Find the coordinates and trigonometric values for an angle on the unit circle

Consider an angle $ \theta = \frac{7\pi}{6} $ on the unit circle. We need to find the coordinates of the point on the unit circle corresponding to this angle, as well as the sine and cosine values.

First, identify the reference angle: $ \theta_{ref} = \pi – \frac{7\pi}{6} = \frac{\pi}{6} $

Next, find the coordinates for the reference angle $ \frac{\pi}{6} $:

$ \left( \cos\left(\frac{\pi}{6}\right), \sin\left(\frac{\pi}{6}\right) \right) = \left( \frac{\sqrt{3}}{2}, \frac{1}{2} \right) $

Since $ \theta = \frac{7\pi}{6} $ is in the third quadrant, both sine and cosine are negative:

$ \left( \cos\left(\frac{7\pi}{6}\right), \sin\left(\frac{7\pi}{6}\right) \right) = \left( -\frac{\sqrt{3}}{2}, -\frac{1}{2} \right) $

Consider an angle $ heta = frac{7pi}{6} $ on the unit circle. To determine the coordinates of the corresponding point and the trigonometric values, follow these steps:

Identify the reference angle:

$ heta_{ref} = pi – frac{7pi}{6} = frac{pi}{6} $

Look up the coordinates for the reference angle $ frac{pi}{6} $:

$ left( cosleft(frac{pi}{6}
ight), sinleft(frac{pi}{6}
ight)
ight) = left( frac{sqrt{3}}{2}, frac{1}{2}
ight) $

Since the angle $ heta = frac{7pi}{6} $ is in the third quadrant, adjust the signs accordingly:

$ left( cosleft(frac{7pi}{6}
ight), sinleft(frac{7pi}{6}
ight)
ight) = left( -frac{sqrt{3}}{2}, -frac{1}{2}
ight) $

For angle $ heta = frac{7pi}{6} $ on the unit circle:

Reference angle:

$ heta_{ref} = frac{pi}{6} $

Coordinates at $ frac{pi}{6} $:

$ left( frac{sqrt{3}}{2}, frac{1}{2}
ight) $

Since $ frac{7pi}{6} $ is in the third quadrant:

$ left( -frac{sqrt{3}}{2}, -frac{1}{2}
ight) $

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