$ ext{Find the tangent for every angle on the unit circle}$

To find the tangent for every angle \(\theta\) on the unit circle, we use the definition of tangent, which is the ratio of the sine of the angle to the cosine of the angle.

The tangent function is given by:

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

To illustrate, let’s take an angle \(\theta = \frac{\pi}{4}\):

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

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

Thus,

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

To determine the tangent for any angle ( heta) on the unit circle, we use the trigonometric identity:

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

Let’s take an angle ( heta = frac{pi}{3}):

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

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

So,

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

To find ( an( heta)) for any angle ( heta) on the unit circle:

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

For example, if ( heta = frac{pi}{6}):

$ anleft(frac{pi}{6}
ight) = frac{1/2}{sqrt{3}/2} = frac{1}{sqrt{3}} = frac{sqrt{3}}{3} $

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