Solveeit Logo
Login

Question

Question: How do you evaluate \({{\sec }^{-1}}(2)\) ?...

How do you evaluate sec⁡−1(2){{\sec }^{-1}}(2) ?

Explanation

Solution

In the above question you were asked to find the value of sec⁡−1(2){{\sec }^{-1}}(2) . You know that sec is the reciprocal of cos. And, since the question is sec⁡−1(2){{\sec }^{-1}}(2) , so you can write it as 1sec⁡(2)\dfrac{1}{\sec (2)} after which you need to convert this into cos(2) . So you can use the above concept to solve this problem.

Complete step-by-step answer:
We have to evaluate sec⁡−1(2){{\sec }^{-1}}(2) .
Let us assume sec⁡−1(2)=x{{\sec }^{-1}}(2)=x
Now, when we move the sec to another side, we get,
If, sec⁡(x)=2\sec (x)=2
We know that sec(x) is reciprocal of cos(x),
⇒1cos⁡x=2\Rightarrow \dfrac{1}{\cos x}=2
Now, when we re-arrange, we get,
⇒cos⁡x=12\Rightarrow \cos x=\dfrac{1}{2}
Now, when we move the cos to another side, we get,
⇒x=cos⁡−1(12)\Rightarrow x={{\cos }^{-1}}(\dfrac{1}{2})
=π3,5π3=\dfrac{\pi }{3},\dfrac{5\pi }{3}
As, the range of sec⁡−1x{{\sec }^{-1}}xis [0,π2)∪(π2,π][0,\dfrac{\pi }{2})\cup (\dfrac{\pi }{2},\pi ]
So, sec⁡−1(2)=π3{{\sec }^{-1}}(2)=\dfrac{\pi }{3}
Therefore, the solution for sec⁡−1(2){{\sec }^{-1}}(2) is π3\dfrac{\pi }{3} and general solution for sec⁡−1(2){{\sec }^{-1}}(2) is π3+2πn\dfrac{\pi }{3}+2\pi n , where n can be any integer.

Note: We know that sec is reciprocal of cos. So, we can say according to the above problem statement that, sec⁡−1(2)=1sec⁡(2)=cos⁡(2){{\sec }^{-1}}(2)=\dfrac{1}{\sec (2)}=\cos (2) because sec is the reciprocal of cos.
Also, cosec is reciprocal of sin and cot is reciprocal of tan. So we can use these reciprocals to solve our problems easily. Also, quadrants help us to evaluate trigonometric identities with angles.