# Question Video: Representing Complex Numbers in Trigonometric Form

Express the complex number π§ = 4π in trigonometric form.

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### Video Transcript

Express the complex number π§ equals four π in trigonometric form.

We do this in three steps. We find π which is the modulus of π§. We find π which is its argument. And we substitute these values into π§ equals π cos π plus π sin π. But first, letβs draw an Argand diagram with four π 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, π is four. How about its argument?

Trying to use a formula involving arctan π over π wonβt work as π is zero. And we canβ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 π§ is π by two. This is a value we have to substitute for π. And we are now ready to substitute. And doing so, we get four times cos π by two plus π sin π by two.