How to find $ an $ on unit circle?

To find the tangent of an angle $ \theta $ on the unit circle, you need to know the coordinates of the point where the terminal side of the angle intersects the unit circle. The coordinates are given by $ ( \cos(\theta), \sin(\theta) ) $.

The tangent of the angle $ \theta $ is given by:

$ \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} $

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

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

and

$ \sin \left( \frac{\pi}{4} \right) = \frac{\sqrt{2}}{2} $

so

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

To find $ an( heta) $ on the unit circle, we use the coordinates of the point at angle $ heta $ which are $ ( cos( heta), sin( heta) ) $.

The formula for tangent is:

$ an( heta) = frac{sin( heta)}{cos( heta)} $

For instance, if $ heta = frac{pi}{3} $, then:

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

and

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

so

$ an left( frac{pi}{3}
ight) = frac{frac{sqrt{3}}{2}}{frac{1}{2}} = sqrt{3} $

To find $ an $ on the unit circle, use the coordinates $ ( cos( heta), sin( heta) ) $:

$ an( heta) = frac{sin( heta)}{cos( heta)} $

For example, for $ heta = frac{pi}{6} $:

$ an left( frac{pi}{6}
ight) = frac{frac{1}{2}}{frac{sqrt{3}}{2}} = frac{1}{sqrt{3}} = frac{sqrt{3}}{3} $

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