Question

Find the value of $\sin \left[ {{\sin }^{-1}}\dfrac{\sqrt{5}}{4}+{{\tan }^{-1}}\sqrt{\dfrac{5}{11}} \right]$.
(a) $\dfrac{\sqrt{5}}{4\sqrt{11}}$
(b) $\dfrac{4}{\sqrt{35}}$
(c) $\dfrac{\sqrt{55}}{8}$
(d) None of these

Answer 1

Hint:First, before proceeding for this, we must know the following fact from the question that it makes the right angle triangle. Then, let us suppose the value of ${{\tan }^{-1}}\sqrt{\dfrac{5}{11}}$as x and then by using the formula of trigonometry which is $\sin 2x=\dfrac{2\tan x}{1+{{\tan }^{2}}x}$. Then, by solving the above expression, we get the desired value.

Complete step by step answer:
In this question, we are supposed to find the value of $\sin \left[ {{\sin }^{-1}}\dfrac{\sqrt{5}}{4}+{{\tan }^{-1}}\sqrt{\dfrac{5}{11}} \right]$.
So, before proceeding for this, we must know the following fact from the question that it makes right angle triangle as:

Now, by using the above figure and also the rules of the trigonometry which says that:
$\begin{align}
  & \sin \theta =\dfrac{\sqrt{5}}{4} \\
 & \theta ={{\sin }^{-1}}\dfrac{\sqrt{5}}{4} \\
\end{align}$ and $\begin{align}
  & \tan \theta =\dfrac{\sqrt{5}}{\sqrt{11}} \\
 & \theta ={{\tan }^{-1}}\sqrt{\dfrac{5}{11}} \\
\end{align}$
So, we can clearly see that we get value of $\theta $ in both cases and now equating them, we get:
${{\sin }^{-1}}\dfrac{\sqrt{5}}{4}={{\tan }^{-1}}\sqrt{\dfrac{5}{11}}$
Then, by using the above relation, we get the expression as:
$\begin{align}
  & \sin \left[ {{\tan }^{-1}}\sqrt{\dfrac{5}{11}}+{{\tan }^{-1}}\sqrt{\dfrac{5}{11}} \right] \\
 & \Rightarrow \sin \left[ 2{{\tan }^{-1}}\sqrt{\dfrac{5}{11}} \right] \\
\end{align}$
Now, let us suppose the value of ${{\tan }^{-1}}\sqrt{\dfrac{5}{11}}$as x and then by using the formula of trigonometry which is $\sin 2x=\dfrac{2\tan x}{1+{{\tan }^{2}}x}$ as:
$\dfrac{2\sqrt{\dfrac{5}{11}}}{1+{{\left( \sqrt{\dfrac{5}{11}} \right)}^{2}}}=\dfrac{2\sqrt{\dfrac{5}{11}}}{1+\dfrac{5}{11}}$
Then, by solving the above expression, we get the value as:
$\begin{align}
  & 2\times \sqrt{\dfrac{5}{11}}\times \left( \dfrac{11}{11+5} \right)=2\times \sqrt{\dfrac{5}{11}}\times \left( \dfrac{11}{16} \right) \\
 & \Rightarrow \dfrac{\sqrt{11\times 5}}{8} \\
 & \Rightarrow \dfrac{\sqrt{55}}{8} \\
\end{align}$
So, we get the value of the required expression as $\dfrac{\sqrt{55}}{8}$.
Hence, the option (c) is correct.

Note:
Now, to solve these type of the questions we need to know some of the basics of the right angled triangle with sides as perpendicular p, base b and hypotenuse h and also by using the concept of trigonometry, we get the formulas for the following figure as:

$\begin{align}
  & \sin \theta =\dfrac{p}{h} \\
 & \cos \theta =\dfrac{b}{h} \\
 & \tan \theta =\dfrac{p}{b} \\
\end{align}$