Determine the angle measures of a point on the unit circle

Given a point $P$ on the unit circle with coordinates $P = (\frac{3}{5}, -\frac{4}{5})$, determine all possible angle measures $\theta$ in degrees.

First, we calculate the reference angle $\alpha$ by using the trigonometric functions. Notice that the coordinates of $P$ give us the cosine and sine of $\theta$:

$\cos(\theta) = \frac{3}{5}, \sin(\theta) = -\frac{4}{5}$

Using the inverse cosine function, we find the reference angle:

$\alpha = \cos^{-1}(\frac{3}{5}) \approx 53.13^\circ$

Since the sine is negative and the cosine is positive, $\theta$ is in the fourth quadrant. Therefore,

$ \theta = 360^\circ – \alpha = 360^\circ – 53.13^\circ \approx 306.87^\circ$

Thus, the possible angle measures are:

$ \theta \approx 306.87^\circ $

Given a point $P$ on the unit circle with coordinates $P = (frac{3}{5}, -frac{4}{5})$, determine all possible angle measures $ heta$ in radians.

First, we recognize that $cos( heta) = frac{3}{5}$ and $sin( heta) = -frac{4}{5}$.

We use the inverse cosine function to find the reference angle $alpha$:

$alpha = cos^{-1}(frac{3}{5}) approx 0.9273 ext{ radians}$

Since the sine is negative and the cosine is positive, the angle $ heta$ is in the fourth quadrant:

$ heta = 2pi – alpha = 2pi – 0.9273 approx 5.3559 ext{ radians}$

Thus, the possible angle measures are:

$ heta approx 5.3559 ext{ radians} $

Given $P = (frac{3}{5}, -frac{4}{5})$ on the unit circle, find $ heta$.

$cos( heta) = frac{3}{5}, sin( heta) = -frac{4}{5}$

$ heta = 360^circ – cos^{-1}(frac{3}{5})$

$ heta approx 306.87^circ$

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