Question

If a trigonometric ratio is given as 3tan$\theta $=4 then the value of $\dfrac{{4\cos \theta - \sin \theta }}{{2\cos \theta + \sin \theta }}$

Answer 1

Hint: In this particular type of question we should use the properties of trigonometric functions on a right angled triangle and thus use the Pythagorean theorem to get the value of hypotenuse and further findings $sin \theta$ and $cos \theta$ . Further we need to use these values to find the desired solution.

Complete step-by-step answer:

Since
$
  3\tan \theta = 4 \\
   \Rightarrow \tan \theta = \dfrac{4}{3} \\
$
We know that $\tan \theta = \dfrac{{perpendicular}}{{base}}$
$
  In{\text{ }}\vartriangle ABC, \\
  AB = perpendicular \\
  BC = base \\
  AC = hypotenuse \\
  A{C^2} = A{B^2} + B{C^2} \\
   \Rightarrow A{C^2} = {4^2} + {3^2} = 25 \\
   \Rightarrow AC = 5 \\
$
 ( Pythagoras theorem )
Therefore $ \sin \theta = \dfrac{{perpendicular}}{{hypotenuse}} = \dfrac{4}{5} \\
  \cos \theta = \dfrac{{base}}{{hypotenuse}} = \dfrac{3}{5} \\
$
On putting the above values in $\dfrac{{4\cos \theta - \sin \theta }}{{2\cos \theta + \sin \theta }}$ and solving we get
\[\dfrac{{4\cos \theta - \sin \theta }}{{2\cos \theta + \sin \theta }} = \dfrac{{4 \times \dfrac{3}{5} - \dfrac{4}{5}}}{{2 \times \dfrac{3}{5} + \dfrac{4}{5}}} = \dfrac{{\dfrac{8}{5}}}{{\dfrac{{10}}{5}}} = \dfrac{8}{{10}} = \dfrac{4}{5}\]

Note: Remember that drawing a figure in this type of question will help us to understand the concept much clearly. Remember to recall the basic formulas of trigonometric functions and also recall Pythagoras theorem to find the third side of the right triangle. This type of question could also be done by dividing the denominator and numerator by cos$\theta $, to directly convert all trigonometric functions into tan$\theta $.