The possible trigonometric function is option (C) [tex]sec \theta = \frac{3}{\sqrt{5} }[/tex] and [tex]tan \theta = \frac{-2}{\sqrt{5} }[/tex] is the correct answer.
What is trigonometry?
Trigonometry is mainly concerned with specific functions of angles and their application to calculations. Trigonometry deals with the study of the relationship between the sides of a triangle (right-angled triangle) and its angles.
For the given situation,
The trigonometric function, sin θ = -2/3
We know that [tex]sin \theta = \frac{opposite}{hypotenuse}[/tex]
So, opposite = -2, hypotenuse = 3
The other side of the right triangle adjacent side can be found by using the Pythagoras theorem,
[tex]adjacent=\sqrt{hypotenuse^{2} -opposite^{2} }[/tex]
⇒ [tex]adjacent=\sqrt{3^{2} -(-2)^{2} }[/tex]
⇒ [tex]adjacent=\sqrt{9-4 }[/tex]
⇒ [tex]adjacent=\sqrt{5}[/tex]
Now, [tex]cos \theta = \frac{adjacent}{hypotenuse}[/tex] and [tex]tan \theta = \frac{opposite}{adjacent}[/tex]
Then, [tex]cos \theta = \frac{\sqrt{5} }{3}[/tex]
[tex]sec \theta = \frac{1}{cos \theta}[/tex]
⇒ [tex]sec \theta = \frac{1}{\frac{\sqrt{5} }{3} }[/tex]
⇒ [tex]sec \theta = \frac{3}{\sqrt{5} }[/tex]
[tex]tan \theta = \frac{-2}{\sqrt{5} }[/tex]
Hence we can conclude that the possible trigonometric function is option (C) [tex]sec \theta = \frac{3}{\sqrt{5} }[/tex] and [tex]tan \theta = \frac{-2}{\sqrt{5} }[/tex] is the correct answer.
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