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

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

Explanation

Solution

In the given problem, we are required to calculate cotangent 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:
In the given problem, we have to find the value of cot⁡(sin⁡−1(14))\cot \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 the 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, cot⁡θ=BasePerpendicular=15xx\cot \theta = \dfrac{{Base}}{{Perpendicular}} = \dfrac{{\sqrt {15} x}}{x}
Cancelling common terms in numerator and denominator, we get,
⇒cot⁡θ=15\Rightarrow \cot \theta = \sqrt {15}
Therefore, the value of cot⁡(sin⁡−1(14))\cot \left( {{{\sin }^{ - 1}}\left( {\dfrac{1}{4}} \right)} \right) is 15\sqrt {15} .

Additional information:
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, lets say θ\theta . Then proceeding further, we have to find the value of a trigonometric function of that unknown angle θ\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.

Note:
The given problem can also be solved by use of some basic trigonometric identities such as cos⁡ec2(θ)=cot⁡2(θ)+1\cos e{c^2}\left( \theta \right) = {\cot ^2}\left( \theta \right) + 1.
This method also provides exposure to the applications of trigonometric identities in various mathematical questions. First we calculate cosecant of the same angle by using trigonometric formula cos⁡ec(θ)=1sin⁡(θ)\cos ec\left( \theta \right) = \dfrac{1}{{\sin \left( \theta \right)}}.
Then, we substitute the cosecant value in the identity cos⁡ec2(θ)=cot⁡2(θ)+1\cos e{c^2}\left( \theta \right) = {\cot ^2}\left( \theta \right) + 1 to find the value of cotangent.