Find the cosine and sine of an angle on the unit circle
Answer 1
Lily Perez
Given an angle $\theta = \frac{5\pi}{6}$ radians, find the coordinates $(\cos \theta, \sin \theta)$ on the unit circle.
Step 1: Recognize that $\theta = \frac{5\pi}{6}$ is an angle in the second quadrant.
Step 2: In the second quadrant, cosine is negative and sine is positive.
Step 3: Use the reference angle, which is $\pi – \frac{5\pi}{6} = \frac{\pi}{6}$.
Step 4: Recall the sine and cosine values for $\frac{\pi}{6}$: $\sin \frac{\pi}{6} = \frac{1}{2}$ and $\cos \frac{\pi}{6} = \frac{\sqrt{3}}{2}$.
Step 5: Apply the signs for the second quadrant: $\cos \frac{5\pi}{6} = -\cos \frac{\pi}{6} = -\frac{\sqrt{3}}{2}$ and $\sin \frac{5\pi}{6} = \sin \frac{\pi}{6} = \frac{1}{2}$.
Thus, the coordinates are $\left(-\frac{\sqrt{3}}{2}, \frac{1}{2}\right)$.
Answer 2
Amelia Mitchell
Given $ heta = frac{5pi}{6}$, find $(cos heta, sin heta)$ on the unit circle.
1. $frac{5pi}{6}$ lies in the second quadrant.
2. Reference angle: $frac{pi}{6}$.
3. Values in the first quadrant: $cos frac{pi}{6} = frac{sqrt{3}}{2}$, $sin frac{pi}{6} = frac{1}{2}$.
4. Adjust for the second quadrant: $cos frac{5pi}{6} = -frac{sqrt{3}}{2}$, $sin frac{5pi}{6} = frac{1}{2}$.
Coordinates: $left(-frac{sqrt{3}}{2}, frac{1}{2}
ight)$
Answer 3
Alex Thompson
For $ heta = frac{5pi}{6}$, find $(cos heta, sin heta)$ on the unit circle.
Second quadrant: $cos = -frac{sqrt{3}}{2}$, $sin = frac{1}{2}$.
Coordinates: $left(-frac{sqrt{3}}{2}, frac{1}{2}
ight)$
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