$Determine the value of tan for given angles on the unit circle$
Answer 1
Mia Harris
$\text{Given an angle of } \theta = \frac{5\pi}{4}$
We know that:
$\tan \theta = \frac{\sin \theta}{\cos \theta}$
On the unit circle, for \(\theta = \frac{5\pi}{4}, \sin \theta = -\frac{\sqrt{2}}{2} \) and \(\cos \theta = -\frac{\sqrt{2}}{2}\)
Therefore,
$\tan \left(\frac{5\pi}{4}\right) = \frac{-\frac{\sqrt{2}}{2}}{-\frac{\sqrt{2}}{2}}$
Simplifying, we get:
$\tan \left(\frac{5\pi}{4}\right) = 1$
Answer 2
Maria Rodriguez
$ ext{Given an angle of } heta = frac{7pi}{6}$
We know that:
$ an heta = frac{sin heta}{cos heta}$
On the unit circle, for ( heta = frac{7pi}{6}, sin heta = -frac{1}{2} ) and (cos heta = -frac{sqrt{3}}{2})
Therefore,
$ an left(frac{7pi}{6}
ight) = frac{-frac{1}{2}}{-frac{sqrt{3}}{2}}$
Simplifying, we get:
$ an left(frac{7pi}{6}
ight) = frac{1}{sqrt{3}} = frac{sqrt{3}}{3}$
Answer 3
Charlotte Davis
$ ext{Given an angle of } heta = frac{11pi}{6}$
We know that:
$ an heta = frac{sin heta}{cos heta}$
On the unit circle, for ( heta = frac{11pi}{6}, sin heta = -frac{1}{2} ) and (cos heta = frac{sqrt{3}}{2})
Therefore,
$ an left(frac{11pi}{6}
ight) = frac{-frac{1}{2}}{frac{sqrt{3}}{2}}$
Simplifying, we get:
$ an left(frac{11pi}{6}
ight) = -frac{1}{sqrt{3}} = -frac{sqrt{3}}{3}$
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