complex number polar formstep by step please c) Use the polar form to prove that1) (1BY3+) = 2+1232) (1+i/3)1º =2¬1'(-1+i/3)||

Name: complex number polar formstep by step please c) Use the polar form to
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Description: complex number polar formstep by step please c) Use the polar form to prove that1) (1BY3+) = 2+1232) (1+i/3)1º =2¬1'(-1+i/3)||

We need to show that, 1+i3-10=2-11-1+i3. To show this, we are using polar form of complex numbers…

Answered: c) Use the polar form to prove that 2)…

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter11: Topics From Analytic Geometry

Section: Chapter Questions

Problem 53RE

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complex number polar form

step by step please

c) Use the polar form to prove that
1) (1
BY3+) = 2+123
2) (1+i/3)1º =2¬1'(-1+i/3)
||

Expert Solution

Step 1

We need to show that, ${(1 + i \sqrt{3})}^{- 10} = 2^{- 11} (- 1 + i \sqrt{3})$ .

To show this, we are using polar form of complex numbers and the below mentioned formulas.

$z = r (c i s θ) x + i y = r (\cos θ + i \sin θ) {(x + i y)}^{n} = {(r (c o s θ + i s i n θ))}^{n} {(x + iy)}^{n} = r^{n} (cosnθ + isinnθ)$

First, Let's consider Left hand side (L.H.S)

$L . H . S = {(1 + i \sqrt{3})}^{- 10}$

We know, a complex number z= x+iy

On comparing,

we have x=1 and y= $\sqrt{3}$ .

& we know,

$\begin{array}{rcl} r & = & \sqrt{x^{2} + y^{2}} \\ = & \sqrt{1 + 3} \\ = & 2 \\ ∴ r & = & 2 \end{array}$

Step 2

Also, The principle amplitude $θ$ ,

$\begin{array}{rcl} θ & = & t a n^{- 1} (\frac{y}{x}) \\ = & \tan^{- 1} (\frac{\sqrt{3}}{1}) \\ = & \tan^{- 1} (\sqrt{3}) \\ = & \frac{π}{3} \\ ∴ θ & = & \frac{π}{3} \end{array}$

Put the values of r, $θ & n$ in the above formula, it becomes

$\begin{array}{rcl} N o w, {(1 + i \sqrt{3})}^{- 10} & = & {(2 (c o s \frac{π}{3} + i s i n \frac{π}{3}))}^{- 10} \\ = & 2^{- 10} {(\cos \frac{π}{3} + i \sin \frac{π}{3})}^{- 10} \\ = & 2^{- 10} (\cos (\frac{- 10 π}{3}) + i \sin (\frac{- 10 π}{3})) (∵ B y D e M o i v r e' s t h e r o r e m) \\ = & 2^{- 10} (- 0.5 + i 0.86602540) \\ = & \frac{2^{- 10} \times 2 (- 0.5 + i 0.86602540)}{2} \\ = & \frac{2^{- 10}}{2} \times (- 0.5 \times 2 + i 0.86602540 \times 2) \\ = & 2^{- 11} (- 1 + 1.73205080 i) \\ = & 2^{- 11} (- 1 + \sqrt{3} i) \end{array}$

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c) Use the polar form to prove that 2) (1+i/3)¬1° = 2-1'(-1+i/3) 2-"(-1+i/3)