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 over 𝑎.
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 𝑖sin of 11𝜋 by six.