Question

If we have an expression as ${\sin ^{ - 1}}\left( {\dfrac{x}{5}} \right) + \cos e{c^{ - 1}}\left( {\dfrac{5}{4}} \right) = \dfrac{\pi }{2}$, then s equals:
(A) $4$
(B) $5$
(C) $1$
(D) $3$

Answer 1

Hint: In the given problem, we are required to calculate the value of variable x from the equation given to us involving inverse trigonometric functions. We first shift the terms and use the fact that secant and cosecant trigonometric functions are complementary ratios of each other. Then, we take sine on both sides of the equation to simplify the expression and find the value of x.

Complete step-by-step solution:
So, In the given problem, we have to find the value of x in ${\sin ^{ - 1}}\left( {\dfrac{x}{5}} \right) + \cos e{c^{ - 1}}\left( {\dfrac{5}{4}} \right) = \dfrac{\pi }{2}$.
We shift the cosecant inverse function to the right side of the equation using the method of transposition. So, we get,
$ \Rightarrow {\sin ^{ - 1}}\left( {\dfrac{x}{5}} \right) = \dfrac{\pi }{2} - \cos e{c^{ - 1}}\left( {\dfrac{5}{4}} \right)$
Now, we know that secant inverse and cosecant inverse functions are complementary functions of each other. So, we have, $\dfrac{\pi }{2} - \cos e{c^{ - 1}}\left( x \right) = {\sec ^{ - 1}}x$. Hence, we get,
$ \Rightarrow {\sin ^{ - 1}}\left( {\dfrac{x}{5}} \right) = {\sec ^{ - 1}}\left( {\dfrac{5}{4}} \right)$
Now, we take sine on both sides of the equation. So, we get,
$ \Rightarrow \sin \left[ {{{\sin }^{ - 1}}\left( {\dfrac{x}{5}} \right)} \right] = \sin \left[ {{{\sec }^{ - 1}}\left( {\dfrac{5}{4}} \right)} \right]$
Now, we know that $\sin \left[ {{{\sin }^{ - 1}}\left( x \right)} \right] = x$. So, we get,
$ \Rightarrow \left( {\dfrac{x}{5}} \right) = \sin \left[ {{{\sec }^{ - 1}}\left( {\dfrac{5}{4}} \right)} \right] - - - - \left( 1 \right)$
Hence, we have to find the sine of the angle whose secant is given to us as $\left( {\dfrac{5}{4}} \right)$.
Let us assume $\theta $to be the concerned angle.
Then, $\theta = {\sec ^{ - 1}}\left( {\dfrac{5}{4}} \right)$
Taking secant on both sides of the equation, we get
 $ \Rightarrow \sec \theta = \left( {\dfrac{5}{4}} \right)$
To evaluate the value of the required expression, we must keep in mind the formulae of basic trigonometric ratios.
We know that, $\sin \theta = \dfrac{{Perpendicular}}{{Hypotenuse}}$and $\sec \theta = \dfrac{{Hypotenuse}}{{Base}}$.
So, $\sec \theta = \dfrac{{Hypotenuse}}{{Base}} = \dfrac{5}{4}$
Let the length of Hypotenuse be $5x$.
Then, length of base $ = 4x$.
Now, applying Pythagoras Theorem,
${\left( {Hypotenuse} \right)^2} = {\left( {Base} \right)^2} + {\left( {Perpendicular} \right)^2}$
Substituting the values of expressions for Hypotenuse and base, we get,
$ \Rightarrow {\left( {5x} \right)^2} = {\left( {4x} \right)^2} + {\left( {Perpendicular} \right)^2}$
Computing squares of the terms,
$ \Rightarrow 25{x^2} = 16{x^2} + {\left( {Perpendicular} \right)^2}$
Shifting terms in the equation,
$ \Rightarrow {\left( {Perpendicular} \right)^2} = 25{x^2} - 16{x^2}$
$ \Rightarrow {\left( {Perpendicular} \right)^2} = 9{x^2}$
Taking square root on both sides of the equation, we get,
$ \Rightarrow Perpendicular = 3x$
So, we get $Perpendicular = 3x$
Hence, $\sin \theta = \dfrac{{Perpendicular}}{{Hypotenuse}} = \dfrac{3}{5}$
So, the value of$\sin \left[ {{{\sec }^{ - 1}}\left( {\dfrac{5}{4}} \right)} \right]$ is $\left( {\dfrac{3}{5}} \right)$.
Substituting this in equation $\left( 1 \right)$, we get,
$ \Rightarrow \left( {\dfrac{x}{5}} \right) = \dfrac{3}{5}$
Cross multiplying the terms and cancelling the terms in numerator and denominator, we get,
$ \Rightarrow x = \dfrac{3}{5} \times 5 = 3$
So, the value of x is $3$. Hence, the correct answer is option (D).

Note: We must take care while doing the calculations as it can change our final answer. Such problems require basic knowledge of trigonometric ratios and formulae. Besides this, knowledge of concepts of inverse trigonometry is extremely essential to answer these questions correctly.