Express the complex number 𝑧
equals four 𝑖 in trigonometric form.
We do this in three steps. We find 𝑟 which is the modulus of
𝑧. We find 𝜃 which is its
argument. And we substitute these values into
𝑧 equals 𝑟 cos 𝜃 plus 𝑖 sin 𝜃. But first, let’s draw an Argand
diagram with four 𝑖 of course lying on the imaginary axis. We can see its modulus, its
distance from the origin, is four. We could have got this using our
formula instead. In any case, 𝑟 is four. How about its argument?
Trying to use a formula involving
arctan 𝑏 over 𝑎 won’t work as 𝑎 is zero. And we can’t divide by zero. But luckily, we have our diagram
where the argument is just this angle here, whose measure is 90 degrees or 𝜋 by two
radians. So the argument of 𝑧 is 𝜋 by
two. This is a value we have to
substitute for 𝜃. And we are now ready to
substitute. And doing so, we get four times cos
𝜋 by two plus 𝑖 sin 𝜋 by two.