Question

If the value of $5\sec \theta = 13$ , then prove that $\sin \theta = \dfrac{{12}}{{13}}$.

Answer 1

Hint- Make use of Pythagoras theorem and also the definition of trigonometric ratios and solve this problem.

Complete step-by-step answer:
In this question, we have been given that $5\sec \theta = 13$
So, now we can write this as $\sec \theta = \dfrac{{13}}{5}$
Now let us make use of the definition of trigonometric ratios,
$\sec \theta = \dfrac{{hyp}}{{adj}} = \dfrac{{13}}{5}$
But , here we have to find out the value of $\sin \theta $
From definition, we know that $\sin \theta = \dfrac{{opp}}{{hyp}}$
But from $\sec \theta $ , we know only the value of hypotenuse and adjacent side , to find out the value of $\sin \theta $ , we also have to know the value of the opposite side
To find out, let's make use of the formula of Pythagoras theorem
Pythagoras theorem states that
\[\begin{gathered}
  hy{p^2} = op{p^2} + ad{j^2} \\
   \Rightarrow opp = \sqrt {hy{p^2} - ad{j^2}} \\
   \Rightarrow opp = \sqrt {{{13}^2} - {5^2}} \\
   \Rightarrow opp = \sqrt {169 - 25} \\
   \Rightarrow opp = \sqrt {144} = 12 \\
\end{gathered} \]
So, now we have the value of hypotenuse as 13 and that of opposite side as 12
So, let's make use of the definition of $\sin \theta = \dfrac{{opp}}{{hyp}}$ and let's find out the value
From this, we get \[\sin \theta = \dfrac{{12}}{{13}}\]
Hence the result is proved.

Note: In this case, we had to prove for $\sin \theta $, we can also be asked to prove for other trigonometric ratios as well, so in that case we have to make use of the appropriate definition for those trigonometric ratios and solve it.