Given that 𝑍 is equal to root
three plus 𝑖, determine the trigonometric form of 𝑍 bar.
For a complex number 𝑍 written in
rectangular form, 𝑎 plus 𝑏𝑖, 𝑍 bar is called the complex conjugate of this
number. And it’s found by changing the sign
between 𝑎 and 𝑏𝑖. So in our example, 𝑍 bar or the
conjugate of 𝑍 is 𝑎 minus 𝑏𝑖. And we can see that if 𝑍 is equal
to root three plus 𝑖, the conjugate 𝑍 bar is root three minus 𝑖.
But what about writing it in its
trigonometric form? In trigonometric form, we write it
as 𝑟 multiplied by cos 𝜃 plus 𝑖 sin 𝜃, where 𝑟 is the modulus and 𝜃 is the
argument, sometimes also called the amplitude. And we can use these two conversion
formulae to find the modulus and argument of our complex number. 𝑟 is equal to the square root of
𝑎 squared plus 𝑏 squared. And 𝜃 is equal to arctan of 𝑏
Since 𝑎 in our rectangular form is
the constant, we can say that 𝑎 of 𝑍 bar must be root three. And 𝑏 is the coefficient of
𝑖. Here, it’s negative one. Remember, 𝑏 is one and not
negative one because we’re comparing 𝑎 plus 𝑏𝑖 with the trigonometric form.
Using our formulae for the modulus,
we say that 𝑟 is the square root of root three squared plus negative one
squared. And of course, root three squared
is simply three. So this is the square root of three
plus one, which is four. And since the square root of four
is two, we can say that the modulus of 𝑍 bar must be two. And the argument is arctan of 𝑏
over 𝑎, which is negative one over root three.
If we rationalize the denominator
of negative one over root three, we’ll be able to find the general result for arctan
of this number. And we do that by multiplying both
the numerator and the denominator by root three. Negative one multiplied by root
three is negative root three. And root three multiplied by root
three is three. So in fact, 𝜃 is found by arctan
of negative root three over three, which is negative 𝜋 by six.
We can add or subtract multiples of
two 𝜋 to get our argument in the form we’re interested in. Here, we want a positive value of
𝜃. So we’re going to add two 𝜋 to
negative 𝜋 by six. That’s the same as adding 12𝜋 by
six. And negative 𝜋 by six plus 12𝜋 by
six is 11𝜋 by six.
All that’s left is for us to
substitute our value for 𝑟 and 𝜃 into a trigonometric form for a complex
number. And when we do, we see that 𝑍 bar
is equal to two multiplied by cos of 11𝜋 by six plus 𝑖 sin of 11𝜋 by six.