Question Video: Converting Complex Numbers from Algebraic to Polar Form

Express the complex number 𝑧 = 4𝑖 in trigonometric form.

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Video Transcript

Express the complex number 𝑧 is equal to four 𝑖 in trigonometric form.

𝑧 is equal to π‘Ž plus 𝑏 𝑖 as known as the rectangular form of the complex number 𝑧. If we compare this form to our complex number 𝑧 is equal to four 𝑖, we can see that π‘Ž must be equal to zero and 𝑏 has a value of four since 𝑏 is the coefficient of 𝑖. When we write a complex number in trigonometric or polar form, we write it as 𝑧 is equal to 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 the argument.

In polar form, πœƒ 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‘Ÿ and πœƒ. In fact, we can use this formula to help us. The modulus π‘Ÿ is the square root of π‘Ž squared plus 𝑏 squared. This is derived from the Pythagorean theorem. And to find πœƒ, we can use tan πœƒ is equal to 𝑏 over π‘Ž.

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

Now that we have values for π‘Ÿ and πœƒ, let’s substitute them into the formula for the trigonometric form of the complex number. Doing so and we can see that 𝑧 is equal to four multiplied by cos of πœ‹ over two plus 𝑖 sin πœ‹ over two.

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