Question Video: Representing Complex Numbers in Trigonometric Form

Express the complex number 𝑧 = 4𝑖 in trigonometric form.

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Video Transcript

Express the complex number 𝑧 equals four 𝑖 in trigonometric form.

We do this in three steps. We find π‘Ÿ which is the modulus of 𝑧. We find πœƒ which is its argument. And we substitute these values into 𝑧 equals π‘Ÿ cos πœƒ plus 𝑖 sin πœƒ. 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, π‘Ÿ is four. How about its argument?

Trying to use a formula involving arctan 𝑏 over π‘Ž won’t work as π‘Ž 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œ‹ by two radians. So the argument of 𝑧 is πœ‹ by two. This is a value we have to substitute for πœƒ. And we are now ready to substitute. And doing so, we get four times cos πœ‹ by two plus 𝑖 sin πœ‹ by two.

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