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Question: How do you evaluate \(\cos \left( {{\sin }^{-1}}\left( \dfrac{\sqrt{3}}{2} \right) \right)\) without...

How do you evaluate cos(sin1(32))\cos \left( {{\sin }^{-1}}\left( \dfrac{\sqrt{3}}{2} \right) \right) without a calculator?

Explanation

Solution

For answering this question we have been asked to evaluate the value of cos(sin1(32))\cos \left( {{\sin }^{-1}}\left( \dfrac{\sqrt{3}}{2} \right) \right) without using a calculator. From the basic concepts we know that the value of sin60\sin {{60}^{\circ }} is 32\dfrac{\sqrt{3}}{2} and the value of cos60\cos {{60}^{\circ }} is 12\dfrac{1}{2} .

Complete step by step solution:
Now considering from the question we need to evaluate the value of cos(sin1(32))\cos \left( {{\sin }^{-1}}\left( \dfrac{\sqrt{3}}{2} \right) \right) .
From the basic concepts of trigonometry we know that the value of sin60\sin {{60}^{\circ }} is 32\dfrac{\sqrt{3}}{2} and the value of cos60\cos {{60}^{\circ }} is 12\dfrac{1}{2} .
Now we will substitute 32\dfrac{\sqrt{3}}{2} with sin60\sin {{60}^{\circ }} , then we will have cos(sin1(sin60))\Rightarrow \cos \left( {{\sin }^{-1}}\left( \sin {{60}^{\circ }} \right) \right) .
From the basics of concepts we know that we have a formula in trigonometry mathematically given as sin1(sinθ)=θ{{\sin }^{-1}}\left( \sin \theta \right)=\theta .
Now we will use this formula and further simplify the expression. After doing that we will have cos(60)\Rightarrow \cos \left( {{60}^{\circ }} \right) .
Now we will use the value of cos(60)\cos \left( {{60}^{\circ }} \right) and simplify it further.
After applying we will have 12\Rightarrow \dfrac{1}{2} .

Therefore we can conclude that the value of the given expression cos(sin1(32))\cos \left( {{\sin }^{-1}}\left( \dfrac{\sqrt{3}}{2} \right) \right) is given as 12\dfrac{1}{2} .

Note: While answering questions of this type we should be sure with our concepts and calculations. This type of question can be answered in a short span of time and very few mistakes are possible. Similarly we can simplify any trigonometric expression for example if we consider the trigonometric expression sin(cos1(12))\sin \left( {{\cos }^{-1}}\left( \dfrac{1}{2} \right) \right) we can simplify this and evaluate it and then we will get sin(cos1(cos60))=sin6032\sin \left( {{\cos }^{-1}}\left( \cos {{60}^{\circ }} \right) \right)=\sin {{60}^{\circ }}\Rightarrow \dfrac{\sqrt{3}}{2} because we know that the value of sin60\sin {{60}^{\circ }} is 32\dfrac{\sqrt{3}}{2} and the value of cos60\cos {{60}^{\circ }} is 12\dfrac{1}{2} .