Question

How do you find the exact value of ${{\tan }^{-1}}\left( -1 \right)$?

Answer 1

From the question given, we have been asked to find the exact value of ${{\tan }^{-1}}\left( -1 \right)$. We can find the exact value of ${{\tan }^{-1}}\left( -1 \right)$ by using the basic functions of trigonometry and some basic values of trigonometry. By using the properties of the trigonometry we can find the exact value for the given inverse trigonometric function.

Complete step by step answer:
Now considering from the question we need to find the exact value of the given expression.
From the basic values of trigonometry, we can write that, $\Rightarrow \sin \left( \dfrac{\pi }{4} \right)=\dfrac{\sqrt{2}}{2}$
Now, we have to find the negative angle value for the above function.
By finding the negative angle value for the above function, we get $\Rightarrow \sin \left( -\dfrac{\pi }{4} \right)=-\dfrac{\sqrt{2}}{2}$
Also, $\Rightarrow \cos \left( \dfrac{\pi }{4} \right)=\dfrac{\sqrt{2}}{2}$
Now, we have to find the negative angle value for the above function.
By finding the negative angle value for the above function, we get $\Rightarrow \cos \left( -\dfrac{\pi }{4} \right)=\dfrac{\sqrt{2}}{2}$
We know that tangent function is the division of sine function with cosine function.
Therefore, $\Rightarrow \tan \left( -\dfrac{\pi }{4} \right)=\dfrac{\sin \left( -\dfrac{\pi }{4} \right)}{\cos \left( -\dfrac{\pi }{4} \right)}=\dfrac{-\dfrac{\sqrt{2}}{2}}{\dfrac{\sqrt{2}}{2}}=-1$
By using this, we can find the exact value for the given inverse trigonometric function.
Note that tangent of $\theta$ is periodic with period $\pi$. So, we find
$\tan \left( k\pi -\dfrac{\pi }{4} \right)=-1$ for any integer $k$.
However, the principal value denoted ${{\tan }^{-1}}$ is chosen to lie in the domain $\left( -\dfrac{\pi }{2},\dfrac{\pi }{2} \right)$, which includes $-\dfrac{\pi }{4}$. So that is the value of ${{\tan }^{-1}}\left( -1 \right)$.
Therefore, ${{\tan }^{-1}}\left( -1 \right)=-\dfrac{\pi }{4}$
Hence, we got the exact value for the given inverse trigonometric function.

Note: We should be well aware of the trigonometric functions and their properties. Also, we should be well known about the basic values of basic angles of the trigonometry. Also, we should be very careful while finding the exact value for the given question. Simply we can solve this question using $\tan \left( -\dfrac{\pi }{4} \right)=-1$ that is generally $\tan \left( k\pi -\dfrac{\pi }{4} \right)=-1$ for any integer $k$ . Therefore we can conclude that the solution of ${{\tan }^{-1}}\left( -1 \right)$ is $\dfrac{-\pi }{4}$ in its domain which is $\left( -\dfrac{\pi }{2},\dfrac{\pi }{2} \right)$ .