To find \(\tan \theta\) using a triangle on the unit circle, we need to recall the definitions of sine and cosine in relation to a right triangle inscribed in the unit circle.
In the unit circle:
- The x-coordinate gives the value of \(\cos \theta\).
- The y-coordinate gives the value of \(\sin \theta\).
- The tangent is defined as \(\tan \theta = \frac{\sin \theta}{\cos \theta}\).
Based on the responses you provided, it seems like we may be considering an angle that has an associated triangle where the sides represent the sine and cosine values.
To find \(\tan \theta\), follow these steps:
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If we have the coordinates of a point on the unit circle (let's say \((\frac{1}{2}, \frac{\sqrt{3}}{2})\) for \(\theta = 60^\circ\) or \(\frac{\pi}{3}\)), we calculate:
\[ \tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{\frac{\sqrt{3}}{2}}{\frac{1}{2}} = \sqrt{3} \]
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If the triangle represents a specific angle, the opposite side (y) and adjacent side (x) indicate the sine and cosine, respectively.
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Using one of the responses and assuming it is \(\sqrt{3}\), then:
\[ \tan \theta = \sqrt{3} \]
Now, if you're trying to select from another set of responses, please clarify which triangle and angle correspond to the options. If those responses relate to specific angles, we could match them based on standard angle values.
If you could specify the angle or further clarify the triangle's sides or coordinates, I would be able to assist you more precisely!
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