Video: Dividing Complex Numbers in Polar Form

Given that 𝑍₁ = 1 and 𝑍₂ = (cos 3πœƒ + 𝑖 sin 3πœƒ)Β², find the trigonometric form of 𝑍₁/𝑍₂.

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Video Transcript

Given that 𝑍 one equals one and 𝑍 two is equal to cos of three πœƒ plus 𝑖 sin of three πœƒ all squared, find the trigonometric form of 𝑍 one divided by 𝑍 two.

There are several formulae we’ll need to consider here. The first is the product formula. And it says for two complex numbers 𝑍 one with the modulus of π‘Ÿ one and an argument of πœƒ one and 𝑍 two with the modulus of π‘Ÿ two and an argument of πœƒ two, their product 𝑍 one 𝑍 two can be found by multiplying the moduli and adding the arguments.

We can extend this into squaring a complex number and say that to find the square of a complex number 𝑍 in polar form, we square the modulus and double the argument. This will allow us to find the value of cos three πœƒ plus 𝑖 sine of three πœƒ all squared. We double the arguments and we get cos of six πœƒ plus 𝑖 sin of six πœƒ.

The quotient formula says that for the two same complex numbers 𝑍 one and 𝑍 two, their quotient 𝑍 one divided by 𝑍 two can be found by dividing the moduli and subtracting the arguments. 𝑍 one is not yet in trigonometric form though; it’s in rectangular form. The rectangular form for a complex number is 𝑍 equals π‘Ž plus 𝑏𝑖.

If we compare the general rectangular form to our complex number 𝑍 one, we can see that the value for π‘Ž is one and the value for 𝑏 is zero. So we need to find a way now to represent the real and complex components of our number in terms of π‘Ÿ and πœƒ, essentially writing in trigonometric form.

The modulus π‘Ÿ is given by the square root of π‘Ž squared plus 𝑏 squared. This comes from the Pythagorean theorem. To find πœƒ, we use the formula tan πœƒ is equal to 𝑏 over π‘Ž. So let’s substitute what we know about our complex number 𝑍 one into these formulae.

The modulus is the square root of one squared plus zero squared, which is just one. tan πœƒ is equal to 𝑏 over π‘Ž, which is zero over one. Zero divided by one is zero. So to solve this equation for πœƒ, we can find the arc tangent of zero, which is simply zero. And now that we have the values for π‘Ÿ and πœƒ let’s substitute them into the formula for the general polar or trigonometric form of the complex number 𝑍 one.

Doing this, we can see that 𝑍 one is equal to one multiplied by cos of zero plus 𝑖 sine of zero. Let’s retain this and clear a little bit of space. We’re looking to find the value of 𝑍 one divided by 𝑍 two. Let’s go back to the quotient formula.

We divide the moduli and we get one over one. We subtract the arguments and we get zero minus six πœƒ. Remember sine and cosine functions are periodic with the period of two πœ‹. So we can add two πœ‹ to the argument and the complex number itself will remain unchanged. And since the modulus of the complex number is one, we actually don’t need to write that.

And we’re done: the quotient 𝑍 one divided by 𝑍 two is equal to cos of two πœ‹ minus six πœƒ plus 𝑖 sin of two πœ‹ minus six πœƒ.

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