Question

Find the value of \[\dfrac{{{{\cos }^{ - 1}}\left( {\dfrac{{41}}{{49}}} \right)}}{{{{\sin }^{ - 1}}\left( {\dfrac{2}{7}} \right)}} \]
A. 4
B. 3
C. 2
D. 1

Answer 1

Hint:
In this question, we need to determine the value of the given trigonometric equation. For this, we will use the trigonometric identities along with the inverse trigonometric function.
Trigonometric functions used in this question:
\[\cos 2\theta = 1 - 2{\sin ^2}\theta \]
\[{\cos ^{ - 1}}\left( {\cos x} \right) = x\]

Complete step by step solution:
We have to determine the value of the trigonometric function \[\dfrac{{{{\cos }^{ - 1}}\left( {\dfrac{{41}}{{49}}} \right)}}{{{{\sin }^{ - 1}}\left( {\dfrac{2}{7}} \right)}} - - (i)\]
Let us expand the numerator of the given trigonometric equation, so this can be written as
 \[{\cos ^{ - 1}}\left( {\dfrac{{41}}{{49}}} \right) = {\cos ^{ - 1}}\left( {1 - \dfrac{8}{{49}}} \right)\]
Now, write 8 as the product of 2 and 4 in the above equation, we get
\[
  {\cos ^{ - 1}}\left( {\dfrac{{41}}{{49}}} \right) = {\cos ^{ - 1}}\left( {1 - \dfrac{{2 \times {2^2}}}{{{7^2}}}} \right) \\
   = {\cos ^{ - 1}}\left( {1 - 2{{\left( {\dfrac{2}{7}} \right)}^2}} \right) - - (ii) \\
 \]
Let us equate the denominator of the given trigonometric equation equal to \[\theta \]; hence we can say
\[
  {\sin ^{ - 1}}\left( {\dfrac{2}{7}} \right) = \theta \\
  \Rightarrow \sin \theta = \dfrac{2}{7} \\
 \]
Since the value of \[\sin \theta = \dfrac{2}{7}\], hence we can write equation (ii) as
\[{\cos ^{ - 1}}\left( {\dfrac{{41}}{{49}}} \right) = {\cos ^{ - 1}}\left( {1 - 2{{\sin }^2}\theta } \right) - - (iii)\]
Now, as we know the trigonometric identity \[\cos 2\theta = 1 - 2{\sin ^2}\theta \], hence we can write equation (iii) as
\[{\cos ^{ - 1}}\left( {\dfrac{{41}}{{49}}} \right) = {\cos ^{ - 1}}\left( {\cos 2\theta } \right) - - - - (iv)\]
Now, using the inverse trigonometric identity \[{\cos ^{ - 1}}\left( {\cos x} \right) = x\], equation (iv) can be simplified as:
\[{\cos ^{ - 1}}\left( {\dfrac{{41}}{{49}}} \right) = 2\theta - - - - (v)\]
Substituting the values from equation (v) and \[{\sin ^{ - 1}}\left( {\dfrac{2}{7}} \right) = \theta \] in the equation (i), we get
\[\dfrac{{{{\cos }^{ - 1}}\left( {\dfrac{{41}}{{49}}} \right)}}{{{{\sin }^{ - 1}}\left( {\dfrac{2}{7}} \right)}} = \dfrac{{2\theta }}{\theta } = 2\]
Hence, \[\dfrac{{{{\cos }^{ - 1}}\left( {\dfrac{{41}}{{49}}} \right)}}{{{{\sin }^{ - 1}}\left( {\dfrac{2}{7}} \right)}} = 2\]

Therefore option (C) is correct

Note:
It is worth noting down here that we have simplified the term inside the numerator such that the equivalent absolute value can be seen similar to the term available in the denominator. This is done so as to use the inverse trigonometric identity of \[\cos 2\theta = 1 - 2{\sin ^2}\theta \] and then, \[{\cos ^{ - 1}}\left( {\cos x} \right) = x\]