Let's go through some of these questions and find the solutions:

1. Radian Measure for 210° $\text{Radian measure} = 210^\circ \times \frac{\pi}{180} = \frac{7\pi}{6}$

2. Degree Measure for $\frac{2\pi}{5}$ $\text{Degree measure} = \frac{2\pi}{5} \times \frac{180}{\pi} = 72^\circ$

3. Value of $\cos(-1800^\circ)$ Since $-1800^\circ$ is equivalent to $0^\circ$ (mod 360), we have: $\cos(-1800^\circ) = \cos(0^\circ) = 1$

4. Value of $\tan \frac{19\pi}{3}$ Reduce $\frac{19\pi}{3}$ modulo $2\pi$: $\frac{19\pi}{3} = 6\pi + \frac{\pi}{3}$ So, $\tan \frac{19\pi}{3} = \tan \frac{\pi}{3} = \sqrt{3}$

5. Expression $\cot^2 \frac{\pi}{6} + \csc \frac{5\pi}{6} + 3\tan^2 \frac{\pi}{6} + 6$ Simplifying each term:

   • $\cot \frac{\pi}{6} = \sqrt{3} \Rightarrow \cot^2 \frac{\pi}{6} = 3$
   • $\csc \frac{5\pi}{6} = 2$
   • $\tan \frac{\pi}{6} = \frac{1}{\sqrt{3}} \Rightarrow 3\tan^2 \frac{\pi}{6} = 1$ $3 + 2 + 1 + 6 = 12$

Would you like further solutions or details on specific problems?

Here are some related questions that can further expand on these topics:

1. How do we convert any given degree measure to radians?
2. What is the importance of trigonometric identities in simplifying expressions?
3. How can we find equivalent angles within standard ranges?
4. What are the differences between degree and radian measures?
5. How can trigonometric values be applied in real-life contexts?

Tip: To convert an angle from degrees to radians, multiply by $\frac{\pi}{180}$. For radians to degrees, multiply by $\frac{180}{\pi}$.