Question

Find the value ${\sec ^{ - 1}}\left[ {\sec \left( { - {{30}^\circ}} \right)} \right]$

Answer 1

Hint: According to given in the question we have to determine the value of ${\sec ^{ - 1}}\left[ {\sec \left( { - {{30}^\circ}} \right)} \right]$. So, first of all we have let the given trigonometric expression be some variable like x, y, or z.
Now, we have to take ${\sin ^{ - 1}}$to the right hand side of the trigonometric expression.
Now, to solve the obtained trigonometric expression we have to use the formula as mentioned below:

Formula used: $ \Rightarrow \sec ( - \theta ) = \sec \theta ..................(A)$
Now, to solve the trigonometric expression further we have to use the formula as mentioned below:
$ \Rightarrow \sec {30^\circ} = \dfrac{2}{{\sqrt 3 }}.........................(B)$
After applying the formula above we just have to convert it into radians which can be obtained by multiplying the obtained angle with $\dfrac{\pi }{{{{180}^\circ}}}$
$ \Rightarrow {\sin ^{ - 1}}(\sin \theta ) = \theta ...........................(C)$

Complete step-by-step solution:
Step 1: First of all we have to let the given trigonometric expression be some variable as x as mentioned in the solution hint. Hence,
$ \Rightarrow {\sec ^{ - 1}}\left[ {\sec \left( { - {{30}^\circ}} \right)} \right] = x$……………………..(1)
Step 2: Now, we have to take ${\sin ^{ - 1}}$to the right hand side of the trigonometric expression as mentioned in the solution hint.
$ \Rightarrow \left[ {\sec \left( { - {{30}^\circ}} \right)} \right] = \sec x$
Now, on rearranging all the terms of the trigonometric expression as obtained just above,
 $ \Rightarrow \sec x = \left[ {\sec \left( { - {{30}^\circ}} \right)} \right]...............................(2)$
Step 3: Now, to solve the expression (2) as obtained in the step 2 we have to use the formula (A) as mentioned in the solution hint.
$ \Rightarrow \sec x = \sec {30^\circ}...............................(3)$
Step 4: Now, to solve the expression (3) as obtained in the solution step 3 we have to use the formula as mentioned in the solution hint.
$ \Rightarrow \sec x = \dfrac{2}{{\sqrt 3 }}...............................(4)$
Now, we have to convert the degree into radians as mentioned in the solution hint,
$
   \Rightarrow \sec x = \sec \dfrac{\pi }{6} \\
   \Rightarrow x = \dfrac{\pi }{6} \\
 $

Hence, with the help of formula (A) and (B) we have determined the value of ${\sec ^{ - 1}}\left[ {\sec \left( { - {{30}^\circ}} \right)} \right] = \dfrac{\pi }{6}$

Note: Another solution:
Step 1: First of all to find the value of given trigonometric expression ${\sec ^{ - 1}}\left[ {\sec \left( { - {{30}^\circ}} \right)} \right]$ we have to use formula (C) as mentioned in the solution hint. Hence,
$ \Rightarrow {\sec ^{ - 1}}\left[ {\sec \left( { - {{30}^\circ}} \right)} \right] = {30^\circ}$
Step 2: Now, we have to convert the obtained degree as in step 1 into radians as mentioned in the solution hint.
$
   = {30^\circ} \times \dfrac{\pi }{{{{180}^\circ}}} \\
   = \dfrac{\pi }{6}
 $
Final solution: Hence, with the help of formula (c) as mentioned in the solution hint we have determined the value of ${\sec ^{ - 1}}\left[ {\sec \left( { - {{30}^\circ}} \right)} \right] = \dfrac{\pi }{6}$