NZ Level 8 (NZC) Level 3 (NCEA) [In development]
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Multiplication of Numbers in Polar Form
Lesson

Using Polar or Modulus Argument forms, multiplication can be distilled down to a very simple process.  

For example, suppose that $z$z$_1=\left(r_{1,}\theta_1\right)$1=(r1,θ1) and $z_2=\left(r_2,\theta_2\right)$z2=(r2,θ2) , then

$z_1z_2$z1z2 $=$= $\left(r_{1,}\theta_1\right)\left(r_2,\theta_2\right)$(r1,θ1)(r2,θ2)
  $=$= $r_1\operatorname{cis}\theta_1r_2\operatorname{cis}\theta_2$r1cisθ1r2cisθ2
  $=$= $r_1\left(\cos\theta_1+i\sin\theta_1\right)\times r_2\left(\cos\theta_2+i\sin\theta_2\right)$r1(cosθ1+isinθ1)×r2(cosθ2+isinθ2)
  $=$= $r_1r_2\left(\cos\theta_1+i\sin\theta_1\right)\left(\cos\theta_2+i\sin\theta_2\right)$r1r2(cosθ1+isinθ1)(cosθ2+isinθ2)
  $=$= $r_1r_2\left(\left(\cos\theta_1\cos\theta_2-\sin\theta_1\sin\theta_2\right)+\left(\sin\theta_1\cos\theta_2+\cos\theta_1\sin\theta_2\right)i\right)$r1r2((cosθ1cosθ2sinθ1sinθ2)+(sinθ1cosθ2+cosθ1sinθ2)i)
   

From here we use what is called the double angle formula,

we can simplify this to

$z_1z_2$z1z2 $=$= $r_1r_2\left(\cos\left(\theta_1+\theta_2\right)+i\sin\left(\theta_1+\theta_2\right)\right)$r1r2(cos(θ1+θ2)+isin(θ1+θ2))

What's happened here is that we have multiplied the moduli and added the arguments

For division, it can be shown that it simplifies to

$\frac{z_1}{z_2}$z1z2 $=$= $\left(r_{1,}\theta_1\right]\left[r_2,\theta_2\right)$(r1,θ1][r2,θ2)
  $=$= $\frac{r_1\operatorname{cis}\theta_1}{r_2\operatorname{cis}\theta_2}$r1cisθ1r2cisθ2
  $=$= $\frac{r_1}{r_2}\operatorname{cis}\left(\theta_1-\theta_2\right)$r1r2cis(θ1θ2)
     

What's happened here is that we have divided the moduli and subtracted the arguments

Activity

Pick any complex number, plot it on an Argand Plane.

a) Multiply it repeatedly by $i$i, plotting the result each time.

b) Multiply it repeatedly by $-i$i, plotting the result each time.

Is there a pattern? 

Generalise the result.

See here for a solution

 

Worked Examples

Question 1

Answer the following.

  1. Which is true?

    When multiplying two complex numbers in trigonometric form, we add their absolute values and add their arguments.

    A

    When multiplying two complex numbers in trigonometric form, we multiply their absolute values and add their arguments.

    B

    When multiplying two complex numbers in trigonometric form, we add their absolute values and multiply their arguments.

    C

    When multiplying two complex numbers in trigonometric form, we multiply their absolute values and multiply their arguments.

    D

    When multiplying two complex numbers in trigonometric form, we add their absolute values and add their arguments.

    A

    When multiplying two complex numbers in trigonometric form, we multiply their absolute values and add their arguments.

    B

    When multiplying two complex numbers in trigonometric form, we add their absolute values and multiply their arguments.

    C

    When multiplying two complex numbers in trigonometric form, we multiply their absolute values and multiply their arguments.

    D
  2. Which is true?

    When dividing two complex numbers in trigonometric form, we subtract their absolute values and divide their arguments.

    A

    When dividing two complex numbers in trigonometric form, we subtract their absolute values and subtract their arguments.

    B

    When dividing two complex numbers in trigonometric form, we divide their absolute values and divide their arguments.

    C

    When dividing two complex numbers in trigonometric form, we divide their absolute values and subtract their arguments.

    D

    When dividing two complex numbers in trigonometric form, we subtract their absolute values and divide their arguments.

    A

    When dividing two complex numbers in trigonometric form, we subtract their absolute values and subtract their arguments.

    B

    When dividing two complex numbers in trigonometric form, we divide their absolute values and divide their arguments.

    C

    When dividing two complex numbers in trigonometric form, we divide their absolute values and subtract their arguments.

    D

Question 2

Find the value $\sqrt{5}\left(\cos240^\circ+i\sin240^\circ\right)\times\sqrt{3}\left(\cos30^\circ+i\sin30^\circ\right)$5(cos240°+isin240°)×3(cos30°+isin30°).

Give your answer in rectangular form with exact values.

Question 3

Consider the multiplication $\left(3\sqrt{3}-3i\right)\times\left(-5i\right)$(333i)×(5i).

  1. Express $3\sqrt{3}-3i$333i in polar form, $r\left(\cos\theta+i\sin\theta\right)$r(cosθ+isinθ), with $r\ge0$r0 and $0\le\theta<2\pi$0θ<2π:

  2. Express $-5i$5i in polar form, $r\left(\cos\theta+i\sin\theta\right)$r(cosθ+isinθ), with $r\ge0$r0 and $0\le\theta<2\pi$0θ<2π:

  3. Use these results to evaluate $\left(3\sqrt{3}-3i\right)\times\left(-5i\right)$(333i)×(5i). Give your final answer in rectangular form.

Outcomes

M8-9

Manipulate complex numbers and present them graphically

91577

Apply the algebra of complex numbers in solving problems

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