Question

(-1 + i √3)
Express the
in polar
complex Number
& Expontial form.​

Answer 1

Answer:

r=|z|=√-1^2+√3^2=√10

x=-1and y=√3

π-tan(y/x)

π-tan(√3/-1)

π-tan-√3{tan√3=60°=π/3}

π+π/3=4π/3

polar form =r(cosx+isinx)

r(cos4π/3+isin4π/3)...solve

Answer 2

$\textbf{\underline{\underline{According\:to\:the\:Question}}}$

Assumption

z = (-1 + i√3)

It is clear that (-1 + i√3) lies in 2 Quadrant

z = r(cosθ + isinθ)

Now,

rcosθ = -1

rsinθ = √3

Now here,

$\textbf{\underline{Squarring\;both\;sides :-}}$

r² = 4

r = √4

r = 2

Hence,

${\boxed{\sf\:{cos\theta=\dfrac{-1}{2}}}}$

Also,

${\boxed{\sf\:{sin\theta=\dfrac{\sqrt{3}}{2}}}}$

tanθ = -√3

tanα = |tanθ| = √3

${\boxed{\sf\:{\alpha=\dfrac{\pi}{3}}}}$

Then,

$\textbf{\underline{It\;lies\;in\;2\;Quadrant}}$

Hence,

θ = (π - α)

${\boxed{\sf\:{\pi-\dfrac{\pi}{3}=\dfrac{2\pi}{3}}}}$

Thus,

$\Large{\boxed{\sf\:{z=2(cos\dfrac{2 \pi}{3}+isin\dfrac{2 \pi}{3}}}}$