Question

If $7\tan \phi =4$, then find the value of $\dfrac{7\sin \phi -3\cos \phi }{7\sin \phi +3\cos \phi }$.

Answer 1

Hint:In order to find the solution of this question, we will start from the expression given and we will try to replace $\sin \phi $ and $\cos \phi $ by $\tan \phi $ and then from the given equality, we will put the value of $\tan \phi $ and then we will simplify it.

Complete step-by-step answer:
In this question, we have been asked to find the value of the expression, $\dfrac{7\sin \phi -3\cos \phi }{7\sin \phi +3\cos \phi }$ if $7\tan \phi =4$. So, to find the answer of this question, we will start from the given expression, that is, $\dfrac{7\sin \phi -3\cos \phi }{7\sin \phi +3\cos \phi }$.
Now, we know that if we multiply the numerator and the denominator of a fraction by a term, the fraction remains the same. So, here we will multiply the numerator and the denominator of the expression by $\dfrac{1}{\cos \phi }$. So, we will get the expression as,
$\dfrac{\left( 7\sin \phi -3\cos \phi \right)\times \dfrac{1}{\cos \phi }}{\left( 7\sin \phi +3\cos \phi \right)\times \dfrac{1}{\cos \phi }}$
Now, we will open the brackets to simplify it. So, we will get,
$\dfrac{7\dfrac{\sin \phi }{\cos \phi }-3\dfrac{\cos \phi }{\cos \phi }}{7\dfrac{\sin \phi }{\cos \phi }+3\dfrac{\cos \phi }{\cos \phi }}$
Now, we know that common terms in the numerator and the denominator will get cancelled out. So, we can write the expression as,
$\dfrac{7\dfrac{\sin \phi }{\cos \phi }-3}{7\dfrac{\sin \phi }{\cos \phi }+3}$
Now, we know that $\dfrac{\sin \phi }{\cos \phi }$ can be written as $\tan \phi $. Therefore, we can write the expression as,
$\dfrac{7\tan \phi -3}{7\tan \phi +3}$
Now, we have been given that $7\tan \phi =4$. So, to obtain the value of the expression, we will substitute the value of $7\tan \phi $ in the above expression. Therefore, we will get,
$\dfrac{4-3}{4+3}$
Which can be further written as, $\dfrac{1}{7}$.
Hence, we can say that the value of the expression, $\dfrac{7\sin \phi -3\cos \phi }{7\sin \phi +3\cos \phi }$ is $\dfrac{1}{7}$ when $7\tan \phi =4$.

Note: While solving this question, one can think of finding the values of $\sin \phi $ and $\cos \phi $ by using the given equality, that is, $7\tan \phi =4$ we know that ratio of opposite to adjacent side gives trigonometric tangent value So we can write the equation as $\tan \phi =\dfrac{4}{7}$,by considering the right angled triangle and using pythagoras theorem we can find values of $\sin \phi $ and $\cos \phi $ . And then simplifying it. This method is also correct but then the solution will become lengthier. So, in order to avoid any mistakes, we have to remember that $\dfrac{\sin \phi }{\cos \phi }=\tan \phi $ because by using this, we will easily find the answer.