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Question: Evaluate \[\cos \left( {{{\sin }^{ - 1}}\left( {\dfrac{1}{4}} \right)} \right)\]....

Evaluate cos⁡(sin⁡−1(14))\cos \left( {{{\sin }^{ - 1}}\left( {\dfrac{1}{4}} \right)} \right).

Explanation

Solution

Hint : In the given problem, we are required to calculate cosine 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 answer :
So, In the given problem, we have to find the value ofcos⁡(sin⁡−1(14))\cos \left( {{{\sin }^{ - 1}}\left( {\dfrac{1}{4}} \right)} \right).
Hence, we have to find the cosine of the angle whose sine is given to us as 14\dfrac{1}{4} .
Let us assume θ\theta to be the concerned angle.
Then, θ=sin⁡−1(14)\theta = {\sin ^{ - 1}}\left( {\dfrac{1}{4}} \right)
Taking sine on both sides of the equation, we get
=sin⁡θ=14= \sin \theta = \dfrac{1}{4}

To evaluate the value of the required expression, we must keep in mind the formulae of basic trigonometric ratios.
We know that, sin⁡θ=PerpendicularHypotenuse\sin \theta = \dfrac{{Perpendicular}}{{Hypotenuse}} and
cos⁡θ=BaseHypotenuse\cos \theta = \dfrac{{Base}}{{Hypotenuse}} .
So,
sin⁡θ=PerpendicularHypotenuse=14\sin \theta = \dfrac{{Perpendicular}}{{Hypotenuse}} = \dfrac{1}{4}
Let length of the perpendicular be xx .
Then, length of hypotenuse =4x= 4x .
Now, applying Pythagoras Theorem,
(Hypotenuse)2=(Base)2+(Perpendicular)2{\left( {Hypotenuse} \right)^2} = {\left( {Base} \right)^2} + {\left( {Perpendicular} \right)^2}
=(4x)2=(Base)2+(x)2= {\left( {4x} \right)^2} = {\left( {Base} \right)^2} + {\left( x \right)^2}
=16x2=(Base)2+x2= 16{x^2} = {\left( {Base} \right)^2} + {x^2}
=(Base)2=15x2= {\left( {Base} \right)^2} = 15{x^2}
=(Base)=15x2= \left( {Base} \right) = \sqrt {15{x^2}}
=(Base)=15x= \left( {Base} \right) = \sqrt {15} x
So, we get Base=15xBase = \sqrt {15} x
Hence, cos⁡θ=15x4x=154\cos \theta = \dfrac{{\sqrt {15} x}}{{4x}} = \dfrac{{\sqrt {15} }}{4}
So, the value of cos⁡(sin⁡−1(14))\cos \left( {{{\sin }^{ - 1}}\left( {\dfrac{1}{4}} \right)} \right) is 154\dfrac{{\sqrt {15} }}{4} .
So, the correct answer is “ 154\dfrac{{\sqrt {15} }}{4} ”.

Note : The given problem can also be solved by use of some basic trigonometric identities such as sin⁡2(θ)+cos⁡2(θ)=1{\sin ^2}\left( \theta \right) + {\cos ^2}\left( \theta \right) = 1 . This method also provides exposure to the applications of trigonometric identities in various mathematical questions.