Question
Question: How do you evaluate \(\cos \left( {{\sin }^{-1}}\left( \dfrac{\sqrt{3}}{2} \right) \right)\) without...
How do you evaluate cos(sin−1(23)) without a calculator?
Solution
For answering this question we have been asked to evaluate the value of cos(sin−1(23)) without using a calculator. From the basic concepts we know that the value of sin60∘ is 23 and the value of cos60∘ is 21 .
Complete step by step solution:
Now considering from the question we need to evaluate the value of cos(sin−1(23)) .
From the basic concepts of trigonometry we know that the value of sin60∘ is 23 and the value of cos60∘ is 21 .
Now we will substitute 23 with sin60∘ , then we will have ⇒cos(sin−1(sin60∘)) .
From the basics of concepts we know that we have a formula in trigonometry mathematically given as sin−1(sinθ)=θ .
Now we will use this formula and further simplify the expression. After doing that we will have ⇒cos(60∘) .
Now we will use the value of cos(60∘) and simplify it further.
After applying we will have ⇒21 .
Therefore we can conclude that the value of the given expression cos(sin−1(23)) is given as 21 .
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(cos−1(21)) we can simplify this and evaluate it and then we will get sin(cos−1(cos60∘))=sin60∘⇒23 because we know that the value of sin60∘ is 23 and the value of cos60∘ is 21 .