Question

if x sinA = 5 and 7 cosecA = 6 SecA then find the valus of x​

Answer 1

$\sin(a) = \frac{5}{x}$

$7 \cosec(a) = 6 \sec(a) \\ \frac{\cosec(a) }{ \sec(a) } = \frac{6}{7} \: \: \: \: \: \: \: \: \: \:$

$\frac{ \cosec(a) }{ \sec(a) } = \cot(a) = \frac{6}{7}$

Using Identity:

$1 + { \cot}^{2} (a) = \cosec {}^{2} (a)$

$1 + {( \frac{6}{7} )}^{2} = \frac{1}{ \sin{ }^{2} (a) }$

$1 + \frac{36}{49} = \frac{1}{ (\frac{5}{x} ) {}^{2} }$

$\frac{49 + 36}{49} = \frac{x {}^{2} }{25}$

$\frac{85}{49} \times 25 = {x}^{2}$

$x = \sqrt{ \frac{85 \times 25}{49} }$

$x = \frac{5}{7} \sqrt{85}$

Answer 2

Given,

$\[\begin{array}{l}x\sin A = 5 - - - - \left( 1 \right)\\7\cos ecA = 6\sec A - - - - \left( 2 \right)\end{array}\]$

Solution,

Consider the equation 2,

$\[\begin{array}{l}7\cos ecA = 6\sec A\\ \Rightarrow \frac{7}{{\sin A}} = \frac{6}{{\cos A}}\\ \Rightarrow \tan A = \frac{7}{6}\end{array}\]$

Calculate the value of $\[\sin A\]$,

$\[\begin{array}{l}\sin A = \frac{7}{{\sqrt {{7^2} + {6^2}} }}\\ \Rightarrow \sin A = \frac{7}{{\sqrt {85} }}\end{array}\]$

Put this value in equation 1,

$\[x\sin A = 5\]$

$\[ \Rightarrow x \times \frac{7}{{\sqrt {85} }} = 5\]$

$\[ \Rightarrow x = \frac{{5\sqrt {85} }}{7}\]$

Hence the value of x is $\[\frac{{5\sqrt {85} }}{7}\]$