Question

plz answer it guys l need help​

Answer 1

$\displaystyle\huge\red{\underline{\underline{Solution}}}$

FORMULA TO BE IMPLEMENTED

${e}^{i \theta} = \cos \theta + i \sin \theta$

TO DETERMINE

1.

$\displaystyle \: \sqrt{3} ( \cos \frac{\pi}{6} + i \sin \frac{\pi}{6} )$

2.

$\displaystyle \: {e}^{ \: \frac{\pi}{3} i\: }$

EVALUATION

1.

$\displaystyle \: \sqrt{3} ( \cos \frac{\pi}{6} + i \sin \frac{\pi}{6} )$

$= \displaystyle \: \sqrt{3} ( \frac{ \sqrt{3} }{2} + i \frac{1}{2} )$

$= \displaystyle \: \frac{3}{2} + \frac{ \sqrt{3} i}{2}$

$\sf { \: Which \: is \: of \: the \: form \: x + iy\: }$

$where \: \: \displaystyle \: x = \frac{3}{2} \: \: and \: \: y = \frac{ \sqrt{3} }{2}$

2.

$\displaystyle \: {e}^{ \: \frac{\pi}{3} i\: }$

$= \displaystyle \: ( \cos \frac{\pi}{3} + i \sin \frac{\pi}{3} )$

$= \displaystyle \: \frac{1}{2} + i \frac{ \sqrt{3} }{2}$

$= \displaystyle \: \frac{1}{2} + \frac{ \sqrt{3}i }{2}$

$\sf { \: Which \: is \: of \: the \: form \: x + iy\: }$

$where \: \: \displaystyle \: x = \frac{3}{2} \: \: and \: \: y = \frac{ \sqrt{3} }{2}$

Answer 2

Step-by-step explanation:

\displaystyle\huge\red{\underline{\underline{Solution}}}

Solution

FORMULA TO BE IMPLEMENTED

{e}^{i \theta} = \cos \theta + i \sin \thetae

iθ

=cosθ+isinθ

TO DETERMINE

1.

\displaystyle \: \sqrt{3} ( \cos \frac{\pi}{6} + i \sin \frac{\pi}{6} )

3

(cos

6

π

+isin

6

π

)

2.

\displaystyle \: {e}^{ \: \frac{\pi}{3} i\: }e

3

π

i

EVALUATION

1.

\displaystyle \: \sqrt{3} ( \cos \frac{\pi}{6} + i \sin \frac{\pi}{6} )

3

(cos

6

π

+isin

6

π

)

= \displaystyle \: \sqrt{3} ( \frac{ \sqrt{3} }{2} + i \frac{1}{2} )=

3

(

2

3

+i

2

1

)

= \displaystyle \: \frac{3}{2} + \frac{ \sqrt{3} i}{2}=

2

3

+

2

3

i

\sf { \: Which \: is \: of \: the \: form \: x + iy\: }Whichisoftheformx+iy

where \: \: \displaystyle \: x = \frac{3}{2} \: \: and \: \: y = \frac{ \sqrt{3} }{2}wherex=

2

3

andy=

2

3

2.

\displaystyle \: {e}^{ \: \frac{\pi}{3} i\: }e

3

π

i

= \displaystyle \: ( \cos \frac{\pi}{3} + i \sin \frac{\pi}{3} )=(cos

3

π

+isin

3

π

)

= \displaystyle \: \frac{1}{2} + i \frac{ \sqrt{3} }{2}=

2

1

+i

2

3

= \displaystyle \: \frac{1}{2} + \frac{ \sqrt{3}i }{2}=

2

1

+

2

3

i

\sf { \: Which \: is \: of \: the \: form \: x + iy\: }Whichisoftheformx+iy

where \: \: \displaystyle \: x = \frac{3}{2} \: \: and \: \: y = \frac{ \sqrt{3} }{2}wherex=

2

3

andy=

2

3