DIVISION OF COMPLEX NUMBERS IN POLAR FORM

Find the trigonometric form of the quotient.

Example 1  :

z₁  =  2(cos 30˚ + i sin 30˚)

z₂  =  3(cos 60˚ + i sin 60˚)

Solution  :

By using the z₁/z₂ formula, we get

z₁/z₂  =  (2/3)[cos (30˚ - 60˚) + i sin (30˚ - 60˚)]

z₁/z₂  =  (2/3)[cos (-30˚) + i sin (-30˚)]

Example 2  :

z₁  =  5(cos 220˚ + i sin 220˚)

z₂  =  2(cos 115˚ + i sin 115˚)]

Solution  :

By using the z₁/z₂ formula, we get

z₁/z₂  =  5/2[cos (220˚ - 115˚) + i sin (220˚ - 115˚)]

z₁/z₂  =  5/2(cos 105˚ + i sin 105˚)

Example 3  :

z₁  =  6(cos 5π + i sin 5π)

z₂  =  3(cos 2π + i sin 2π)

Solution  :

z₁/z₂  =  6/3[cos (5π - 2π) + i sin (5π - 2π)]

z₁/z₂  =  2(cos 3π + i sin 3π)

Example 4  :

z₁  =  cos (π/2) + i sin (π/2)

z₂  =  cos (π/4 + i sin (π/4)

Solution  :

z₁/z₂  =  cos (π/2 - π/4) + i sin (π/2 - π/4)

Taking the least common multiple, we get

z₁/z₂  =  cos ((2π - π)/4) + i sin ((2π - π)/4) 

z₁/z₂  =  cos (π/4) + i sin (π/4)

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