Finding Trigonometric Values at Specific Angles on the Unit Circle
Answer 1
Lucas Brown
To find the trigonometric values of an angle of 150 degrees on the unit circle, we can use the angle relationship and symmetry of the unit circle.
1. Convert the angle to radians: $150° = \frac{5\pi}{6}$ radians.
2. Recall that on the unit circle, the coordinates of a point corresponding to an angle θ in radians are $(\cos θ, \sin θ)$.
3. Since $\frac{5\pi}{6}$ is in the second quadrant, where the cosine value is negative and the sine value is positive, we can use the reference angle of $\pi – \frac{5\pi}{6} = \frac{\pi}{6}$.
4. Therefore, $\cos \frac{5\pi}{6} = -\cos \frac{\pi}{6} = – \frac{\sqrt{3}}{2}$ and $\sin \frac{5\pi}{6} = \sin \frac{\pi}{6} = \frac{1}{2}$.
Hence, the coordinates for 150 degrees on the unit circle are $(-\frac{\sqrt{3}}{2}, \frac{1}{2})$.
Answer 2
Joseph Robinson
To determine the trigonometric values at 120 degrees:
1. Convert the angle: $120° = frac{2pi}{3}$ radians.
2. On the unit circle, coordinates for angle θ are $(cos θ, sin θ)$.
3. The angle $frac{2pi}{3}$ is in the second quadrant where $cos$ is negative and $sin$ is positive. The reference angle is $pi – frac{2pi}{3} = frac{pi}{3}$.
4. Thus, $cos frac{2pi}{3} = -cos frac{pi}{3} = -frac{1}{2}$ and $sin frac{2pi}{3} = sin frac{pi}{3} = frac{sqrt{3}}{2}$.
The coordinates for 120 degrees are $(-frac{1}{2}, frac{sqrt{3}}{2})$.
Answer 3
Sophia Williams
To find the values at 210 degrees:
$210° = frac{7pi}{6}$ radians. Coordinates are $(cos frac{7pi}{6}, sin frac{7pi}{6})$.
In the third quadrant, $cos$ and $sin$ are negative. Reference angle: $frac{7pi}{6} – pi = frac{pi}{6}$.
$cos frac{7pi}{6} = -cos frac{pi}{6} = -frac{sqrt{3}}{2}$ and $sin frac{7pi}{6} = -sin frac{pi}{6} = -frac{1}{2}$.
Coordinates: $(-frac{sqrt{3}}{2}, -frac{1}{2})$.
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