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4.3 Polar Form & de Moivre’s Th Study Book App B, Sec 8.3

4.3 Polar Form & de Moivre’s Th Study Book App B, Sec 8.3. Objectives: know what the modulus ( r ) & argument ( t ) of z = a + ib are how to convert a + i b to polar form r (cos t + i sin t ) & vice versa

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4.3 Polar Form & de Moivre’s Th Study Book App B, Sec 8.3

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  1. 4.3 Polar Form & de Moivre’s ThStudy Book App B, Sec 8.3 Objectives: know • what the modulus (r) & argument (t) of z = a + ib are • how to convert a + i b to polar form r (cos t + i sin t ) & vice versa • multiplying complex numbers causes their moduli to multiply, & their arguments (angles) to add • dividing complex numbers causes their moduli to divide, & their arguments tosubtract • (r cos t + i r sin t) n = rn( cos nt + i sin nt ) ie, de Moivre’sTheorem.

  2. To change z = a + ib, to polar form, plot z and find its polar coordinates: distance r , ie modulus | z | = | a + ib | = √ a 2 + b 2 . and angle t, the argument. a + ib r t Then a = r cos t and b = r sin t . Hence a + ib = r cos t + i r sin t = r ( cos t + i sin t ) the polar form of a+ ib.

  3. To convert from standard form a + ib to polar form: First plota+ib to see its quadrant, then use Pythagoras & trig to get r & t. Example: - 1 - sqrt(3) i -1 Pythagoras gives r = sqrt(1 + 3) = 2 r Angle is (180 + 60) degrees = 4pi/3 Polar form is 2 ( cos 4pi/3 + i sin 4pi/3 ) -1 - sqrt3 i To convert from polar form to form a + ib, find a and b using a = r cos t , b = r sin t. Example: 3 (cos 3pi/4 + i sin 3pi/4) = 3 ( -1/sqrt2 + i 1/ sqrt 2) = - 3 / sqrt 2 + i 3/ sqrt 2

  4. Multiplying 2 numbers in polar form gives: R (cosA + i sinA) S(cos B + i sin B) = RS { cos (A+B)+ i sin(A+B) } ie their moduli multiply, but their arguments add! • Dividing 2 numbers in polar form gives … R (cosA + i sinA) / S (cos B + i sin B) = R/S { cos (A-B)+ i sin(A-B) } ie their moduli divide, but their arguments subtract! Hence squaring in polar form gives { R( cos  + i sin) } 2 = R2 { cos 2+ i sin 2 } And cubing in polar form gives ( cos  + i sin) 3 = cos 3+ i sin 3

  5. Then de Moivre’s Theorem (Th 8.5, p 449) follows, ( cos  + i sin)n = cos n+ i sin n True not only for positive integers n, but also for nnegative or rational. Example: (cos pi/6 + sin pi/6 )12 = cos 2pi + sin 2pi = 1 + 0 i = 1 Appendix B, Ex 3 & 4 (p 448 -9) use this rule. But doing them in Euler Form is easier. So master Sec 4.4 first, Euler Form, then come back to these. Nth roots (pp 450 - 2) are also easier done in Euler form.

  6. Homework Appendix B, p 438, Sec 8.3 • Master a few of Q 1 - 25 • Do more of these once you have mastered Section 4.4, converting z to Euler Form. • Write full solutions to Q 1 - 4, 5, 7, 9, 11, 37, 43.

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