Question

If $4\cos \theta = 11\sin \theta$, find value of $\dfrac{{11\cos \theta - 7\sin \theta }}{{11\cos \theta + 7\sin \theta }}$.

Answer 1

We can substitute the value of $\cos \theta$ from$4\cos \theta = 11\sin \theta$ in terms of $\sin \theta$ in $\dfrac{{11\cos \theta - 7\sin \theta }}{{11\cos \theta + 7\sin \theta }}$ in order to find the value of $\dfrac{{11\cos \theta - 7\sin \theta }}{{11\cos \theta + 7\sin \theta }}$ or we can do the vice versa, that is, we can substitute the value of $\sin \theta$ from $4\cos \theta = 11\sin \theta$ in terms of $\cos \theta$.

Complete step-by-step answer:
We are given that $4\cos \theta = 11\sin \theta$ and we are required to find the value of $\dfrac{{11\cos \theta - 7\sin \theta }}{{11\cos \theta + 7\sin \theta }}$.
Since, $4\cos \theta = 11\sin \theta$, we get
$\Rightarrow 4\cos \theta = 11\sin \theta \\\ \Rightarrow \cos \theta = \dfrac{{11}}{4}\sin \theta \\\$ ..$\left( 1 \right)$
Now we will substitute this value of $\cos \theta$ from equation $1$ in $\dfrac{{11\cos \theta - 7\sin \theta }}{{11\cos \theta + 7\sin \theta }}$. That is,
$\Rightarrow \dfrac{{11\cos \theta - 7\sin \theta }}{{11\cos \theta + 7\sin \theta }} \\\ \Rightarrow \dfrac{{11\left( {\dfrac{{11}}{4}\sin \theta } \right) - 7\sin \theta }}{{11\left( {\dfrac{{11}}{4}\sin \theta } \right) + 7\sin \theta }} \\\ \Rightarrow \dfrac{{\left( {\dfrac{{121}}{4}\sin \theta } \right) - 7\sin \theta }}{{\left( {\dfrac{{121}}{4}\sin \theta } \right) + 7\sin \theta }} \\\ \Rightarrow \dfrac{{121\sin \theta - 28\sin \theta }}{{121\sin \theta + 28\sin \theta }} \\\$
On simplifying this and taking out $\sin \theta$common from both numerator and denominator, we will get
$\Rightarrow \dfrac{{\sin \theta \left( {121 - 28} \right)}}{{\sin \theta \left( {121 + 28} \right)}}$
By cancelling, $\sin \theta$ from both numerator and denominator, we will get
$\Rightarrow \dfrac{{\left( {121 - 28} \right)}}{{\left( {121 + 28} \right)}} \\\ \Rightarrow \dfrac{{93}}{{149}} \\\$
Hence, the value of $\dfrac{{11\cos \theta - 7\sin \theta }}{{11\cos \theta + 7\sin \theta }} = \dfrac{{93}}{{149}}$.
Therefore the answer is $\dfrac{{93}}{{149}}$.

Note: We could have also substituted the value of $\sin \theta$ in terms of $\cos \theta$ and the answer would have been the same.
Alternate Method:
$4\cos \theta = 11\sin \theta$
Dividing by $4$ on both sides, we get
$\cos \theta = \dfrac{{11}}{4}\sin \theta$
Taking $\sin \theta$ to left hand side, we get
$\dfrac{{\cos \theta }}{{\sin \theta }} = \dfrac{{11}}{4}$
Now, multiply by $11$ and divide by $7$on both sides, we will get
$\dfrac{{11\cos }}{{7\sin }} = \dfrac{{11 \times 11}}{{7 \times 4}} = \dfrac{{121}}{{28}}$
Applying $\dfrac{{Numerator - denominator}}{{numerator + denominator}}$on both sides, we will get
$\dfrac{{11\cos \theta - 7\sin \theta }}{{11\cos \theta + 7\sin \theta }} = \dfrac{{121 - 28}}{{121 + 28}} = \dfrac{{93}}{{149}}$
Hence, the answer is $\dfrac{{93}}{{149}}$.