91Ó°ÊÓ

To convert \(300^{\circ}\) to radians, we multiply the degree value by the conversion factor \(\frac{\pi}{180^{\circ}}\): \[300^{\circ} \times \frac{\pi}{180^{\circ}} = \frac{5}{3}\pi\] So, the equivalent angle in radians is \(\frac{5}{3}\pi\).

We need to visualize the unit circle, which is a circle with a radius of 1, with its center at the origin of the coordinate plane. Observe that an angle of \(300^{\circ}\) corresponds to an angle of \(360^{\circ}-300^{\circ} = 60^{\circ}\) in the clockwise direction (as the first quadrant includes angles between \(0^{\circ}\) and \(90^{\circ}\)). Now, we know from the special right triangle corresponding to a \(30^{\circ}\)-\(60^{\circ}\)-\(90^{\circ}\) triangle, the coordinates of the point on the unit circle corresponding to an angle of \(60^{\circ}\) are: \((x,y) = (\frac{1}{2}, \frac{\sqrt{3}}{2})\) Since moving clockwise from \(360^{\circ}\) to \(300^{\circ}\) flips the sign of the y-coordinate, the point on the unit circle corresponding to an angle of \(300^{\circ}\) is: \((x,y) = (\frac{1}{2}, -\frac{\sqrt{3}}{2})\)

Now that we have the coordinates of the point on the unit circle, we can find the values of the trigonometric functions. 1. sin(300°): \[\sin(300^{\circ}) = y = -\frac{\sqrt{3}}{2}\] 2. cos(300°): \[\cos(300^{\circ}) = x = \frac{1}{2}\] 3. tan(300°): \[\tan(300^{\circ}) = \frac{\sin(300^{\circ})}{\cos(300^{\circ})} = \frac{-\sqrt{3}/2}{1/2} = -\sqrt{3}\] 4. csc(300°) (the reciprocal of sin): \[\csc(300^{\circ}) = \frac{1}{\sin(300^{\circ})} = \frac{1}{-\sqrt{3}/2} = -\frac{2\sqrt{3}}{3}\] 5. sec(300°) (the reciprocal of cos): \[\sec(300^{\circ}) = \frac{1}{\cos(300^{\circ})} = \frac{1}{1/2} = 2\] 6. cot(300°) (the reciprocal of tan): \[\cot(300^{\circ}) = \frac{1}{\tan(300^{\circ})} = \frac{1}{-\sqrt{3}} = -\frac{\sqrt{3}}{3}\] Now we have found all six trigonometric function values for an angle of 300°: \[\sin(300^{\circ}) = -\frac{\sqrt{3}}{2}\] \[\cos(300^{\circ}) = \frac{1}{2}\] \[\tan(300^{\circ}) = -\sqrt{3}\] \[\csc(300^{\circ}) = -\frac{2\sqrt{3}}{3}\] \[\sec(300^{\circ}) = 2\] \[\cot(300^{\circ}) = -\frac{\sqrt{3}}{3}\]