Given a point on the unit circle, find its cosine and sine values.
Answer 1
Sophia Williams
Given a point \((\cos\theta, \sin\theta)\) on the unit circle, determine the coordinates when \(\theta = \frac{\pi}{4}\).
The unit circle has a radius of 1. At \(\theta = \frac{\pi}{4}\), both x and y coordinates are equal:
$\cos\frac{\pi}{4} = \sin\frac{\pi}{4} = \frac{\sqrt{2}}{2}$
Therefore, the coordinates are:
$(\cos\frac{\pi}{4}, \sin\frac{\pi}{4}) = \left(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right)$
Answer 2
Isabella Walker
Given a point ((cos heta, sin heta)) on the unit circle, find the coordinates for ( heta = frac{5pi}{6}).
For ( heta = frac{5pi}{6}), we use reference angle (frac{pi}{6}) and quadrant considerations:
$cosfrac{pi}{6} = frac{sqrt{3}}{2}$ and $sinfrac{pi}{6} = frac{1}{2}$
Since (frac{5pi}{6}) is in the second quadrant,
$cosfrac{5pi}{6} = -frac{sqrt{3}}{2}$ and $sinfrac{5pi}{6} = frac{1}{2}$
Therefore, the coordinates are:
$(cosfrac{5pi}{6}, sinfrac{5pi}{6}) = left(-frac{sqrt{3}}{2}, frac{1}{2}
ight)$
Answer 3
Michael Moore
Given a point ((cos heta, sin heta)) on the unit circle, compute the coordinates for ( heta = frac{3pi}{2}).
At ( heta = frac{3pi}{2}), the coordinates are:
$cosfrac{3pi}{2} = 0$ and $sinfrac{3pi}{2} = -1$
Therefore, the coordinates are:
$(cosfrac{3pi}{2}, sinfrac{3pi}{2}) = (0, -1)$
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