Find the tangent values at $ 0 $, $ frac{pi}{4} $, and $ frac{pi}{2} $ on the unit circle
Answer 1
To find the tangent values at specific points on the unit circle, we use the definition of the tangent function, which is $ \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} $.
1. At $ \theta = 0 $:
$ \tan(0) = \frac{\sin(0)}{\cos(0)} = \frac{0}{1} = 0 $
2. At $ \theta = \frac{\pi}{4} $:
$ \tan\left(\frac{\pi}{4}\right) = \frac{\sin\left(\frac{\pi}{4}\right)}{\cos\left(\frac{\pi}{4}\right)} = \frac{\frac{\sqrt{2}}{2}}{\frac{\sqrt{2}}{2}} = 1 $
3. At $ \theta = \frac{\pi}{2} $:
$ \tan\left(\frac{\pi}{2}\right) = \frac{\sin\left(\frac{\pi}{2}\right)}{\cos\left(\frac{\pi}{2}\right)} = \frac{1}{0} $
Since division by zero is undefined, $ \tan\left(\frac{\pi}{2}\right) $ does not exist.
Answer 2
For $ heta = 0 $, $ an(0) = frac{sin(0)}{cos(0)} = 0 $.
For $ heta = frac{pi}{4} $, $ anleft(frac{pi}{4}
ight) = frac{sinleft(frac{pi}{4}
ight)}{cosleft(frac{pi}{4}
ight)} = 1 $.
For $ heta = frac{pi}{2} $, $ anleft(frac{pi}{2}
ight) $ does not exist as $ cosleft(frac{pi}{2}
ight) = 0 $.
Answer 3
1. $ an(0) = 0 $
2. $ anleft(frac{pi}{4}
ight) = 1 $
3. $ anleft(frac{pi}{2}
ight) $ does not exist.
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