Question

How do you find the exact values of ${\cos ^{ - 1}}\left( {\dfrac{1}{2}} \right)$ ?

Answer 1

Hint : The inverse trigonometric functions are used to find the missing angles. Here we will use the inverse cosine and cosine function to solve the given expression.

Complete step-by-step answer :
Take the given expression –
${\cos ^{ - 1}}\left( {\dfrac{1}{2}} \right)$
Referring the trigonometric table for the values for cosine function, it implies that $\cos 60^\circ = \dfrac{1}{2}$
Place the cosine angle in the given expression –
${\cos ^{ - 1}}\left( {\dfrac{1}{2}} \right) = {\cos ^{ - 1}}(\cos 60^\circ )$
Cosine inverse and cosine function cancel each other.
${\cos ^{ - 1}}\left( {\dfrac{1}{2}} \right) = 60^\circ$
The above expression can be re-written as –
${\cos ^{ - 1}}\left( {\dfrac{1}{2}} \right) = \dfrac{\pi }{3}$
This is the required solution.
So, the correct answer is “ $\dfrac{\pi }{3}$ ”.

Note : Remember the trigonometric formulas and the correlation between the trigonometric functions to find the unknowns. Remember the trigonometric table for the reference values for different angles for sine, cosine and tangent functions for direct substitution.