Video: Writing a Complex Number in Polar Form given Its Modulus and Principal Argument

Given that |𝑍| = 9 and the argument of 𝑍 is πœƒ = πœ‹/6, find 𝑍, giving your answer in trigonometric form.

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Video Transcript

Given that the modulus of 𝑍 is equal to nine and the argument of 𝑍 is πœƒ equals πœ‹ over six, find 𝑍, giving your answer in trigonometric form.

Let’s recap. The polar or trigonometric form of a complex number is given as 𝑍 equals π‘Ÿ multiplied by cos πœƒ plus 𝑖 sin πœƒ, where π‘Ÿ is the modulus, sometimes written as shown, and πœƒ is the argument of the complex number 𝑍.

When writing a number in trigonometric form, we can use either degrees or radians, though radians tend to be preferred. Remember though, if we were being asked to write this in exponential form, radians is a must. And the convention for the argument is that it’s greater than negative πœ‹ and less than or equal to πœ‹. This is sometimes called the principal value.

Notice that the value specified for πœƒ here, the argument of this complex number, is already given in radian form and is indeed between negative πœ‹ and πœ‹. So all that remains is to substitute what we’re given into the polar form of a complex number. 𝑍 is therefore equal to nine multiplied by cos of πœ‹ over six plus 𝑖 sin of πœ‹ over six.

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