Find the value of $ an( heta)$ at three specific angles on the unit circle: $ heta = frac{pi}{4}$, $frac{3pi}{4}$, and $frac{5pi}{6}$.
Answer 1
Ava Martin
For $\theta = \frac{\pi}{4}$:
On the unit circle, at $\theta = \frac{\pi}{4}$, the coordinates are $(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2})$.
The tangent function is given by $\tan(\theta) = \frac{y}{x}$.
Thus,
$ \tan(\frac{\pi}{4}) = \frac{\frac{\sqrt{2}}{2}}{\frac{\sqrt{2}}{2}} = 1 $
For $\theta = \frac{3\pi}{4}$:
On the unit circle, at $\theta = \frac{3\pi}{4}$, the coordinates are $(-\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2})$.
Thus,
$ \tan(\frac{3\pi}{4}) = \frac{\frac{\sqrt{2}}{2}}{-\frac{\sqrt{2}}{2}} = -1 $
For $\theta = \frac{5\pi}{6}$:
On the unit circle, at $\theta = \frac{5\pi}{6}$, the coordinates are $(-\frac{\sqrt{3}}{2}, \frac{1}{2})$.
Thus,
$ \tan(\frac{5\pi}{6}) = \frac{\frac{1}{2}}{-\frac{\sqrt{3}}{2}} = -\frac{1}{\sqrt{3}} = -\frac{\sqrt{3}}{3} $
Answer 2
Alex Thompson
For $ heta = frac{pi}{4}$:
Coordinates: $(frac{sqrt{2}}{2}, frac{sqrt{2}}{2})$
$ an(frac{pi}{4}) = frac{frac{sqrt{2}}{2}}{frac{sqrt{2}}{2}} = 1 $
For $ heta = frac{3pi}{4}$:
Coordinates: $(-frac{sqrt{2}}{2}, frac{sqrt{2}}{2})$
$ an(frac{3pi}{4}) = frac{frac{sqrt{2}}{2}}{-frac{sqrt{2}}{2}} = -1 $
For $ heta = frac{5pi}{6}$:
Coordinates: $(-frac{sqrt{3}}{2}, frac{1}{2})$
$ an(frac{5pi}{6}) = frac{frac{1}{2}}{-frac{sqrt{3}}{2}} = -frac{1}{sqrt{3}} = -frac{sqrt{3}}{3} $
Answer 3
Emily Hall
$ an(frac{pi}{4}) = 1 $
$ an(frac{3pi}{4}) = -1 $
$ an(frac{5pi}{6}) = -frac{sqrt{3}}{3} $
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