Question

How do you find the exact value of ${\tan ^{ - 1}}\left( {\dfrac{{ - \sqrt 3 }}{3}} \right)$?

Answer 1

First see whether the value of $\theta$ is known or not if not, then try to solve, so that we get the known value of $\theta$. And then check whether we have a proper sign in the number or not, if not then try to convert it into another angle so that the value matched with the $\theta$ value.

Complete step by step answer:
To solve this equation, first check whether we know the value of $\theta$ or not. In this problem, it seems we have to simplify more to get the solution. So, first let us consider the bracketed term,
$\left( {\dfrac{{ - \sqrt 3 }}{3}} \right)$ $=$ $\left( {\dfrac{{ - \sqrt 3 }}{{\sqrt 3 \times \sqrt 3 }}} \right) = \left( {\dfrac{{ - 1}}{{\sqrt 3 }}} \right)$
Now find the value of $\theta$, for which the value is $\left( {\dfrac{{ - 1}}{{\sqrt 3 }}} \right)$ and the value of $\theta$ is in negative value. And$\tan \theta$ is negative in the first, second and fourth quadrant. We know the value for $\tan \left( {\dfrac{\pi }{6}} \right)$ is $\left( {\dfrac{1}{{\sqrt 3 }}} \right)$. And the solution will be $\theta = \dfrac{\pi }{6}$, but the value of $\tan \dfrac{\pi }{6}$ is equal to $\left( {\dfrac{1}{{\sqrt 3 }}} \right)$. But we have $\left( {\dfrac{{ - 1}}{{\sqrt 3 }}} \right)$, the angle cannot be negative. So, we are considering $\tan \left( {\pi - \theta } \right) = - \tan \theta$, substituting the value of $\theta$ and we get,
$\tan \left( {\pi - \dfrac{\pi }{6}} \right) = \tan \left( {\dfrac{{5\pi }}{6}} \right)$
And the value of $\theta$ will be equal to $\dfrac{{5\pi }}{6}$.
Additional information: Here, ${180^ \circ }$ is expressed as $\pi$ because the circumference of a triangle is $2\pi r$.

Note: To solve this problem we must know the trigonometric ratios for related triangles. Because this is where many students will struggle. And we must know the value of $\sin \theta ,\cos \theta ,\tan \theta ,\cos ec\theta ,\sec \theta ,$ and $\cot \theta$ for the values ${0^ \circ },{30^ \circ },{45^ \circ },{60^ \circ },$ and ${90^ \circ }$. Knowing these values, we can solve any problem by converting them into known values.