Given a point on the unit circle where the $sec$ of the angle is 3, find the angle in radians and degrees, and determine the corresponding coordinates on the unit circle.

Given that $\sec \theta = 3$, we know that:

$\sec \theta = \frac{1}{\cos \theta}$

Solving for $\cos \theta$, we get:

$\cos \theta = \frac{1}{3}$

Using $\cos^{-1}(\frac{1}{3})$, we find:

$\theta = \cos^{-1}(\frac{1}{3})$

Converting to degrees:

$\theta \approx 70.5288°$

Since $\sec \theta = \sec (360° – \theta)$, the other solution is:

$\theta = 360° – 70.5288° \approx 289.4712°$

In radians, this is:

$\theta \approx 1.23095 \text{ radians or } 5.05224 \text{ radians}$

The corresponding coordinates on the unit circle are:

$ (\cos (1.23095), \sin (1.23095)) = (\frac{1}{3}, \sqrt{1 – \frac{1}{9}}) = (\frac{1}{3}, \sqrt{\frac{8}{9}}) = (\frac{1}{3}, \frac{2\sqrt{2}}{3}) $

and

$ (\cos (5.05224), \sin (5.05224)) = (\frac{1}{3}, -\frac{2\sqrt{2}}{3}) $

Given that $sec heta = 3$, this implies:

$frac{1}{cos heta} = 3$

Therefore:

$cos heta = frac{1}{3}$

Using $arccos(frac{1}{3})$, we have:

$ heta approx 1.23096 ext{ radians}$

To find the angle in degrees:

$ heta approx 70.5288°$

The corresponding coordinates are:

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

Another solution in radians is:

$2pi – 1.23096 approx 5.0522 ext{ radians}$

The coordinates are:

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

Given that $sec heta = 3$, then:

$cos heta = frac{1}{3}$

Using $arccos(frac{1}{3})$, we find $ heta$:

$ heta approx 70.5288° ext{ or } 1.23096 ext{ radians}$

The coordinates are:

$ (frac{1}{3}, frac{2sqrt{2}}{3}) ext{ and } (frac{1}{3}, -frac{2sqrt{2}}{3}) $

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