4'3 Polar Form

1
4.3 Polar Form de Moivres 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
• multiplying complex numbers causes their moduli
  to multiply, their arguments (angles) to add
• dividing complex numbers causes their moduli to
  divide,
• their arguments to subtract
• (r cos t i r sin t) n rn ( cos nt i
  sin nt )
• ie, de Moivres Theorem.

2

• To change z a ib, to polar form,
• plot z and find its polar coordinates
• distance r , ie modulus z a
  ib v 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 plot aib 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 (AB) i sin(AB)
• 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 Moivres 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 n negative 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.

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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.