Express the number root three 𝑖 in trigonometric form.
We have been given a complex number in rectangular or algebraic form. In general, we can say that the complex number in rectangular form is 𝑎 plus 𝑏𝑖. 𝑎 is the real part. And 𝑏 is the imaginary component of our complex number.
We’re looking to write this number in trigonometric form. That’s 𝑟 cos 𝜃 plus 𝑖 sin 𝜃, where 𝑟 is the modulus of our complex number and 𝜃 is its argument. And there are some conversion formulae we can use to save us some time. But let’s look at the Argand diagram to see where these come from.
On this Argand diagram, the horizontal axis represents the real component. And the vertical axis represents the imaginary component of our complex number. We can see that root three 𝑖 would lie somewhere on this vertical axis whereas a general complex number in our algebraic form with positive values of 𝑎 and 𝑏 would lie somewhere in this first quadrant.
We can add in a right-angle triangle. And we can also add a vertical height of 𝑏 and a horizontal width of 𝑎 units. The modulus is the hypotenuse of this triangle. It’s sometimes called the magnitude of a complex number. And we can use the Pythagorean theorem to find an equation for 𝑟. It’s found by finding the square root of 𝑎 squared plus 𝑏 squared.
And next, we consider the argument. It’s measured in a counterclockwise direction from the horizontal axis as shown. We can label the triangle with respect to this included angle. Labelling the opposite adjacent and hypotenuse with respect to the included angle, we can see that we can form an equation for 𝜃 by using the tan ratio.
Tan 𝜃 is equal to opposite over adjacent. In this case, that’s 𝑏 over 𝑎. And we can solve this equation for 𝜃 by finding the inverse tan or 𝑟 arctan of both sides. So 𝜃 is equal to arctan of 𝑏 over 𝑎. And now, we have the two conversion formulae we can use to express a complex number in rectangular form in trigonometric form.
𝑎 is a constant. In this case, it’s zero. And 𝑏 is the coefficient of 𝑖. It’s root three. So the modulus of our complex number is the square root of zero plus root three squared, which is simply root three. And 𝜃 we can work out by considering the position on the Argand diagram. But let’s use the formula. It’s arctan of root three over zero.
And of course, root three divided by zero is undefined. And we know that the tan function is undefined at the point where 𝜃 is equal to 𝜋 by two. Now, this makes a lot of sense since we said that 𝜃 is measured in an anticlockwise direction from the horizontal axis. Alternatively, we could measure it in degrees. And we get 90 degrees.
All that’s left is to substitute what we know about our complex number into the formula. We get 𝑧 is equal to root three multiplied by cos of 90 degrees plus 𝑖 sin of 90 degrees.