Define the unit circle in trigonometry

To calculate the tangent values for specific angles on the unit circle, you will use the formula:

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

Consider the angles $ heta = frac{pi}{4}, frac{2pi}{3},$ and $frac{7pi}{6}$. First, we need to find the sine and cosine values for each angle:

For $ heta = frac{pi}{4}$:

$ sin(frac{pi}{4}) = frac{sqrt{2}}{2}, cos(frac{pi}{4}) = frac{sqrt{2}}{2} $

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

For $ heta = frac{2pi}{3}$:

$ sin(frac{2pi}{3}) = frac{sqrt{3}}{2}, cos(frac{2pi}{3}) = -frac{1}{2} $

$ an(frac{2pi}{3}) = frac{sqrt{3}/2}{-1/2} = -sqrt{3} $

For $ heta = frac{7pi}{6}$:

$ sin(frac{7pi}{6}) = -frac{1}{2}, cos(frac{7pi}{6}) = -frac{sqrt{3}}{2} $

$ an(frac{7pi}{6}) = frac{-1/2}{-sqrt{3}/2} = frac{1}{sqrt{3}} = frac{sqrt{3}}{3} $

Using the unit circle, calculate $ an( heta)$ for the angles $ heta = frac{pi}{4}, frac{2pi}{3},$ and $frac{7pi}{6}$.

For $ heta = frac{pi}{4}$, we have:

$ an(frac{pi}{4}) = 1 $

For $ heta = frac{2pi}{3}$, we have:

$ an(frac{2pi}{3}) = -sqrt{3} $

For $ heta = frac{7pi}{6}$, we have:

$ an(frac{7pi}{6}) = frac{sqrt{3}}{3} $

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