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Graphing Points on Polar Paper POLE: the common centre, O, of a series of concentric circles POLAR AXIS: a horizontal ray drawn from the pole in a positive direction. Polar Coordinates POLAR COORDINATES - consist of an ordered pair, P where is the distance from the pole to point P, and is the measure of the angle formed by the polar axis and the terminal arm, OP. Plot on polar paper. (+,+) An angle rotated counterclockwise is positive. An angle rotated clockwise is negative. (+,-) (2,-60) (-,+) (-,-) http://www.analyzemath.com/polarcoordinates/plot_polar_coordinates.html Converting from Rectangular to Polar Form Express (3,4) as a polar coordinate. (3,4) First - determine r. How can we determine r? Converting Between Rectangular Form and Polar Form (3,4) r To understand the relationship between rectangular coordinates, (x, y) and polar coordinates, (r, θ) Pythagorean Theorem r 4 3 Now we need to determine 5 4 3 So the rectangular coordinate (3,4) would be (5, 53 )in polar form. (3,4) = (5, 53 ) Quadrants 2 1 3 4 Express ( -4, -4) in polar form. ­4 (-4,-4) in rectangular form is (5.66, 225 ) in polar form. ­4 (­4,­4) BUT is not the whole rotation angle. It is the reference or related angle. (5.66, 225 ) = + 180 = (45 + 180) = 225 So (­4,­4) in polar form is (5.66, 225 ) Express (8,-15) in polar form. In which quadrant would (8, -15) be found? What about points in the other two quadrants? Express the (-5,12) in polar form. First find r (8,-15) x=8 y = -15 (­5,12) 2nd quadrant 4th r Now find Now to find the angle = 360 - 62 = 298 = 180 ­ 67.4 = 112.6 (­5, 12 ) is ( 13, 112.6 ) in polar form. (8,-15) = (17, 298 ) Converting from Polar Form to Rectangular Form (4, 30 ) To find x 4 y Converting from Polar Form to Rectangular Form 30 x To find y (4, 150 ) x= ­3.46 (-3.46, 2) Does this make sense? So ( 4, 30 ) in polar form is (3.46, 2) in rectangular form. first quadrant Polar Form of a Complex Number Converting between Rectangular (complex #) and Polar Complex number Consider the Argand plane which is used to plot complex numbers of the form a + bi. The point is plotted as (a,b) as opposed to (x,y) Therefore a= where r cos a + bi a = r cos b = r sin + ri sin This can be factored as r(cos + i sin ) This is usually shortened to r cis 4 cos 120 + 4 i sin 120 b= 4 cis 120 polar form of a complex number Convert the complex number -6 + 8i into polar form. Convert 4-3i into polar form Find r 4th quadrant ALWAYS CONSIDER THE QUADRANT Now find In the Argand plane (-6+8i is in quadrant 2) Find 2nd quadrant 4-3i would be in the 4th quadrant. 360-37=323 -6 + 8i would be 10 cis 127 in polar form 4-3i would be 5 cis 323 in polar form. Convert to rectangular form. Convert 7cis 120 into rectangular form b = -2.5 -3.5+6.1i -4.3 - 2.5i Multiplying Complex Numbers in Polar Form Example 1: convert to polar form first REVIEW Convert into rectangular form a=5cos 300 a= 2.5 b=5sin300 b= ­4.33 (1+i)(-2+2i) -2+2i -2i +2i2 -2-2 -4 Quadrant 1 Quadrant 2 1+i-2 + 2i 2.5­4.3i In polar form 4cis180 Convert into polar form 4cis 180 a+bi 10cos 233.1 +10isin233.1 Multiply the radii, or moduli (the r values), and add the angles, or arguments. What is 4cis 180 in rectangular form? a=4cos 180 b=4sin 180 a=-4 b=0 a+bi=-4 This checks EXAMPLE 2: On Handout (√2cis45)(2√2cis135) illustrated on previous pages 10√3+10i illustrated on previous page 3+11i (√13cis56.3)(√10cis18.4) √130cis74.7 Multiply the radii, or moduli (the r values), and add the angles, or arguments. 20cis30 a=20cos 30 b=20sin 30 a=17.32 b=10 17.32+10i This checks Multiply the radii, or moduli (the r values), and add the angles, or arguments. DIVIDING COMPLEX NUMBERS IN POLAR FORM Work it out in rectangular form. Work it out in polar form. To multiply complex numbers in polar form: Quadrant 2 r1r2cis (α+β) Multiply the radii, or moduli (the r values), and add the angles, or arguments. NOTE When dividing complex numbers in polar form,divide the moduli and subtract the arguments. Now convert back to rectangular: This checks with -0.5i r1 cis(α-β) r2 √2cis45 2√2cis135 -0.2i 0.9+0.7i When dividing complex numbers in polar form,divide the moduli and subtract the arguments. √13cis56.3 √10cis18.4 1.14cis37.9 When dividing complex numbers in polar form,divide the moduli and subtract the arguments. #6 page 302 Complete for homework To multiply out using complex numbers is time consuming. Is there another way? Evaluate by converting to polar form: #6 page 302 (1+ i)8 4√2 cis 135 2cis 300 10cis 210 Is there a faster way to solve this question? Now evaluate using the rules for multiplying polar coordinates Now convert back to rectangular complex form 2√2cis 315 34.64 ­20i Rectangular Form (1+ i)8 Now let's apply DeMoivre's Theorem to this question. DeMoivre's Theorem note complex number must be in polar form (rcisθ)n = rn cis nθ Example 1: (4cis225)3 4 cis(3x225) 64cis 675 675-360 64cis 315 First convert to polar form r = √2 θ=45 (√2 cis 45)8 Apply DeMoivre's Theorem 3 Example 2 (3cis50)4 34 cis (4x50) 81 cis 200 (√2)8 cis ( 8 x 45) 16 cis 360 Convert back to rectangular complex form a = 16 cos 360 b = 16 sin 360 a = 16 b=0 16 (1+i)8 = 16 Powers of Complex Numbers Example 2: Example 1 (-8+15i)2 First convert to polar form (2nd quadrant) (3+4i)4 Convert to polar form (17 cis 118.07)2 (5cis 53.13)4 Apply DeMoivre's Theorem 2 17 cis(2x118.07) 289cis236.14 Now convert back to rectangular complex form a = 289 cos 236.14 a=-161.02 (-8+15i)2 b = 289sin 236.14 b=-239.99 = -161.02 -239.99i THE END Apply DeMoivre's Theorem 54 cis(4x53.13) 625 cis 212.52 Convert back to complex form a=625cos212.52b=625sin 212.52 a=-526.94b=-336 (3+4i)4 = -526.94 - 336i