### Video Transcript

Express the complex number π§ is equal to four π in trigonometric form.

π§ is equal to π plus π π as known as the rectangular form of the complex number π§. If we compare this form to our complex number π§ is equal to four π, we can see that π must be equal to zero and π has a value of four since π is the coefficient of π. When we write a complex number in trigonometric or polar form, we write it as π§ is equal to π multiplied by cos π plus π sin π, where π is known as the modulus of the complex number π§ and π is the argument.

In polar form, π can be in degrees or radians. The radians is often preferred, whereas in exponential form it does need to be in radians. So we need to find a way to represent the real and complex components of our number in terms of π and π. In fact, we can use this formula to help us. The modulus π is the square root of π squared plus π squared. This is derived from the Pythagorean theorem. And to find π, we can use tan π is equal to π over π.

So letβs substitute what we know about our complex number into these formula: π is equal to the square root of π squared plus π squared, which is the square root of zero squared plus four squared, which is simply four. Tan π is equal to four divided by zero. Now this is actually undefined. However, we do know that the tangent function is undefined at π over two and then at intervals of two π. So π must be π over two for this to be true.

Now that we have values for π and π, letβs substitute them into the formula for the trigonometric form of the complex number. Doing so and we can see that π§ is equal to four multiplied by cos of π over two plus π sin π over two.