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Polar Equations Project by Brenna Nelson, Stewart Foster, Kathy Huynh Converting From Polar to Rectangular Coordinates A point P in a polar coordinate system is represented by an ordered pair of numbers (r, θ) (r, θ): polar coordinates r: radius θ: angle measure (in radians) A point with the polar coordinates (r, θ) can also be represented by either of the following: (r, θ ± 2kπ) or (-r, θ + π + 2kπ) where k is any integer Polar coordinates of the pole are (0, θ) where θ can be any angle Polar to Rectangular Coordinates If P is a point with polar coordinates (x,y) of P are given by x = rcosθ y = rsinθ cosθ = x/r sinθ = y/r tanθ = y/x r² = x² + y² where r is the hypotenuse and x and y are the corresponding sides to the triangle Plug the values of r and θ into the x and y equations to find the values of the rectangular coordinates Converting from Polar to Rectangular Polar Coordinates (r, θ) Given: (6, π/6) r=6 θ = π/6 Use the equations x=rcosθ and y=rsinθ to find the values for x and y by plugging in the given values of r and θ Insert the values for r and x=rcosθ y=rsinθ x=(6)cos(π/6) y=(6)sin(π/6) θ into the equations x=(6) · ( 3 / 2 ) y=(6) · (1/2) Find the numerical values from solving the found x=3 3 y=3 equations (x, y) = (3 3, 3) The found values for x and y are the rectangular coordinates Rectangular to Polar Coordinates r = x2 y2 tanθ = y/x so θ = 1 tan ( y / x ) Plug the values of the x and y coordinates into the equations to find the values of the polar coordinates Steps for conversion: Step 1) Always plot the point (x,y) first Step 2) If x=0 or y=0, use your illustration to find (r, θ) polar coordinates Step 3) If x does not equal zero and y does not equal zero, then r= x 2 y 2 Step 4) To find θ, first determine the quadrant that the point lies in Converting from Rectangular to Polar Rectangular Coordinates (x, y) Given (2, -2) x=2 y = -2 By plugging the values of x and y into the polar coordinate equations r = x 2 y 2 and tan 1 , you can thus find the values of r and θ. 2 2 r = (2) ( 2) 4 4 8 2 2 1 1 ( 2 / 2) tan ( 1) / 4 θ = tan Polar coordinates (r, θ) = ( 2 2 , -π/4) Try these on your own: Convert r = 4sinθ from the polar equation the rectangular equation. Convert 4xy=9 from the rectangular equation to the polar equation. Solutions: Example 1 Convert r = 4sinθ from the polar equation the rectangular equation. r = 4sinθ r² = 4rsinθ r² = 4y x² + y² = 4y x² + (y² - 4y) = 0 x² + (y² - 4y + 4) = 4 x² + (y – 2)² = 4 Given equation Multiply each side by r y = rsinθ r² = x² + y² Equation of a circle Subtract 4y from each side Complete the square in y Factor y This is the standard form of the equation of a circle with center (0,2) and radius 2. Solutions: Example 2 Convert 4xy = 9 from the rectangular equation to the polar equation. Use x =rcosθ and y = rsinθ to substitute into the equation 4(rcosθ)(rsinθ) = 9 x = rcosθ, y =rsinθ 4r²cosθsinθ = 9 2r²(2sinθcosθ) = 9 Double Angle 2sinθcosθ = sin(2θ) Formula 2r²sin(2θ) = 9 This is the standard polar equation for the rectangular equation 4xy = 9 Polar Equations Limaçons: Gen. equation: (0 < a, 0 < b) r = a ± bcosθ r = a ± bsinθ Rose Curves: Gen. equation: r = a ± acos(nθ) r = a ± asin(nθ) (n petals if n is odd, 2n petals if n is even) Polar Equations Circles: Gen. equation: r=a r = cos(θ) Lemniscates: Gen. equation: r2 = a ± a2cos(2θ) r2 = a ± a2sin(2θ) How to Sketch Polar Equations Sketch the graph of the polar equation: r = 2 + 3cosθ The function is a graph of a limaçon because it matches the general formula: r = a ± bcosθ Method 1 r = 2 + 3cosθ 1. Convert the equation from polar to rectangular ~ x = rcosθ ~ y = rsinθ x = (2 + 3cosθ)cosθ x y = (2 + 3cosθ)sinθ 0 5 Substitute different values 6 3.982 for θ to find the remaining 1.75 3 coordinates 0 2 2 0 2.299 3.031 2 3 -.25 .433 6 .518 -.299 1 0 5 y Method 2 2. r = 2 + 3cosθ Substitute values of θ and use radial lines to plot points Use a number of radial lines to ensure that the entire graph of the polar function is sketched Radial line: the lines that extends from the origin, forming an angle equivalent to the radian value Ex. Because 2 = 90 , the radial line for 2 is… REMEMBER: draw arrows to show in which direction the polar function is being sketched Method 2 (con’t.) r = 2 + 3cosθ Method 2 is used to sketch the polar equation The work is shown below: r 0 5 6 4.598 3 3.5 2 2 3 .5 6 -.598 2 5 -1 Because you know that the equation is a limaçon, you can roughly sketch the rest of the graph. NOTE: this method is only an approximation; it should not be used for calculations. Method 3 r = 2 + 3cosθ 3. Using a calculator The 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 = 3cosθ, using Method 1 Graph the polar equation, r = 2, using Method 2 Solutions: Graph the polar equation, r = 3cosθ, using Method 1 x = 3cosθ(cosθ) y = 3cosθ(sinθ) x y 0 3 6 3 2.25 2 .75 0 0 1.299 1.299 6 3 .75 0 5 2 2.25 3 -1.299 -1.299 0 Graph the polar equation, r = 2, using Method 2 r 0 2 6 3 2 2 2 2 5 2 6 3 2 2 2 Finding Polar Intersection Points Method 1: Set equations equal to each other. Solve for θ. Method 2, for θ 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(θ) r = sin(θ) sin( ) cos( ) = cos( ) cos( ) tanθ = 1 θ = 45º , 225º 5 θ= and 4 8 Try these on your own: Find the intersection points of the equations using Method 1: r = 3 + 3sin(θ) r = 2 – cos(2 θ) Solutions: Find the intersection points of the equations using Method 1: r = 3 + 3sin(θ) r = 2 – cos(2θ) 3 + 3sinθ = 2 – cos(2θ) 1 + 3sinθ = −cos(2θ) 1 + 3sinθ = −2cos2θ +1 3sinθ + 2(1 – sin2θ) = 0 3sinθ + 2 – 2sin2θ = 0 Double Angle cos(2θ) = 2cos2θ + 1 Formula Trig Property cos2θ = 1 – sin2θ Factor Solutions: 3sinθ + 2 – 2sin2θ = 0 Factor 2sin2θ – 3sinθ – 2 = 0 (2sinθ + 1)(sinθ – 2) = 0 2sinθ = −1 sinθ = 2 Doesn’t exist sinθ = −1/2 θ= 5 3 and 6 Use unit circle to solve for θ Method 2 r = 1 + 3cosθ r=2 Bibliography Sullivan, 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.gif http://curvebank.calstatela.edu/index/limacon.gif http://curvebank.calstatela.edu/index/rose.gif http://www.libraryofmath.com/pages/graphing-polarequations/Images/graphing-polar-equations_gr_3.gif