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)

Simplify. Express answers in both polar form and in rectangular form. Match angle measurement units to the problem, where 0° < θ < 360° or 0 ≤ θ ≤ 2π

Example 5 :

6(cos 𝜋/2 + 𝑖 sin 𝜋/2) ∙ 4(cos 𝜋/4 + 𝑖 sin 𝜋/4)

Solution :

= 6(cos 𝜋/2 + 𝑖 sin 𝜋/2) ∙ 4(cos 𝜋/4 + 𝑖 sin 𝜋/4)

= 6(4) (cos (𝜋/2 + 𝜋/4) + 𝑖 sin (𝜋/2 + 𝜋/4))

= 24 (cos (2𝜋+𝜋)/4 + 𝑖 sin (2𝜋+𝜋)/4)

Polar form :

= 24 (cos (3𝜋/4) + 𝑖 sin (3𝜋/4))

Rectangular form :

= 24 [cos (𝜋-𝜋/4) + 𝑖 sin (𝜋-𝜋/4)]

= 24 [cos (𝜋/4) + 𝑖 sin (𝜋/4)]

= 24 [-√2/2 + 𝑖√2/2]

= 24 [(-√2 + 𝑖√2)/2]

= 12 (-√2 + 𝑖√2)

Example 6 :

5(cos 135 + 𝑖 sin 135) ∙ 2(cos 45 + 𝑖 sin 45)

Solution :

= 5(cos 135 + 𝑖 sin 135) ∙ 2(cos 45 + 𝑖 sin 45)

= 5(2) (cos (135 + 45) + 𝑖 sin (135 + 45))

= 10 (cos 180 + 𝑖 sin 180)

Polar form :

= 10 (cos 180 + 𝑖 sin 180)

Rectangular form :

= 10 (-1 + 𝑖 (0))

= 10(-1)

= -10

Example 7 :

3(cos (3𝜋/4) + 𝑖 sin (3𝜋/4)) ÷ (1/2)(cos 𝜋 + 𝑖 sin 𝜋)

Solution :

= 3(cos (3𝜋/4) + 𝑖 sin (3𝜋/4)) ÷ (1/2)(cos 𝜋 + 𝑖 sin 𝜋)

= 3/(1/2)[(cos (3𝜋/4) + 𝑖 sin (3𝜋/4) ÷ (cos 𝜋 + 𝑖 sin 𝜋)]

= 6[cos ((3𝜋/4) - 𝜋) + 𝑖 sin ((3𝜋/4) - 𝜋)]

= 6[cos ((3𝜋-4𝜋)/4) + 𝑖 sin ((3𝜋-4𝜋)/4)]

= 6[cos (-𝜋/4) + 𝑖 sin (-𝜋/4)]

Polar form :

6[cos (𝜋/4) - 𝑖 sin (𝜋/4)]

Rectangular form :

= 6[cos (𝜋/4) - 𝑖 sin (𝜋/4)]

= 6[(√2/2) - 𝑖 (√2/2)]

= 6[(√2 - 𝑖 √2)/2]

= 3(√2 - 𝑖 √2)

Example 8 :

3(cos (𝜋/6) + 𝑖 sin (3𝜋/4)) ÷ 4(cos 2𝜋/3 + 𝑖 sin 2𝜋/3)

Solution :

= 3(cos (𝜋/6) + 𝑖 sin (3𝜋/4)) ÷ 4(cos 2𝜋/3 + 𝑖 sin 2𝜋/3)

= (3/4)(cos ((𝜋/6)-(2𝜋/3)) + 𝑖 sin ((𝜋/6)-(2𝜋/3)))

= (3/4)[cos ((𝜋/6)-(4𝜋/6)) + 𝑖 sin ((𝜋/6)-(4𝜋/6))]

= (3/4)[cos (𝜋-4𝜋)/6 + 𝑖 sin (𝜋-4𝜋)/6)]

= (3/4)[cos (-3𝜋/6) + 𝑖 sin (-3𝜋/6)]

= (3/4)[cos (-𝜋/2) + 𝑖 sin (-𝜋/2)]

Polar form :

= (3/4)[cos (𝜋/2) - 𝑖 sin (𝜋/2)]

Rectangular form :

= (3/4)[cos (𝜋/2) - 𝑖 sin (𝜋/2)]

= (3/4)[0 - 𝑖 (1)]

= (3/4)(-1)

= -3/4

Example 9 :

4(cos (9𝜋/4) + 𝑖 sin (9𝜋/4)) ÷ 2(cos 3𝜋/2 + 𝑖 sin 3𝜋/2)

Solution :

= 4(cos (9𝜋/4) + 𝑖 sin (9𝜋/4)) ÷ 2(cos 3𝜋/2 + 𝑖 sin 3𝜋/2)

= (4/2)[cos ((9𝜋/4)-(3𝜋/2)) + 𝑖 sin ((9𝜋/4)-(3𝜋/2))]

= 2[cos ((9𝜋-3𝜋)/4)) + 𝑖 sin ((9𝜋-3𝜋)/4))]

= 2[cos (6𝜋/4) + 𝑖 sin (6𝜋/4)]

= 2[cos (3𝜋/2) + 𝑖 sin (3𝜋/2)]

Polar form :

= 2[cos (3𝜋/2) + 𝑖 sin (3𝜋/2)]

Rectangular form :

= 2[cos (3𝜋/2) + 𝑖 sin (3𝜋/2)]

= 2[0 + i(-1)]

= 2(-i)

= -2i

Example 10 :

6(cos (3𝜋/4) + 𝑖 sin (3𝜋/4)) ÷ 2(cos 𝜋/4 + 𝑖 sin 𝜋/4)

Solution :

= 6(cos (3𝜋/4) + 𝑖 sin (3𝜋/4)) ÷ 2(cos 𝜋/4 + 𝑖 sin 𝜋/4)

= 6/2[cos ((3𝜋/4) - (𝜋/4)) + 𝑖 sin ((3𝜋/4) - (𝜋/4))]

= 3[cos (3𝜋 - 𝜋)/4 + 𝑖 sin (3𝜋 - 𝜋)/4]

= 3[cos (2𝜋/4) + 𝑖 sin (2𝜋/4)]

= 3[cos (𝜋/2) + 𝑖 sin (𝜋/2)]

Polar form :

= 3[cos (𝜋/2) + 𝑖 sin (𝜋/2)]

Rectangular form :

= 3[cos (𝜋/2) + 𝑖 sin (𝜋/2)]

= 3[0 + i(1)]

= 3i

Example 11 :

(1/2)(cos (𝜋/3) + 𝑖 sin (𝜋/3)) ÷ 3(cos 𝜋/6 + 𝑖 sin 𝜋/6)

Solution :

= (1/2)(cos (𝜋/3) + 𝑖 sin (𝜋/3)) ÷ 3(cos 𝜋/6 + 𝑖 sin 𝜋/6)

= ((1/2)/3)[cos ((𝜋/3)-(𝜋/6)) + 𝑖 sin ((𝜋/3)-(𝜋/6))]

= (1/6)[cos ((2𝜋-𝜋)/6)) + 𝑖 sin ((2𝜋-𝜋)/6))]

= (1/6)[cos (𝜋/6) + 𝑖 sin (𝜋/6)]

Polar form :

= (1/6)[cos (𝜋/6) + 𝑖 sin (𝜋/6)]

Rectangular form :

= (1/6)[cos (𝜋/6) + 𝑖 sin (𝜋/6)]

= (1/6) [√3/2 + i(1/2)]

= (1/6)[(√3 + i)/2]

= (√3 + i)/12

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