A. Express in the form , where and and . c. On an Argand diagram, , and are…

MATHEMATICS · Multi-part · CBSE · Mathematics · Complex Numbers and Quadratic Equations · Complex Numbers and Quadratic Equations

Problem

a. Express

-3+\sqrt3i

in the form

re^{i\theta}

, where

r>0

and

-\pi<\theta\le\pi

. b. Let the roots of

z^3=-3+\sqrt3i

u

v

and

w

. Find

u

v

and

w

in the form

re^{i\theta}

, where

r>0

and

-\pi<\theta\le\pi

. c. On an Argand diagram,

u

v

and

w

are represented by the points

U

V

and

W

, respectively. Find the area of triangle

UVW

. d. By considering the sum of the roots

u

v

and

w

, show that

\cos\frac{5\pi}{18}+\cos\frac{7\pi}{18}+\cos\frac{17\pi}{18}=0

. (a) Express

-3+\sqrt3i

in the form

re^{i\theta}

, where

r>0

and

-\pi<\theta\le\pi

. (b) Let the roots of

z^3=-3+\sqrt3i

u

v

and

w

. Find

u

v

and

w

in the form

re^{i\theta}

, where

r>0

and

-\pi<\theta\le\pi

. (c) On an Argand diagram,

u

v

and

w

are represented by the points

U

V

and

W

, respectively. Find the area of triangle

UVW

. (d) By considering the sum of the roots

u

v

and

w

, show that

\cos\frac{5\pi}{18}+\cos\frac{7\pi}{18}+\cos\frac{17\pi}{18}=0