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

Given that |𝑍| = 5 and the argument of 𝑍 is πœƒ = 2πœ‹ + 2π‘›πœ‹, where 𝑛 ∈ β„€, find 𝑍, giving your answer in trigonometric form.

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Video Transcript

Given that the modulus of 𝑍 is equal to five and the argument of 𝑍 is πœƒ equals two πœ‹ plus two π‘›πœ‹, where 𝑛 is an integer, find 𝑍, giving your answer in trigonometric form.

When we write a complex number in trigonometric or polar form, we write it as 𝑍 equals π‘Ÿ multiplied by cos πœƒ plus 𝑖 sin πœƒ, where π‘Ÿ is known as the modulus of the complex number 𝑍 and πœƒ is its argument. In polar form, πœƒ can be in degrees or radians whereas in exponential form, it does need to be in radians.

Let’s substitute what we know about the complex number 𝑍 into this formula. The modulus of 𝑍 was five and the argument was two πœ‹ plus two π‘›πœ‹. So we get 𝑍 equals five multiplied by cos of two πœ‹ plus two π‘›πœ‹ plus 𝑖 sin of two πœ‹ plus two π‘›πœ‹.

Next, we can recall what we know about the cosine and sine functions. They are periodic. That is to say, they repeat. And their period is two πœ‹ radians. That means that for any value of πœƒ, sin of πœƒ plus some multiple of two πœ‹ is equal to sin πœƒ. And cos of πœƒ plus some multiple of two πœ‹ is equal to cos of πœƒ.

This means that cos of two πœ‹ plus two π‘›πœ‹ is simply cos of two πœ‹. And sin of two πœ‹ plus two π‘›πœ‹ is equal to sin of two πœ‹. Our complex number can therefore be written as five multiplied by cos of two πœ‹ plus 𝑖 sin of two πœ‹, in trigonometrical polar form.

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