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7.7 Polar Coordinates 7.8 De Moivre's Theorem Name: _______________ Objectives: Students will be able to convert points and equations from polar coordinates to rectangular and vice versa. Students will be able to perform computations with complex numbers in polar form and use De Moivre's Theorem. When surveyors record the location of objects using distances and angles, they are using ____________________. Mar 237:45 AM Examples: Graph each point. 1.) P(3, 60o) 2.) Q(-1.5, 7π/6) 3.) R(-2,-135o) Mar 238:00 AM 1 Examples The polar coordinates of a point are given. Find its rectangular coordinates. (-4, 5π/4) Jan 226:26 PM Let the point P have polar coordinates (r, θ) and rectangular coordinates (x, y). Then: x = rcosθ y = rsinθ tan = y/x r = x2 + y2 Example Find the rectangular coordinates of the point with the given polar coordinates. P(3, 5π/6) Jan 226:30 PM 2 Example Polar coordinates of P are given. Find all of its polar coordinates. P = (1, -π/4) Jan 226:34 PM Examples Convert the rectangular equation to polar form. 1.) x + y = 1 2.) x2 + y2 - 4x + 6y = 12 Example Convert r = 4sinθ to rectangular form. Graph. Jan 226:48 PM 3 Examples: Graph each polar equation. Convert to rectangular form. 1.) r = 3 2.) θ=3π/4 Mar 238:07 AM Example The location of two ships from Mays Landing Lighthouse, given in polar coordinates, are (3 mi, 170o) and (5 mi, 150o). Find the distance between the two ships. Jan 226:52 PM 4 Symmetry To test for symmetry 1.) about the x-axis 2.) about the y-axis 3.) about the origin replace by Jan 237:57 AM Examples Use the polar symmetry tests to determine symmetry. 1.) r = 1 + 2cosθ 2.) r = 7sin3θ Jan 2310:26 AM 5 Types of Polar Curves Curve Rose Lemniscate Limacon spiral of Archimedes Polar Equation r = acosnθ r = asinnθ r2 = a2cos2θ r2 = a2sin2θ r = a ± bcosθ r = a ± bsinθ r = aθ n is a positive integer Cardioid General Graph Jan 231:17 PM Examples Graph the following polar curves. Identify the type of curve. 2.) r = 6 -5cosθ 1.) r = -3cos4θ Jan 232:15 PM 6 4.) r = 3 - 4sinθ 3.) r = 5 - 5sinθ 5.) r = θ/4 6.) r2 = 9cos2θ Jan 232:15 PM Complex Number and Complex Plane: Standard (rectangular) form of a complex number: Examples Plot u = 1 + 3i, v = 2i , w = -3 - 6i and x = -2 + 4i in the complex plane. Mar 119:40 PM 7 The __________ _____ or ______ of a complex number z = a + bi is z = a + bi = √a2 + b2 . In the complex plane, a + bi is the distance of a + bi from the origin. Example Let z = 2 - 4i. Find z . Jan 276:25 PM The _______________ _______ or (polar form) of the complex number z = a + bi is _________________, where a = _______, b = _______, r = __________, and tanθ = _____. The number r is the absolute value or modulus of z, and θ is the __________ of z. An angle θ for the trig form of z can always be chosen so that 0 ≤ θ ≤ 2π, although any angle coterminal with θ could be used. Consequently, the angle θ and the argument of a complex number z are not unique. Therefore, the trig form of a complex number is also not unique. Jan 276:28 PM 8 Examples Find the trig (polar) form of the complex number where the argument satisfies 0 ≤ θ < 2π. 1.) √3 + i 2.) 45o 4 z Jan 276:34 PM Examples Write the complex number in standard (rectangular) form a + bi. 1.) 8(cos210o + isin210o) 2.) √7(cos5π/6 + isin5π/6) Jan 276:35 PM 9 Product and Quotient of Complex Numbers Let z1 = r1(cosθ1 + isinθ1) and z2 = r2(cosθ2 + isinθ2). Then 1.) 2.) Jan 276:40 PM Examples Find the product and quotient of z1 and z2. 1.) z1 = √2(cos118o + isin118o) and z2 = 0.5[cos(-19o) + isin(-19o)]. 2.) z1 = 3 + 3i and z2 = 2 - 2√3 i Mar 119:59 PM 10 De Moivre's Theorem Let z = r(cosθ + isinθ) and let n be a positive integer. Then zn = [r(cosθ + isinθ)]n = rn(cosnθ + isinnθ). Examples 1.) Find [3(cos(3π/2) + isin(3π/2))]5. 2.) (3 + 4i)20 Jan 276:47 PM A complex number v = a + bi is an _____________ if vn = z. If z = 1, then vn = 1 and v is said to be an _____________. If z = r(cosθ + isinθ), then the n distinct complex numbers n ) ) √r cos(θ + 2πk) + isin(θ + 2πk) , where k = 0, 1, 2, ..., n-1, are n n the nth roots of the complex number z. Example Find the cube roots of 2(cosπ/4 + isinπ/4). Jan 276:54 PM 11 2.) Solve v3 = -1. Jan 277:01 PM 3.) Find the sixth roots of unity. Jan 277:05 PM 12 Assignments: 7.7: Pages 693-694: #7-21 odd (Just plot the points. Use the graph paper on the following page.) , 23-75 every other odd, 105-108 all 7.8: Pages 703-704: #7-59 every other odd, 69, 70 Jan 277:07 PM Mar 119:57 PM 13 Jan 227:02 PM Jan 227:02 PM 14