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 $(\frac{1}{2},\frac{\sqrt{3}}{2})$.
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.