Question

If sin x + sin² x + ............ = 4 + $\sf{\sqrt{3}}$ , 0 < x < π and x ≠ $\sf{\dfrac{\pi}{2}}$ , then x =

a] 30° , 60°

b] 60° , 120°

c] 90° , 120°

d] 30° , 45°

Please include the correct answer with proper explanation. ​

Answer 1

Correct Question :–

If 1+sin x + sin² x + ............ = 4 + $\sf{2\sqrt{3}}$ , 0 < x < π and x ≠ $\sf{\dfrac{\pi}{2}}$ , then x =

a] 30° , 60°

b] 60° , 120°

c] 90° , 120°

d] 30° , 45°

ANSWER :–

GIVEN :–

$\\ \implies\bf 1+ \sin x + \sin^{2} x + ............ \infty = 4 + 2{\sqrt{3}}$

• And –

$\\ \implies \bf 0 < x < π \: \: and \: \: x \ne\sf{\dfrac{\pi}{2}}$

TO FIND :–

• Value of 'x' = ?

SOLUTION :–

$\\ \implies\bf 1+\sin x + \sin^{2} x + ............ \infty = 4 + 2{\sqrt{3}}$

• It's an infinite G.P. , So that –

$\\ \implies\bf \dfrac{1}{1 - \sin x}= 4 +2 {\sqrt{3}}$

$\\ \implies\bf 1= (4 +2 {\sqrt{3})(1 - \sin x)}$

$\\ \implies\bf 1= (4 + 2{\sqrt{3})(1) - ( \sin x)(4 + 2{\sqrt{3})}}$

$\\ \implies\bf 1=4 + 2{\sqrt{3} -4\sin x- 2{\sqrt{3} \sin x}}$

$\\ \implies\bf 1+ 4\sin x +2 {\sqrt{3} \sin x}=4 + 2{\sqrt{3}}$

$\\ \implies\bf 4\sin x + 2{\sqrt{3} \sin x}=4 +2 {\sqrt{3}} -1$

$\\ \implies\bf (4+2{\sqrt{3}) \sin x}=3+ 2{\sqrt{3}}$

$\\ \implies\bf \sin x= \dfrac{3 + 2\sqrt{3}}{4+2\sqrt{3}}$

• Now rationalization –

$\\ \implies\bf \sin x= \dfrac{3+2\sqrt{3}}{4+2 \sqrt{3}} \times \dfrac{4-2 \sqrt{3} }{4- 2\sqrt{3} }$

$\\ \implies\bf \sin x= \dfrac{(3+2 \sqrt{3})(4-2\sqrt{3} )}{(4+2\sqrt{3})(4-2 \sqrt{3} )}$

$\\ \implies\bf \sin x= \dfrac{12 - 6 \sqrt{3} + 8\sqrt{3} - 12}{(4)^{2} -(2 \sqrt{3} )^{2} }$

$\\ \implies\bf \sin x= \dfrac{2 \sqrt{3}}{16-12}$

$\\ \implies\bf \sin x= \dfrac{2 \sqrt{3}}{4}$

$\\ \implies\bf \sin x= \dfrac{ \sqrt{3}}{2}$

$\\ \implies\bf x= \sin^{-1}\bigg(\dfrac{ \sqrt{3}}{2}\bigg)$

• We know that –

$\\ \longrightarrow\bf \sin(60^{\circ}) =\sin(120^{\circ})=\dfrac{ \sqrt{3}}{2}$

$\\ \implies\bf x=60^{\circ},120^{\circ}$

• Hence , Option (b) is correct.

Answer 2

Answer:

Option b

Option b X=60° or 120°

Step-by-step explanation:

Used formulae:-

• If a is the first term and the common ratio is r then sum of infinity terms of the GP is a/1-r.
• (a+b)+a-b)=a²-b²
• Sin(90°+A)=Cos A
• Sin²X=(Sin X)²