Use the triangle shown on the unit circle to find tanπ3 . (1 point) Responses 1 1 12 Start Fraction 1 over 2 End Fraction 3–√ Start Root 3 End Root 3√2

Answer 1

To find \(\tan\left(\frac{\pi}{3}\right)\) using the unit circle and the properties of a triangle, we can proceed as follows:

1. \(\frac{\pi}{3}\) radians corresponds to \(60^\circ\).
2. On the unit circle, the coordinates for \(\frac{\pi}{3}\) are \(\left(\frac{1}{2}, \frac{\sqrt{3}}{2}\right)\).
3. The tangent function is defined as the ratio of the sine to the cosine:

\[ \tan\left(\frac{\pi}{3}\right) = \frac{\sin\left(\frac{\pi}{3}\right)}{\cos\left(\frac{\pi}{3}\right)} = \frac{\frac{\sqrt{3}}{2}}{\frac{1}{2}}. \]

4. Simplifying this gives:

\[ \tan\left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2} \cdot \frac{2}{1} = \sqrt{3}. \]

So, the value of \(\tan\left(\frac{\pi}{3}\right)\) is \(\sqrt{3}\).

From the choices provided in your question, it seems there's a formatting issue, but the correct response for \(\tan\left(\frac{\pi}{3}\right)\) is \(\sqrt{3}\). Please clarify or correct the choices if necessary.