar equation, r = 3cos,using Method 1
Graph the polar equation, r = 2, usingMethod 2
Solutions:Graph the polar equation, r = 3cos,using Method 1x = 3cos(cos)y = 3cos(sin)
Graph the polar equation, r = 2, usingMethod 2
Finding Polar Intersection PointsMethod 1:Set equations equal to each other.Solve for .
Method 2, for values not on unit circle:Set calculator mode to polar.Graph equations.Find approximate intersection points using TRACE and then find exact intersection points using method 1.
Use Method 1 to find the intersection points for the two polar equations. r = cos()r = sin()
tan = 1 = 45 , 225 = and
Try these on your own:Find the intersection points of the equations using Method 1:r = 3 + 3sin()r = 2 cos(2 )
Solutions:Find the intersection points of the equations using Method 1:r = 3 + 3sin()r = 2 cos(2)
3 + 3sin = 2 cos(2)1 + 3sin = cos(2)1 + 3sin = 2cos2 +13sin + 2(1 sin2) = 03sin + 2 2sin2 = 0 Factor Double Angle Formulacos(2) = 2cos2 + 1
Solutions:3sin + 2 2sin2 = 02sin2 3sin 2 = 0(2sin + 1)(sin 2) = 02sin = 1 sin = 2sin = 1/2 = and
Use unit circleto solve for Doesnt exist Factor
Method 2r = 1 + 3cosr = 2
BibliographySullivan,Michael.Precalculus.Upper Saddle River:Pearson Education,2006. Foerster,Paul.Calculus: Concepts and Applications.Emeryville:Key Curriculum Press,2005. http://curvebank.calstatela.edu/index/lemniscate.gifhttp://curvebank.calstatela.edu/index/limacon.gifhttp://curvebank.calstatela.edu/index/rose.gifhttp://www.libraryofmath.com/pages/graphing-polar-equations/Images/graphing-polar-equations_gr_3.gif