Let's solve the problems step by step.

1. Решите уравнения:

a) $\cos x = -\frac{\sqrt{2}}{2}$

The cosine function equals $-\frac{\sqrt{2}}{2}$ at the following points: $x = \pi - \frac{\pi}{4} + 2n\pi \quad \text{and} \quad x = \pi + \frac{\pi}{4} + 2n\pi$ Simplifying: $x = \frac{3\pi}{4} + 2n\pi \quad \text{and} \quad x = \frac{5\pi}{4} + 2n\pi, \; n \in \mathbb{Z}.$

b) $\sin\left(\frac{x}{2} - \frac{\pi}{4}\right) = 0$

The sine function equals 0 at: $\frac{x}{2} - \frac{\pi}{4} = n\pi, \; n \in \mathbb{Z}.$ Solving for $x$: $x = 2n\pi + \frac{\pi}{2}.$

c) $\cos(3x) = \frac{\sqrt{3}}{2}$

The cosine function equals $\frac{\sqrt{3}}{2}$ at: $3x = 2n\pi \pm \frac{\pi}{6}, \; n \in \mathbb{Z}.$ Solving for $x$: $x = \frac{2n\pi}{3} \pm \frac{\pi}{18}.$

2. Найдите значения других трех основных тригонометрических функций:

Given $\sin\alpha = -\frac{12}{13}$ and $\frac{3\pi}{2} < \alpha < 2\pi$:

1. Compute $\cos\alpha$ using $\sin^2\alpha + \cos^2\alpha = 1$: $\cos^2\alpha = 1 - \sin^2\alpha = 1 - \left(-\frac{12}{13}\right)^2 = 1 - \frac{144}{169} = \frac{25}{169}.$ \cos\alpha = -\frac{5}{13}, \quad \text{(negative because \alpha is in the 4th quadrant)}.

2. Compute $\tan\alpha$: $\tan\alpha = \frac{\sin\alpha}{\cos\alpha} = \frac{-\frac{12}{13}}{-\frac{5}{13}} = \frac{12}{5}.$

Given $\cos\alpha = -0.8$ and $\frac{\pi}{2} < \alpha < \pi$:

1. Compute $\sin\alpha$ using $\sin^2\alpha + \cos^2\alpha = 1$: $\sin^2\alpha = 1 - \cos^2\alpha = 1 - (-0.8)^2 = 1 - 0.64 = 0.36.$ \sin\alpha = 0.6, \quad \text{(positive because \alpha is in the 2nd quadrant)}.

2. Compute $\tan\alpha$: $\tan\alpha = \frac{\sin\alpha}{\cos\alpha} = \frac{0.6}{-0.8} = -\frac{3}{4}.$

Would you like me to go into more detail for any specific part?

Expanding Questions:

1. Can you derive the periodic solutions for trigonometric equations?
2. Would you like to verify the ranges of $\alpha$ for sine and cosine properties?
3. How do transformations in sine or cosine affect their zeros?
4. Can you visualize the solutions on the unit circle for better understanding?
5. Should I help plot or tabulate these values for clarity?

Tip: Always use the unit circle to confirm signs of trigonometric values in different quadrants.