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