Find the value of $ an(frac{7pi}{6}) $ and explain using the unit circle
Answer 1
John Anderson
To find the value of $ \tan(\frac{7\pi}{6}) $ using the unit circle:
1. Locate the angle $\frac{7\pi}{6}$ on the unit circle. This angle is in the third quadrant.
2. The reference angle for $\frac{7\pi}{6}$ is $\frac{\pi}{6}$.
3. In the third quadrant, both sine and cosine are negative. Knowing the coordinates for $\frac{\pi}{6}$ are $(\frac{\sqrt{3}}{2}, \frac{1}{2})$:
The coordinates for $\frac{7\pi}{6}$ are $(-\frac{\sqrt{3}}{2}, -\frac{1}{2})$.
4. Finally, calculate the tangent value:
$ \tan(\frac{7\pi}{6}) = \frac{\sin(\frac{7\pi}{6})}{\cos(\frac{7\pi}{6})} = \frac{-\frac{1}{2}}{-\frac{\sqrt{3}}{2}} = \frac{1}{\sqrt{3}} = \frac{\sqrt{3}}{3} $
Answer 2
Abigail Nelson
To calculate $ cos(frac{5pi}{4}) $ using the unit circle:
1. Locate the angle $frac{5pi}{4}$ on the unit circle. This angle is in the third quadrant.
2. The reference angle for $frac{5pi}{4}$ is $frac{pi}{4}$.
3. The coordinates for $frac{pi}{4}$ are $(frac{sqrt{2}}{2}, frac{sqrt{2}}{2})$.
4. In the third quadrant, cosine is negative. Therefore, the coordinates for $frac{5pi}{4}$ are $(-frac{sqrt{2}}{2}, -frac{sqrt{2}}{2})$.
5. The cosine value is:
$ cos(frac{5pi}{4}) = -frac{sqrt{2}}{2} $
Answer 3
James Taylor
To evaluate $ sin(frac{11pi}{6}) $ using the unit circle:
1. Locate the angle $frac{11pi}{6}$ on the unit circle. This angle is in the fourth quadrant.
2. The reference angle for $frac{11pi}{6}$ is $frac{pi}{6}$.
3. The coordinates for $frac{pi}{6}$ are $(frac{sqrt{3}}{2}, frac{1}{2})$.
4. In the fourth quadrant, sine is negative. Therefore, the coordinates for $frac{11pi}{6}$ are $(frac{sqrt{3}}{2}, -frac{1}{2})$.
5. The sine value is:
$ sin(frac{11pi}{6}) = -frac{1}{2} $
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