Question Video: Converting Complex Numbers from Polar to Algebraic Form Mathematics

Express 12 [cos (5πœ‹/6) + 𝑖 sin (5πœ‹/6)] in algebraic form.

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Video Transcript

Express 12 multiplied by cos five πœ‹ over six plus 𝑖 sin five πœ‹ over six in algebraic form.

This complex number is given in polar or trigonometric form. It has a modulus of 12 and an argument of five πœ‹ over six. When we write a complex number in algebraic or rectangular form, we write it as π‘Ž plus 𝑏𝑖, where π‘Ž is the real component and 𝑏 is the imaginary component of this complex number.

We can use these conversion formulae for converting the polar coordinates with the modulus of π‘Ÿ and an argument of πœƒ into the corresponding rectangular form. π‘Ž is equal to π‘Ÿ cos πœƒ and 𝑏 is equal to π‘Ÿ sin πœƒ.

Let’s substitute what we know about our complex number, that’s π‘Ÿ and πœƒ, into the conversion formulae. π‘Ž is equal to 12 multiplied by cos five πœ‹ over six, which is negative six root three. 𝑏 is equal to 12 multiplied by sin of five πœ‹ over six, which is six. Now that we have the values for π‘Ž and 𝑏, we can write our complex number in algebraic form.

Since π‘Ž is negative six root three and 𝑏 is six, our complex number 𝑍 can be written as negative six root three plus six 𝑖.

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