### Video Transcript

Given that the modulus of π is equal to five and the argument of π is π equals two π plus two ππ, where π is an integer, find π, giving your answer in trigonometric form.

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

Letβs substitute what we know about the complex number π into this formula. The modulus of π was five and the argument was two π plus two ππ. So we get π equals five multiplied by cos of two π plus two ππ plus π sin of two π plus two ππ.

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 radians. That means that for any value of π, sin of π plus some multiple of two π is equal to sin π. And cos of π plus some multiple of two π is equal to cos of π.

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