Question

Plz solve the above question​

Answer 1

Answer:

13/21

hope you get the solution

Answer 2

Given :

4tanθ = 3

To find :

$\sf Value \: of \: \: \dfrac{4 \sin( \theta) \: - \: \cos( \theta) \: + 1}{4 \sin( \theta) \: + \: \cos( \theta) \: - 1}$

Identity used :

$\sf \bullet \: \dfrac{ \sin( \theta) }{ \cos(\theta) } \: = \: \tan( \theta)$

$\sf \bullet \: \ \sec^{2} (\theta) \: = \: \tan^{2} ( \theta) + 1$

Solution :

According to question,

4tanθ = 3

➝ tanθ = 3/4

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We know that sec²θ = tan²θ + 1

$\sf \implies \: \ \sec^{2} (\theta) \: = \: \bigg( \dfrac{3}{4} \bigg)^{2} + 1$

$\sf \implies \: \ \sec^{2} (\theta) \: = \: \dfrac{9}{16} + 1$

$\sf \implies \: \ \sec^{2} (\theta) \: = \: \dfrac{(9 \times 1) + (1 \times 16)}{16}$

$\sf \implies \: \ \sec^{2} (\theta) \: = \: \dfrac{(9 + 16)}{16} = \dfrac{25}{16}$

$\sf \implies \: \ \sec (\theta) \: = \: \sqrt{\dfrac{25}{16} }$

$\sf \implies \: \ \sec (\theta) \: = \: \dfrac{5}{4}$

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$\dfrac{4 \sin( \theta) \: - \: \cos( \theta) \: + 1}{4 \sin( \theta) \: + \: \cos( \theta) \: - 1}$

• Divide numerator and denominator by cosθ

$\sf \implies \: \dfrac{ \dfrac{ \: 4 \sin( \theta) \: - \: \cos( \theta) \: + 1 \: \:}{ \cos( \theta) } }{ \: \dfrac{4 \sin( \theta) \: + \: \cos( \theta) \: - 1 \:}{ \cos( \theta) } }$

$\sf \implies \: \dfrac{ \dfrac{ \: 4 \sin( \theta) }{ \cos( \theta) } - \dfrac{ \cos( \theta) }{ \cos( \theta) } + \dfrac{1}{ \cos( \theta) } }{ \: \dfrac{4 \sin( \theta) \: \:}{ \cos( \theta) } + \dfrac{ \cos( \theta) }{ \cos( \theta) } - \dfrac{1}{ \cos( \theta) } }$

$\sf \implies \: \dfrac{ \: 4 \tan( \theta) - 1 + \sec( \theta) }{4 \tan( \theta) + 1 - \sec( \theta) }$

Now , put value of 4tanθ, secθ

$\sf \implies \: \dfrac{ \: 3 - 1 + \dfrac{5}{4} }{ \: 3 + 1 - \dfrac{5}{4} }$

$\sf \implies \: \dfrac{ \: 2+ \dfrac{5}{4} }{ \: 4 - \dfrac{5}{4} }$

Take LCM in numerator and denominator

$\sf \implies \: \dfrac{ \dfrac{(2 \times 4) + (5 \times 1)}{4} }{ \: \dfrac{(4 \times 4) - (5 \times 1)}{4} }$

$\sf \implies \: \dfrac{ \dfrac{8 + 5 }{4} }{ \: \dfrac{16 - 5}{4} }$

$\sf \implies \: \dfrac{ \dfrac{13}{4} }{ \: \dfrac{11}{4} }$

$\sf \implies \: \dfrac{ 13 }{ \: 11 }$

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ANSWER : 13/11