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Question: How do you find the exact value of \({\sec ^{ - 1}}\left( 2 \right)\) ?...

How do you find the exact value of sec⁡−1(2){\sec ^{ - 1}}\left( 2 \right) ?

Explanation

Solution

In order to find the solution of a trigonometric equation start by taking the inverse trigonometric function such as inverse sine, inverse cosine, inverse tangent on both sides of the equation and then set up reference angles to find the rest of the answers.

Complete step by step answer:
The principal value branch for sin⁡−1{\sin ^{ - 1}} is [−π2,π2]\left[ { - \dfrac{\pi }{2},\dfrac{\pi }{2}} \right].The principal value branch for cos⁡−1{\cos ^{ - 1}} is [0,π]\left[ {0,\pi } \right].The principal value branch for tan⁡−1{\tan ^{ - 1}} is (−π2,π2)\left( { - \dfrac{\pi }{2},\dfrac{\pi }{2}} \right)
In the given question, we are required to find the value of sec⁡−1(2){\sec ^{ - 1}}\left( 2 \right). According to the definition of inverse ratio, if sec⁡(x)=2\sec \left( x \right) = 2, then x=sec⁡−1(2)x = {\sec ^{ - 1}}\left( 2 \right).Now, we know that secant function is positive in first and fourth quadrants only. Also, we know that sec⁡(π3)=2\sec \left( {\dfrac{\pi }{3}} \right) = 2.
So, We have sec⁡(π3)=2\sec \left( {\dfrac{\pi }{3}} \right) = 2.

Hence x=sec⁡−1(2)=(π3)x = {\sec ^{ - 1}}\left( 2 \right) = \left( {\dfrac{\pi }{3}} \right).

Additional information:
The basic inverse trigonometric functions are used to find the missing angles in right triangles. While the regular trigonometric functions are used to determine the missing sides of the right-angled triangles, using the following formulae:
sin⁡θ=Opposite SideHypotenuse\sin \theta = \dfrac{{{\text{Opposite Side}}}}{{{\text{Hypotenuse}}}}
cos⁡θ=Adjacent SideHypotenuse\cos \theta = \dfrac{{{\text{Adjacent Side}}}}{{{\text{Hypotenuse}}}}
tan⁡θ=(Opposite SideAdjacent Side)\tan \theta = \left( {\dfrac{{{\text{Opposite Side}}}}{{{\text{Adjacent Side}}}}} \right)

Note: To find answers to such questions use reference triangles, draw your triangle on the set of axes in the proper quadrant. The inverse trigonometric functions are used to find the missing angles. Besides the trigonometric formulae and identities, we also have some trigonometric rules such as sine rule and cosine rule. The given question requires us to find the exact value of sec⁡−1(2){\sec ^{ - 1}}\left( 2 \right). We can also find the exact value of sec⁡−1(2){\sec ^{ - 1}}\left( 2 \right) by converting the inverse secant function into an inverse cosine function.