91Ó°ÊÓ

Recall that for an angle \(\alpha\), the coordinates on the unit circle can be found using the sine and cosine trigonometric functions: - x-coordinate = \(\cos(\alpha)\) - y-coordinate = \(\sin(\alpha)\) For \(\alpha = \frac{1}{3}\pi\), let's find the x and y coordinates: - x-coordinate = \(\cos\left(\frac{1}{3}\pi\right)\) - y-coordinate = \(\sin\left(\frac{1}{3}\pi\right)\)

We have: - \(\sin\left(\frac{1}{3}\pi\right) = \frac{\sqrt{3}}{2}\) (as it corresponds to sine of \(60^\circ\), which is a well-known value) - \(\cos\left(\frac{1}{3}\pi\right) = \frac{1}{2}\) (as it corresponds to cosine of \(60^\circ\), which is a well-known value) So, the coordinates on the unit circle for the angle \(\frac{1}{3}\pi\) are \(\left(\frac{1}{2}, \frac{\sqrt{3}}{2}\right)\).

Now that we have the sine and cosine values, we can find the values of the other trigonometric functions: - \(\tan(\alpha) = \frac{\sin(\alpha)}{\cos(\alpha)}\) - \(\csc(\alpha) = \frac{1}{\sin(\alpha)}\) - \(\sec(\alpha) = \frac{1}{\cos(\alpha)}\) - \(\cot(\alpha) = \frac{1}{\tan(\alpha)} = \frac{\cos(\alpha)}{\sin(\alpha)}\) Substitute the sine and cosine values we found in Step 2: - \(\tan\left(\frac{1}{3}\pi\right) = \frac{\frac{\sqrt{3}}{2}}{\frac{1}{2}} = \sqrt{3}\) - \(\csc\left(\frac{1}{3}\pi\right) = \frac{1}{\frac{\sqrt{3}}{2}} = \frac{2}{\sqrt{3}}\cdot\frac{\sqrt{3}}{\sqrt{3}} = \frac{2\sqrt{3}}{3}\) - \(\sec\left(\frac{1}{3}\pi\right) = \frac{1}{\frac{1}{2}} = 2\) - \(\cot\left(\frac{1}{3}\pi\right) = \frac{\frac{1}{2}}{\frac{\sqrt{3}}{2}} = \frac{1}{\sqrt{3}}\cdot\frac{\sqrt{3}}{\sqrt{3}} = \frac{\sqrt{3}}{3}\) To summarize, here are the values of the six trigonometric functions at the point \(\frac{1}{3}\pi\): - \(\sin\left(\frac{1}{3}\pi\right) = \frac{\sqrt{3}}{2}\) - \(\cos\left(\frac{1}{3}\pi\right) = \frac{1}{2}\) - \(\tan\left(\frac{1}{3}\pi\right) = \sqrt{3}\) - \(\csc\left(\frac{1}{3}\pi\right) = \frac{2\sqrt{3}}{3}\) - \(\sec\left(\frac{1}{3}\pi\right) = 2\) - \(\cot\left(\frac{1}{3}\pi\right) = \frac{\sqrt{3}}{3}\)