Question

plz answer it guys l need help​

Answer 1

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

FORMULA TO BE IMPLEMENTED

1.

A number is said to be purely real if

Imaginary part of the number = 0

2.

The general Solution of

$\sin \theta = 0 \: \: is$

$\theta \: = n \pi \: \: where \: \: n \in \mathbb{Z}$

TO DETERMINE

$\sf{ \: The \: real \: values \: of \: \theta \: for \: which \: \: \: }$

$\displaystyle \: \frac{4 + 3i \sin \theta}{1 - 2i \sin \theta} \: \: is \: purely \: real \:$

EVALUATION

$\displaystyle \: \frac{4 + 3i \sin \theta}{1 - 2i \sin \theta}$

$= \displaystyle \: \frac{4( \: 1 - 2i \sin \theta \: ) + 11 \: i \sin \theta}{1 - 2i \sin \theta}$

$= \displaystyle \: 4 + \frac{11 \: i \sin \theta}{1 - 2i \sin \theta}$

Which is purely real when

$\sin \theta = 0 \:$

$\implies \: \theta \: = n \pi \: \: where \: \: n \in \mathbb{Z}$

$\sf{ \:Which \: is \: the \: required \: values \: of \: \theta \: \: }$

Answer 2

Answer:

$\huge{\underline{\mathtt{\red{❥A}\pink{N}\green{S}\blue{W}\purple{E}\orange{R}}}}$

FORMULA TO BE IMPLEMENTED

1.

A number is said to be purely real if

Imaginary part of the number = 0

2.

The general Solution of

\sin \theta = 0 \: \: issinθ=0is

\theta \: = n \pi \: \: where \: \: n \in \mathbb{Z}θ=nπwheren∈Z

TO DETERMINE

\sf{ \: The \: real \: values \: of \: \theta \: for \: which \: \: \: }Therealvaluesofθforwhich

\displaystyle \: \frac{4 + 3i \sin \theta}{1 - 2i \sin \theta} \: \: is \: purely \: real \:

1−2isinθ

4+3isinθ

ispurelyreal

EVALUATION

\displaystyle \: \frac{4 + 3i \sin \theta}{1 - 2i \sin \theta}

1−2isinθ

4+3isinθ

= \displaystyle \: \frac{4( \: 1 - 2i \sin \theta \: ) + 11 \: i \sin \theta}{1 - 2i \sin \theta}=

1−2isinθ

4(1−2isinθ)+11isinθ

= \displaystyle \: 4 + \frac{11 \: i \sin \theta}{1 - 2i \sin \theta}=4+

1−2isinθ

11isinθ

Which is purely real when

\sin \theta = 0 \:sinθ=0

\implies \: \theta \: = n \pi \: \: where \: \: n \in \mathbb{Z}⟹θ=nπwheren∈Z

\sf{ \:Which \: is \: the \: required \: values \: of \: \theta \: \: }Whichistherequiredvaluesofθ