Calculate the value of $ an( heta)$ for $ heta = frac{7pi}{4}$ using the unit circle.
Answer 1
Mia Harris
To find the value of $\tan(\theta)$ for $\theta = \frac{7\pi}{4}$ using the unit circle, we first need to determine the coordinates of the point on the unit circle corresponding to $\theta = \frac{7\pi}{4}$.
$\theta = \frac{7\pi}{4}$ corresponds to an angle of $315^\circ$ in standard position.
In the unit circle, this point is $\left( \cos\left(\frac{7\pi}{4}\right), \sin\left(\frac{7\pi}{4}\right) \right)$.
The coordinates at $\frac{7\pi}{4}$ are $( \frac{\sqrt{2}}{2}, -\frac{\sqrt{2}}{2} )$.
Therefore, $\tan\left(\frac{7\pi}{4}\right) $ can be calculated as:
$ \tan\left(\frac{7\pi}{4}\right) = \frac{\sin\left(\frac{7\pi}{4}\right)}{\cos\left(\frac{7\pi}{4}\right)} = \frac{-\frac{\sqrt{2}}{2}}{\frac{\sqrt{2}}{2}} = -1 $
Answer 2
Samuel Scott
To determine $ an( heta)$ for $ heta = frac{7pi}{4}$ using the unit circle, we start by identifying the point on the unit circle for $ heta = frac{7pi}{4}$.
The angle $ heta = frac{7pi}{4}$ is equivalent to $360^circ – 45^circ = 315^circ$.
Therefore, the coordinates on the unit circle for $ heta = frac{7pi}{4}$ are $( frac{sqrt{2}}{2}, -frac{sqrt{2}}{2} )$.
Thus, $ anleft(frac{7pi}{4}
ight)$ is given by:
$ anleft(frac{7pi}{4}
ight) = frac{sinleft(frac{7pi}{4}
ight)}{cosleft(frac{7pi}{4}
ight)} = frac{-frac{sqrt{2}}{2}}{frac{sqrt{2}}{2}} = -1 $
Answer 3
Charlotte Davis
To find $ an( heta)$ for $ heta = frac{7pi}{4}$ using the unit circle:
Angle $ heta = frac{7pi}{4}$ corresponds to $315^circ$.
The coordinates are $( frac{sqrt{2}}{2}, -frac{sqrt{2}}{2} )$.
Therefore:
$ anleft(frac{7pi}{4}
ight) = frac{sinleft(frac{7pi}{4}
ight)}{cosleft(frac{7pi}{4}
ight)} = frac{-frac{sqrt{2}}{2}}{frac{sqrt{2}}{2}} = -1 $
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