Determine the exact values of $sin$, $cos$, and $ an$ for a given angle on the unit circle

Let $\theta$ be an angle on the unit circle such that $\theta = \frac{7\pi}{6}$. Determine the exact values of $\sin(\theta)$, $\cos(\theta)$, and $\tan(\theta)$.

The angle $\theta = \frac{7\pi}{6}$ is in the third quadrant. The reference angle is $\pi – \frac{\pi}{6} = \frac{\pi}{6}$. In the third quadrant, both sine and cosine are negative.

Therefore, $\sin\left(\frac{7\pi}{6}\right) = -\sin\left(\frac{\pi}{6}\right) = -\frac{1}{2}$

$\cos\left(\frac{7\pi}{6}\right) = -\cos\left(\frac{\pi}{6}\right) = -\frac{\sqrt{3}}{2}$

and $\tan\left(\frac{7\pi}{6}\right) = \frac{\sin\left(\frac{7\pi}{6}\right)}{\cos\left(\frac{7\pi}{6}\right)} = \frac{-\frac{1}{2}}{-\frac{\sqrt{3}}{2}} = \frac{1}{\sqrt{3}} = \frac{\sqrt{3}}{3}$

Consider the angle $ heta = frac{7pi}{6}$ on the unit circle. Compute the exact values of $sin( heta)$, $cos( heta)$, and $ an( heta)$.

Since $ heta = frac{7pi}{6}$, this angle is in the third quadrant where sine and cosine values are negative. The reference angle is $frac{pi}{6}$.

We have $sinleft(frac{pi}{6}
ight) = frac{1}{2}$ and $cosleft(frac{pi}{6}
ight) = frac{sqrt{3}}{2}$

Thus, $sinleft(frac{7pi}{6}
ight) = -frac{1}{2}$, $cosleft(frac{7pi}{6}
ight) = -frac{sqrt{3}}{2}$

and $ anleft(frac{7pi}{6}
ight) = frac{sinleft(frac{7pi}{6}
ight)}{cosleft(frac{7pi}{6}
ight)} = frac{-frac{1}{2}}{-frac{sqrt{3}}{2}} = frac{sqrt{3}}{3}$

Find the exact trigonometric values for $ heta = frac{7pi}{6}$ on the unit circle.

$sinleft(frac{7pi}{6}
ight) = -frac{1}{2}$, $cosleft(frac{7pi}{6}
ight) = -frac{sqrt{3}}{2}$

$ anleft(frac{7pi}{6}
ight) = frac{sqrt{3}}{3}$

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