Multiplication of Numbers in Polar Form
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((cosθ1cosθ2−sinθ1sinθ2)+(sinθ1cosθ2+cosθ1sinθ2)i) | |
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From here we use what is called the double angle formula, we can simplify this to |
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| $z_1z_2$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(θ1−θ2) | |
What's happened here is that we have divided the moduli and subtracted the arguments.
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.
Which is true?
When multiplying two complex numbers in trigonometric form, we add their absolute values and add their arguments.
AWhen multiplying two complex numbers in trigonometric form, we multiply their absolute values and add their arguments.
BWhen multiplying two complex numbers in trigonometric form, we add their absolute values and multiply their arguments.
CWhen multiplying two complex numbers in trigonometric form, we multiply their absolute values and multiply their arguments.
DWhich is true?
When dividing two complex numbers in trigonometric form, we subtract their absolute values and divide their arguments.
AWhen dividing two complex numbers in trigonometric form, we subtract their absolute values and subtract their arguments.
BWhen dividing two complex numbers in trigonometric form, we divide their absolute values and divide their arguments.
CWhen 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 $\left(3\sqrt{3}-3i\right)\times\left(-5i\right)$(3√3−3i)×(−5i).
Express $3\sqrt{3}-3i$3√3−3i in polar form, $r\left(\cos\theta+i\sin\theta\right)$r(cosθ+isinθ), with $r\ge0$r≥0 and $0\le\theta<2\pi$0≤θ<2π:
Express $-5i$−5i in polar form, $r\left(\cos\theta+i\sin\theta\right)$r(cosθ+isinθ), with $r\ge0$r≥0 and $0\le\theta<2\pi$0≤θ<2π:
Use these results to evaluate $\left(3\sqrt{3}-3i\right)\times\left(-5i\right)$(3√3−3i)×(−5i). Give your final answer in rectangular form.