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

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

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### Video Transcript

Express 12 multiplied by cos five 𝜋 over six plus 𝑖 sin five 𝜋 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 over six. When we write a complex number in algebraic or rectangular form, we write it as 𝑎 plus 𝑏𝑖, where 𝑎 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 𝑟 and an argument of 𝜃 into the corresponding rectangular form. 𝑎 is equal to 𝑟 cos 𝜃 and 𝑏 is equal to 𝑟 sin 𝜃.

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

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