Find the value of $ an( heta) $ on the unit circle for $ heta = frac{5pi}{4} $.
Answer 1
Benjamin Clark
To find the value of $ \tan(\theta) $ on the unit circle for $ \theta = \frac{5\pi}{4} $, we first determine the coordinates of the point on the unit circle that corresponds to this angle.
The angle $ \frac{5\pi}{4} $ is in the third quadrant, where both sine and cosine values are negative. The reference angle is $ \pi/4 $.
The coordinates for $ \theta = \frac{5\pi}{4} $ are $ (-\frac{\sqrt{2}}{2}, -\frac{\sqrt{2}}{2}) $.
Since $ \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} $, we have:
$ \tan\left(\frac{5\pi}{4}\right) = \frac{\sin\left(\frac{5\pi}{4}\right)}{\cos\left(\frac{5\pi}{4}\right)} = \frac{-\frac{\sqrt{2}}{2}}{-\frac{\sqrt{2}}{2}} = 1 $
Answer 2
Amelia Mitchell
To find $ an( heta) $ at $ heta = frac{5pi}{4} $, identify the corresponding point on the unit circle.
The angle $ frac{5pi}{4} $ maps to the third quadrant where $ sin $ and $ cos $ are negative. The reference angle is $ pi/4 $.
The coordinates at $ heta = frac{5pi}{4} $ are $ (-frac{sqrt{2}}{2}, -frac{sqrt{2}}{2}) $.
Thus, $ an( heta) = frac{sin( heta)}{cos( heta)} $:
$ anleft( frac{5pi}{4}
ight) = frac{sinleft( frac{5pi}{4}
ight)}{cosleft( frac{5pi}{4}
ight)} = 1 $
Answer 3
John Anderson
Given $ heta = frac{5pi}{4} $ in the third quadrant, the coordinates are $ (-frac{sqrt{2}}{2}, -frac{sqrt{2}}{2}) $.
So,
$ anleft( frac{5pi}{4}
ight) = 1 $
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