Given that the modulus of 𝑧 is
five and the argument of 𝑧 is 𝜃, which is equal to 270 degrees, find 𝑧, giving
your answer in algebraic form.
When we write a complex number in
algebraic or rectangular form, we write it as 𝑎 plus 𝑏𝑖, where 𝑎 is the real
component of the complex number and 𝑏 is the imaginary component.
We can use these conversion
formulae to convert the polar coordinates with a modulus of 𝑟 and an argument 𝜃
into the corresponding rectangular form. 𝑎 is equal to 𝑟 cos 𝜃 and 𝑏 is
equal to 𝑟 sin 𝜃. The modulus of our complex number
is five. And 𝜃, the argument, is 270
Often, the argument will be given
in radians. But since we’re converting a
complex number into rectangular form, this doesn’t really matter, as long as we
remember to make sure our calculator is working in degrees.
Let’s substitute these values into
the conversion formulae for 𝑎 and 𝑏. 𝑎 is equal to five multiplied by
cos of 270 degrees, which is zero. And 𝑏 is equal to five multiplied
by sin of 270 degrees, which is negative five.
In rectangular form, 𝑎 plus 𝑏𝑖
then, we can write our complex number as zero plus negative five 𝑖, which is simply
negative five 𝑖.
We have expressed our complex
number in algebraic or rectangular form. 𝑧 is equal to negative five