Question

Find the principal value of ${{\sec }^{-1}}\left( 2\tan \dfrac{3\pi }{4} \right)$.

Answer 1

Hint:For the above question we have to know about the principal values of an inverse trigonometric function. Principal value of an inverse trigonometric function is a value that belongs to the principal branch of the range of the function. We know that the principal branch of range of ${{\sec }^{-1}}x$ is \left[ 0,\pi \right]-\left\\{ \dfrac{\pi }{2} \right\\}.

Complete step-by-step answer:
We have been given the expression ${{\sec }^{-1}}\left( 2\tan \dfrac{3\pi }{4} \right)$.
Now we know that $\tan \dfrac{3\pi }{4}=-1$.
On substituting the value of $\tan \dfrac{3\pi }{4}$ in the given expression we get as follows:
${{\sec }^{-1}}\left( 2\tan \dfrac{3\pi }{4} \right)={{\sec }^{-1}}\left( 2\times -1 \right)={{\sec }^{-1}}\left( -2 \right)$
So we have ${{\sec }^{-1}}\left( 2\tan \dfrac{3\pi }{4} \right)={{\sec }^{-1}}\left( -2 \right)$
Now we know that the principal value means the value which lies between the defined range of the function.
For ${{\sec }^{-1}}x$ the range is \left[ 0,\pi \right]-\left\\{ \dfrac{\pi }{2} \right\\}.
Since the value of $\sec \left( \dfrac{2\pi }{3} \right)=-2$
So by substituting the value of (-2) in the above expression, we get as follows:
${{\sec }^{-1}}\left( 2\tan \dfrac{3\pi }{4} \right)={{\sec }^{-1}}\left[ \sec \left( \dfrac{2\pi }{3} \right) \right]$
We know that ${{\sec }^{-1}}\sec \theta =\theta$ where ‘$\theta$' must lies between the range of ${{\sec }^{-1}}x$, i.e. \theta \in \left[ 0,\pi \right]-\left\\{ \dfrac{\pi }{2} \right\\}.
So, ${{\sec }^{-1}}\left( 2\tan \dfrac{3\pi }{4} \right)=\dfrac{2\pi }{3}$
Therefore, the principal values of ${{\sec }^{-1}}\left( 2\tan \dfrac{3\pi }{4} \right)$ is equal to $\dfrac{2\pi }{3}$.

Note: Be careful while finding the principal value of inverse trigonometric functions and do check once that the value must lie between the principal branch of range of the function. Sometimes by mistake we might forget the number ‘2’ which is multiplied by $\tan \dfrac{3\pi }{4}$ in the given expression and we only substitute the value of $\tan \dfrac{3\pi }{4}$ and thus we get the incorrect answer so be careful while calculating it.Also be careful while finding the values of $\tan\dfrac{3\pi }{4}$, by mistake we may write $\tan\dfrac{3\pi }{4}=1$ which is wrong since $\tan\dfrac{3\pi }{4}=-1$ , because $\tan\left( \pi -\dfrac{\pi }{4} \right)=-\tan \dfrac{\pi }{4}$ not $\left( \tan \dfrac{\pi }{4} \right)$.