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