Question

How do you use the properties of inverse trigonometric functions to evaluate $ \tan \left( {{{\sin }^{ - 1}}\left( {0.31} \right)} \right) $ ?

Answer 1

Hint: In the given problem, we are required to calculate tangent of an angle whose sine is given to us. 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.

Complete step by step solution:
So, In the given problem, we have to find the value of $ \tan \left( {{{\sin }^{ - 1}}\left( {0.31} \right)} \right) $ .
Hence, we have to find the tangent of the angle whose sine is given to us as $ \left( {0.31} \right) $ .
Let us assume $ \theta $ to be the concerned angle.
Then, $ \theta = {\sin ^{ - 1}}\left( {0.31} \right) $
Taking sine on both sides of the equation, we get
  $ \Rightarrow \sin \theta = 0.31 $
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 $ \tan \theta = \dfrac{{Perpendicular}}{{Base}} $ .
So, $ \sin \theta = \dfrac{{Perpendicular}}{{Hypotenuse}} = 0.31 $
Let the length of hypotenuse be $ x $ .
Now, using the transposition rule, we get, length of perpendicular $ = 0.31x $ .
Now, applying Pythagoras Theorem,
 $ {\left( {Hypotenuse} \right)^2} = {\left( {Base} \right)^2} + {\left( {Perpendicular} \right)^2} $
 $ \Rightarrow {\left( x \right)^2} = {\left( {Base} \right)^2} + {\left( {0.31x} \right)^2} $
 $ \Rightarrow {x^2} = {\left( {Base} \right)^2} + 0.0961{x^2} $
Shifting the terms in the equation in order to find the value of base of the right angled triangle, we get,
 $ \Rightarrow {\left( {Base} \right)^2} = {x^2} - 0.0961{x^2} $
 $ \Rightarrow {\left( {Base} \right)^2} = 0.9039{x^2} $
 $ \Rightarrow \left( {Base} \right) = \sqrt {0.9039{x^2}} = x\sqrt {0.9039} $
So, we get $ Base = x\sqrt {0.9039} $
Hence, $ \tan \theta = \dfrac{{Perpendicular}}{{Base}} = \dfrac{{0.31x}}{{x\sqrt {0.9039} }} $
Cancelling the common factors in numerator and denominator, we get,
 $ \Rightarrow \tan \theta = \dfrac{{0.31}}{{\sqrt {0.9039} }} $
Finding approximate value of the square root in the denominator, we get,
 $ \Rightarrow \tan \theta = \dfrac{{0.31}}{{0.95}} $
So, the value of $ \tan \left( {{{\sin }^{ - 1}}\left( {0.31} \right)} \right) $ is $ \dfrac{{0.31}}{{0.95}} $ .
So, the correct answer is “ $ \dfrac{{0.31}}{{0.95}} $ ”.

Note: For finding a trigonometric ratio for an angle given in terms of an inverse trigonometric ratio, we have to first assume that angle to be some unknown, let's say $ \theta $ . Then proceeding further, we have to find the value of a trigonometric function of that unknown angle $ \theta $ . Then we find the required trigonometric ratio with help of basic trigonometric formulae and definitions of trigonometric ratios. Such questions require clarity of basic concepts of trigonometric functions as well as their inverse.