James used the unit circle to find tan 4pi/3. First he foubd the point on the unit circle that corresponds to 0 = 4pi/3. Next he used the ordered pair (-1/2, -v3/2) to write the radio tan 0 = x/y. How should james finish the problem? Include the final answer for 4pi/3.

Answer 1

To find \(\tan\left(\frac{4\pi}{3}\right)\), James is on the right track by identifying the point on the unit circle corresponding to the angle \(\frac{4\pi}{3}\).

1. Finding the Coordinates: The angle \(\frac{4\pi}{3}\) radians is located in the third quadrant of the unit circle. The coordinates associated with this angle are \((-1/2, -\sqrt{3}/2)\). Here, the x-coordinate is \(-\frac{1}{2}\) and the y-coordinate is \(-\frac{\sqrt{3}}{2}\).

2. Finding the Tangent: The tangent function \(\tan(\theta)\) is defined as the ratio of the y-coordinate to the x-coordinate at that angle: \[ \tan\left(\frac{4\pi}{3}\right) = \frac{y}{x} = \frac{-\sqrt{3}/2}{-1/2} \]

3. Calculating the Ratio: When we simplify the ratio: \[ \tan\left(\frac{4\pi}{3}\right) = \frac{-\sqrt{3}/2}{-1/2} = \frac{-\sqrt{3}/2 \cdot (-2)}{-1} = \sqrt{3} \]

The negatives cancel out, so: \[ \tan\left(\frac{4\pi}{3}\right) = \sqrt{3} \]

Final Answer

Thus, \(\tan\left(\frac{4\pi}{3}\right) = \sqrt{3}\).