Compute $ an left( frac{4pi}{3}
ight)$ using the unit circle
Answer 1
Chloe Evans
To find $\tan \left( \frac{4\pi}{3} \right)$, we use the unit circle.
The angle $\frac{4\pi}{3}$ is in the third quadrant where tangent is positive since both sine and cosine are negative and $\tan\theta = \frac{\sin\theta}{\cos\theta}$
The reference angle for $\frac{4\pi}{3}$ is $\frac{4\pi}{3} – \pi = \frac{\pi}{3}. $
Therefore, $\sin\left(\frac{4\pi}{3}\right) = -\sin\left( \frac{\pi}{3} \right) = -\frac{\sqrt{3}}{2}$ and $\cos\left(\frac{4\pi}{3}\right) = -\cos\left( \frac{\pi}{3} \right) = -\frac{1}{2}.$
Hence,
$ \tan \left( \frac{4\pi}{3} \right) = \frac{\sin \left( \frac{4\pi}{3} \right)}{\cos \left( \frac{4\pi}{3} \right)} = \frac{-\frac{\sqrt{3}}{2}}{-\frac{1}{2}} = \sqrt{3} $
Answer 2
Michael Moore
First, locate $frac{4pi}{3}$ on the unit circle. This angle lies in the third quadrant.
The coordinates of the point corresponding to $frac{4pi}{3}$ are $(-frac{1}{2}, -frac{sqrt{3}}{2})$.
The tangent function is defined as the ratio of the y-coordinate to the x-coordinate.
Therefore,
$ an left( frac{4pi}{3}
ight) = frac{y}{x} = frac{-frac{sqrt{3}}{2}}{-frac{1}{2}} = sqrt{3} $
Answer 3
Christopher Garcia
Angle $frac{4pi}{3}$ is in the third quadrant.
The coordinates are $(-frac{1}{2}, -frac{sqrt{3}}{2})$.
So,
$ an left( frac{4pi}{3}
ight) = sqrt{3} $
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