Find the $sin$, $cos$, and $ an$ of an angle using the unit circle.

To find the sine, cosine, and tangent of an angle $\theta$ on the unit circle, use the following steps:

1. Identify the coordinates $(x, y)$ on the unit circle corresponding to $\theta$.

2. The $x$-coordinate is $\cos(\theta)$.

3. The $y$-coordinate is $\sin(\theta)$.

4. The tangent of the angle is $\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}$.

For example, consider $\theta = \frac{\pi}{4}$:

1. The coordinates are $(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2})$.

2. $\cos(\frac{\pi}{4}) = \frac{\sqrt{2}}{2}$.

3. $\sin(\frac{\pi}{4}) = \frac{\sqrt{2}}{2}$.

4. $\tan(\frac{\pi}{4}) = \frac{\frac{\sqrt{2}}{2}}{\frac{\sqrt{2}}{2}} = 1$.

To solve for $sin$, $cos$, and $ an$ using the unit circle:

1. Locate the point on the unit circle for angle $ heta$.

2. The $x$-value is $cos( heta)$.

3. The $y$-value is $sin( heta)$.

4. Calculate $ an( heta)$ by dividing $sin( heta)$ by $cos( heta)$.

For $ heta = frac{pi}{3}$:

1. The coordinates are $(frac{1}{2}, frac{sqrt{3}}{2})$.

2. $cos(frac{pi}{3}) = frac{1}{2}$.

3. $sin(frac{pi}{3}) = frac{sqrt{3}}{2}$.

4. $ an(frac{pi}{3}) = frac{frac{sqrt{3}}{2}}{frac{1}{2}} = sqrt{3}$.

To find $sin$, $cos$, and $ an$ using the unit circle:

1. Use the point $(x, y)$ on the unit circle for angle $ heta$.

2. $x = cos( heta)$

3. $y = sin( heta)$

4. $ an( heta) = frac{y}{x}$

For $ heta = frac{pi}{6}$:

1. $(frac{sqrt{3}}{2}, frac{1}{2})$

2. $cos(frac{pi}{6}) = frac{sqrt{3}}{2}$

3. $sin(frac{pi}{6}) = frac{1}{2}$

4. $ an(frac{pi}{6}) = frac{frac{1}{2}}{frac{sqrt{3}}{2}} = frac{1}{sqrt{3}} = frac{sqrt{3}}{3}$

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