The required points are \(\left(\dfrac{\pi}{3},\sqrt{3}\right)\) and \(\left(\dfrac{7\pi}{4},-1\right)\). So, the second and fifth options are correct.
Important information:
The given equation is \(y=\tan x\).
We need to find the points that are lying on the graph of the given equation.
Trigonometry:
Substitute \(x=-\dfrac{5\pi}{6}\) in the given equation.
\(y=\tan \left(-\dfrac{5\pi}{6}\right)\)
\(y=-\tan \left(\dfrac{5\pi}{6}\right)\)
\(y=-\tan \left(\pi-\dfrac{\pi}{6}\right)\)
\(y=\tan \left(\dfrac{\pi}{6}\right)\)
\(y=\dfrac{1}{\sqrt{3}}
eq \dfrac{1}{\sqrt{3}}\)
Substitute \(x=\dfrac{\pi}{3}\) in the given equation.
\(y=\tan \left(\dfrac{\pi}{3}\right)\)
\(y=\sqrt{3}\)
Substitute \(x=\dfrac{\pi}{3}\) in the given equation.
\(y=\tan \left(0\right)\)
\(y=0\)
Substitute \(x=\dfrac{3\pi}{4}\) in the given equation.
\(y=\tan \left(\dfrac{3\pi}{4}\right)\)
\(y=\tan \left(\pi-\dfrac{\pi}{4}\right)\)
\(y=-\tan \left(\dfrac{\pi}{4}\right)\)
\(y=-1\)
Substitute \(x=\dfrac{7\pi}{4}\) in the given equation.
\(y=\tan \left(\dfrac{7\pi}{4}\right)\)
\(y=\tan \left(2\pi-\dfrac{\pi}{4}\right)\)
\(y=-\tan \left(\dfrac{\pi}{4}\right)\)
\(y=-1\)
Only \(\left(\dfrac{\pi}{3},\sqrt{3}\right)\) and \(\left(\dfrac{7\pi}{4},-1\right)\) are on the graph of given equation. Therefore, the second and fifth options are correct.
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