Given that the modulus of 𝑍 is equal to five and the argument of 𝑍 is 𝜃 equals two 𝜋 plus two 𝑛𝜋, 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 𝑟 multiplied by cos 𝜃 plus 𝑖 sin 𝜃, where 𝑟 is known as the modulus of the complex number 𝑍 and 𝜃 is its argument. In polar form, 𝜃 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 plus two 𝑛𝜋. So we get 𝑍 equals five multiplied by cos of two 𝜋 plus two 𝑛𝜋 plus 𝑖 sin of two 𝜋 plus two 𝑛𝜋.
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 𝜃, sin of 𝜃 plus some multiple of two 𝜋 is equal to sin 𝜃. And cos of 𝜃 plus some multiple of two 𝜋 is equal to cos of 𝜃.
This means that cos of two 𝜋 plus two 𝑛𝜋 is simply cos of two 𝜋. And sin of two 𝜋 plus two 𝑛𝜋 is equal to sin of two 𝜋. Our complex number can therefore be written as five multiplied by cos of two 𝜋 plus 𝑖 sin of two 𝜋, in trigonometrical polar form.