start by graphing 5√3 - 5i
as (5√3, -5) in the argand plane
which would be in quadrant IV
r^2 = (5√3)^2 + (-5)^2 = 100
r = 10
we also know that tanŲ = -5/(5√3) = -1/√3
( I know that tan 30° = tan π/6 = 1/√3)
but my angle is in IV, so Ų = 11π/6 or 330°
so 5√3 - 5i = 10(cos 11π/6 + i sin 11π/6)
RS = 10( √3/2 + i(-1/2)
= 5√3 - 5i
= original complex number
so in summary
for a + bi
1. sketch (a,b) to see which quadrant you are in
2. evaluate r,
with r^2 = a^2 + b^2 ---> r = | √(a^2 + b^2)
3. from tan Ų = |b/a| find the acute angel Ų
- for I, Ų is that acute angle
- for II , Ų = π - acute angle
- for III , Ų = π + acute angle
- for IV, Ų = 2π - acute angle
a + bi = r( cosŲ + isinŲ)
use your calculator to verfiy your answer.
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