To find the exact value of $\sec 300^\circ$, we can follow these steps:

1. Recall the definition of secant: $\sec \theta = \frac{1}{\cos \theta}$

2. Determine $\cos 300^\circ$: $300^\circ$ is in the fourth quadrant. In the fourth quadrant, the cosine of an angle is positive. The reference angle for $300^\circ$ is: $360^\circ - 300^\circ = 60^\circ$ Therefore, $\cos 300^\circ$ is the same as $\cos 60^\circ$, but with a positive value since $\cos$ is positive in the fourth quadrant: $\cos 300^\circ = \cos 60^\circ = \frac{1}{2}$

3. Calculate $\sec 300^\circ$: Using the definition of secant: $\sec 300^\circ = \frac{1}{\cos 300^\circ} = \frac{1}{\frac{1}{2}} = 2$

So, the exact value of $\sec 300^\circ$ is: $\sec 300^\circ = 2$

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Related Questions

1. What is the value of $\sec 60^\circ$?
2. How do we determine reference angles in different quadrants?
3. What is the exact value of $\sec 240^\circ$?
4. How do we find the exact value of $\cos 330^\circ$?
5. What is the value of $\sec 0^\circ$?

Tip: Remember that in trigonometry, secant, cosecant, and cotangent can be calculated using their reciprocal definitions with sine, cosine, and tangent, respectively.