$ ext{Understanding the Unit Circle: An Advanced Problem}$

To understand the unit circle at an advanced level, consider the problem of determining the exact value of trigonometric functions given a point on the unit circle. Suppose a point $P$ on the unit circle corresponds to an angle $\theta$. Given $P = \left(-\frac{3}{5}, -\frac{4}{5}\right)$, find $\sin \theta$, $\cos \theta$, $\tan \theta$, and the corresponding coordinates for $\theta + 2\pi$.

First, recall that for any point $(x, y)$ on the unit circle:

$ x = \cos \theta, \quad y = \sin \theta $

Thus, for $P = \left(-\frac{3}{5}, -\frac{4}{5}\right)$, we have:

$ \cos \theta = -\frac{3}{5}, \quad \sin \theta = -\frac{4}{5} $

Next, compute $\tan \theta$:

$ \tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{-\frac{4}{5}}{-\frac{3}{5}} = \frac{4}{3} $

Lastly, for the angle $\theta + 2\pi$, the coordinates remain the same since $2\pi$ represents a full rotation around the unit circle:

$ P_{\theta + 2\pi} = \left(-\frac{3}{5}, -\frac{4}{5}\right) $

Consider the unit circle centered at the origin, and a point $P$ at $left(frac{1}{sqrt{2}}, -frac{1}{sqrt{2}}
ight)$. Determine $sin heta$, $cos heta$, $ an heta$, and the points corresponding to $ heta + pi$.

Recall:

$ cos heta = frac{x}{r}, quad sin heta = frac{y}{r}, quad r = 1 ext{ (radius of the unit circle)} $

Thus:

$ cos heta = frac{1}{sqrt{2}}, quad sin heta = -frac{1}{sqrt{2}} $

Next, calculate $ an heta$:

$ an heta = frac{sin heta}{cos heta} = frac{-frac{1}{sqrt{2}}}{frac{1}{sqrt{2}}} = -1 $

For $ heta + pi$, the coordinates will be:

$ left( cos( heta + pi), sin( heta + pi)
ight) = left( -frac{1}{sqrt{2}}, frac{1}{sqrt{2}}
ight) $

Given a point $P$ on the unit circle at $left(-frac{1}{2}, frac{sqrt{3}}{2}
ight)$, find $sin heta$, $cos heta$, and $ an heta$.

We know that:

$ cos heta = -frac{1}{2}, quad sin heta = frac{sqrt{3}}{2} $

Therefore,

$ an heta = frac{sin heta}{cos heta} = frac{frac{sqrt{3}}{2}}{-frac{1}{2}} = -sqrt{3} $

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