Question

If $24\tan \theta = 7$ then find
(i) $\sin \theta $
(ii) $\cos \theta $

Answer 1

Hint:
 Divide the given equation by 24 to find the value of $\tan \theta $, which is also equal to $\tan \theta = \dfrac{P}{B}$, where $P$ is perpendicular and $B$ is base of a right triangle. Next, use Pythagoras theorem to find the value of hypotenuse. At last, substitute the values in the formulas, $\sin \theta = \dfrac{P}{H}$ and $\cos \theta = \dfrac{B}{H}$ to find the required values.

Complete step by step solution:
We are given that $24\tan \theta = 7$
We will first find the value of \[\tan \theta \] by dividing the equation by 24.
$\tan \theta = \dfrac{7}{{24}}$
Now, we know that $\tan \theta = \dfrac{P}{B}$, where $P$ is perpendicular and $B$is the base of a right triangle.
Therefore,
\[P = 7,B = 24\]
Also, in a right triangle, the sum of square of perpendicular and base is equal to the square of hypotenuse by Pythagoras theorem.
${H^2} = {P^2} + {B^2}$
Now, we will substitute the value of $P$ and $B$ in the above equation to find the value of $H$
$
  {H^2} = {\left( 7 \right)^2} + {\left( {24} \right)^2} \\
   \Rightarrow {H^2} = 49 + 576 \\
   \Rightarrow {H^2} = 625 \\
$
Take square root on both sides, $H = 25$
In part (i), we have to find the value of $\sin \theta $
We know that $\sin \theta = \dfrac{P}{H}$
On substituting the value of $P$ and $H$ in the above formula, we will get,
$\sin \theta = \dfrac{7}{{25}}$
Similarly, In part (ii), we have to find the value of $\cos \theta $
We know that $\cos \theta = \dfrac{B}{H}$
On substituting the value of $B$ and $H$ in the above formula, we will get,

$\cos \theta = \dfrac{{24}}{{25}}$

Note:
After calculating the value of $\sin \theta $, we can use the identity ${\sin ^2}\theta + {\cos ^2}\theta = 1$ to find the value of $\cos \theta $ by substituting the value of $\sin \theta $ and then solving the equation to find the value of $\cos \theta $. Students must know the basic formula of ratios of trigonometry to do these types of questions.