Video Transcript
Consider the complex number 𝑧 is
equal to one plus root three 𝑖. Part one, find the modulus of
𝑧. Part two, find the argument of
𝑧. Part three, hence, use the
properties of multiplication of complex numbers in polar form to find the modulus
and argument of 𝑧 cubed. Part four, hence, find the value of
𝑧 cubed.
In this question, we’re given a
complex number 𝑧 given in algebraic form, and we need to answer four parts of the
question to determine the value of 𝑧 cubed. The first part of this question
wants us to determine the modulus of 𝑧. So, let’s start by recalling how we
determine the modulus of a complex number given in algebraic form. It’s the distance the complex
number is from the origin on an Argand diagram. In other words, it’s the square
root of the sum of the squares of its real and imaginary parts.
For a complex number given in
algebraic form 𝑎 plus 𝑏𝑖, its modulus is the square root of 𝑎 squared plus 𝑏
squared. In our case, the real part of 𝑧 is
one, and the imaginary part is root three. So, the modulus of 𝑧 is equal to
the square root of one squared plus root three squared, which simplifies to give us
the square root of four which we can evaluate as two. So, the modulus of 𝑧 is equal to
two.
Let’s now move on to the second
part of this question. We want to determine the argument
of this complex number 𝑧. And we can do this by recalling the
argument of a complex number 𝑧 is the angle the line between the origin and 𝑧 on
an Argand diagram makes with the positive real axis measured counterclockwise. And there’s many different ways of
determining the argument of a complex number. We’re first going to determine
which quadrant 𝑧 lies in by looking at its algebraic form.
The real part of 𝑧 is positive and
the imaginary part of 𝑧 is also positive. So, 𝑧 lies in the first quadrant
of an Argand diagram. Now, we can connect 𝑧 to the
origin with a line segment. And then we know the argument of 𝑧
is the angle this line segment makes with the positive real axis. And since 𝑧 lies in the first
quadrant, this angle will be given by the inverse tangent of the imaginary part of
𝑧 divided by the real part of 𝑧. This is because we can construct a
right triangle on our Argand diagram, where the argument of 𝑧 is one of the angles,
the length of the opposite side is the imaginary part of 𝑧, and the length of the
adjacent side is the real part of 𝑧.
So, the argument of 𝑧 is equal to
the inverse tan of root three divided by one, which simplifies to give us the
inverse tan of root three. And we can evaluate this; it’s
equal to 𝜋 by three.
The next part of this question
wants us to use the properties of multiplication of complex numbers in polar form to
determine the modulus and argument of 𝑧 cubed. And since this is an integer value,
this can remind us of de Moivre’s theorem, which we can recall tells us for a
complex number written in polar form, 𝑟 times the cos of 𝜃 plus 𝑖 sin of 𝜃 and
any integer value of 𝑛 that 𝑟 times the cos of 𝜃 plus 𝑖 sin of 𝜃 all raised to
the 𝑛th power is equal to 𝑟 to the 𝑛th power multiplied by the cos of 𝑛 𝜃 plus
𝑖 sin of 𝑛 𝜃. And we can determine the modulus
and argument of this complex number by noticing its modulus is 𝑟 to the 𝑛th power
and its argument is 𝑛 multiplied by 𝜃.
So, in particular, we get the
following two results. For any integer value of 𝑛, the
argument of 𝑧 to the 𝑛th power is equal to 𝑛 times the argument of 𝑧. And the modulus of 𝑧 to the 𝑛th
power is equal to the modulus of 𝑧 all raised to the 𝑛th power. So, let’s apply this to determine
the modulus and argument of 𝑧 cubed.
Let’s start with the argument of 𝑧
cubed. It’s three times the argument of
𝑧, which is three times 𝜋 by three. And if we cancel the shared factor
of three in the numerator and denominator, we get 𝜋. Similarly, we can determine the
modulus of 𝑧 cubed. It’s equal to the modulus of 𝑧 all
cubed. We showed that the modulus of 𝑧
was two, so we get two cubed is equal to eight. Therefore, the modulus of 𝑧 cubed
is equal to eight, and the argument of 𝑧 cubed is equal to 𝜋.
In the final part of this question,
we need to determine the value of 𝑧 cubed. Well, we’ve already found the
modulus of 𝑧 cubed and the argument of 𝑧 cubed. So, we can substitute the modulus
and the argument of this into its polar form to determine its value. So, we substitute 𝑟 is equal to
eight and 𝜃 is equal to 𝜋 into the polar form of a complex number. This gives us 𝑧 cubed is equal to
eight multiplied by the cos of 𝜋 plus 𝑖 sin of 𝜋. But we can evaluate these. The cos of 𝜋 is negative one, and
the sin of 𝜋 is zero. So, this simplifies to give us
eight multiplied by negative one, which is negative eight.
Therefore, by finding the modulus
and argument of 𝑧 and using de Moivre’s theorem, we were able to show if 𝑧 is
equal to one plus root three 𝑖, then 𝑧 cubed is equal to negative eight.
