Question Video: Converting Complex Numbers from Algebraic to Polar Form | Nagwa Question Video: Converting Complex Numbers from Algebraic to Polar Form | Nagwa

# Question Video: Converting Complex Numbers from Algebraic to Polar Form Mathematics • Third Year of Secondary School

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Express the complex number π§ = 4π in trigonometric form.

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### 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 letβs 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, letβs 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.

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