Find the value of $ an( heta)$ given specific conditions on the unit circle.

Given that $\theta$ is a point on the unit circle where the coordinates are $( \cos(\theta), \sin(\theta) )$ and $\theta$ lies in the second quadrant, find $\tan(\theta)$.

Since $\theta$ is in the second quadrant, $\cos(\theta)$ is negative and $\sin(\theta)$ is positive. Assume $\cos(\theta) = -3/5$, we use the Pythagorean identity to find $\sin(\theta)$:

$\cos^2(\theta) + \sin^2(\theta) = 1$

$(-3/5)^2 + \sin^2(\theta) = 1$

$9/25 + \sin^2(\theta) = 1$

$\sin^2(\theta) = 1 – 9/25$

$\sin^2(\theta) = 16/25$

Since $\sin(\theta)$ is positive in the second quadrant:

$\sin(\theta) = 4/5$

Now, we can find $\tan(\theta)$:

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

Given that $ heta$ is on the unit circle and $ heta$ is in the third quadrant, find $ an( heta)$ assuming $sin( heta) = -5/13$.

In the third quadrant, both $sin( heta)$ and $cos( heta)$ are negative. Using the Pythagorean identity:

$cos^2( heta) + sin^2( heta) = 1$

$cos^2( heta) + (-5/13)^2 = 1$

$cos^2( heta) + 25/169 = 1$

$cos^2( heta) = 1 – 25/169$

$cos^2( heta) = 144/169$

Since $cos( heta)$ is negative in the third quadrant:

$cos( heta) = -12/13$

Now, we can find $ an( heta)$:

$ an( heta) = frac{sin( heta)}{cos( heta)} = frac{-5/13}{-12/13} = frac{5}{12}$

If $ heta$ is in the fourth quadrant and $cos( heta) = 3/5$, find $ an( heta)$.

In the fourth quadrant, $cos( heta)$ is positive and $sin( heta)$ is negative. Using the identity:

$sin^2( heta) + cos^2( heta) = 1$

$sin^2( heta) + (3/5)^2 = 1$

$sin^2( heta) + 9/25 = 1$

$sin^2( heta) = 16/25$

Since $sin( heta)$ is negative in the fourth quadrant:

$sin( heta) = -4/5$

Therefore,

$ an( heta) = frac{sin( heta)}{cos( heta)} = frac{-4/5}{3/5} = -frac{4}{3}$

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