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

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

Explanation

Solution

According to given in the question we have to determine the value of sec⁡−1[sec⁡(−30∘)]{\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{\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: ⇒sec⁡(−θ)=sec⁡θ..................(A) \Rightarrow \sec ( - \theta ) = \sec \theta ..................(A)
Now, to solve the trigonometric expression further we have to use the formula as mentioned below:
⇒sec⁡30∘=23.........................(B)\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 π180∘\dfrac{\pi }{{{{180}^\circ}}}
⇒sin⁡−1(sin⁡θ)=θ...........................(C)\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,
⇒sec⁡−1[sec⁡(−30∘)]=x\Rightarrow {\sec ^{ - 1}}\left[ {\sec \left( { - {{30}^\circ}} \right)} \right] = x……………………..(1)
Step 2: Now, we have to take sin⁡−1{\sin ^{ - 1}}to the right hand side of the trigonometric expression as mentioned in the solution hint.
⇒[sec⁡(−30∘)]=sec⁡x\Rightarrow \left[ {\sec \left( { - {{30}^\circ}} \right)} \right] = \sec x
Now, on rearranging all the terms of the trigonometric expression as obtained just above,
⇒sec⁡x=[sec⁡(−30∘)]...............................(2)\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.
⇒sec⁡x=sec⁡30∘...............................(3)\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.
⇒sec⁡x=23...............................(4)\Rightarrow \sec x = \dfrac{2}{{\sqrt 3 }}...............................(4)
Now, we have to convert the degree into radians as mentioned in the solution hint,
⇒sec⁡x=sec⁡π6 ⇒x=π6  \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[sec⁡(−30∘)]=π6{\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[sec⁡(−30∘)]{\sec ^{ - 1}}\left[ {\sec \left( { - {{30}^\circ}} \right)} \right] we have to use formula (C) as mentioned in the solution hint. Hence,
⇒sec⁡−1[sec⁡(−30∘)]=30∘\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∘×π180∘ =π6 = {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[sec⁡(−30∘)]=π6{\sec ^{ - 1}}\left[ {\sec \left( { - {{30}^\circ}} \right)} \right] = \dfrac{\pi }{6}