Convert the point 18(cos(2pi/3) + i sin(2pi/3)) to the rectangular form of a complex number and simplify.

Question

Convert the point 18(cos(2pi/3) + i sin(2pi/3))

to the rectangular form of a complex number

and simplify. (The answer should have the form a + bi with no trig functions.)

Answer 1

cos(2π/3) (using the unit circle) is -1/2 and sin(2π/3) is √3/2 so the problem becomes:

18(-1/2 + i(√3/2))

distributing, we get: -9 + 9√3i (a = -9 and b = 9√3)

Answer 2

use Euler's formula

z=r(cos(θ)+isin(θ))

given complex number 18[cos(2pi)/3+i sin(2pi)/3

here r=18

θ=2pi/3

To convert rectangular form

z=18[cos(2pi/3)+isin(2pi)/3]

z=18[-1/2+isqrt3/2]

z=-9 + 9√3i

The rectangular form of the given complex number is -9 + i9√3