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Let's work on solving the inequalities from the image. We will proceed step by step. The first inequality to solve is:

### Exercise 626
\[
\sqrt{3} \sin x - \cos x > 0
\]

### Solution:
We start by solving the inequality algebraically.

1. \(\sqrt{3} \sin x - \cos x > 0\)

Rearrange the equation:
\[
\sqrt{3} \sin x > \cos x
\]

Divide both sides by \(\cos x\) (assuming \(\cos x \neq 0\)):
\[
\frac{\sin x}{\cos x} > \frac{1}{\sqrt{3}}
\]

Since \(\frac{\sin x}{\cos x} = \tan x\), we rewrite the inequality:
\[
\tan x > \frac{1}{\sqrt{3}}
\]

We know that \(\tan x = \frac{1}{\sqrt{3}}\) at specific points. This happens at \(x = \frac{\pi}{6} + k\pi\), where \(k\) is an integer.

Now we analyze when \(\tan x > \frac{1}{\sqrt{3}}\), which is satisfied in the following intervals:
\[
x \in \left( \frac{\pi}{6} + k\pi, \frac{\pi}{2} + k\pi \right), \quad k \in \mathbb{Z}
\]

Thus, the solution for inequality 626 is:
\[
x \in \left( \frac{\pi}{6} + k\pi, \frac{\pi}{2} + k\pi \right), \quad k \in \mathbb{Z}
\]

---

Would you like to proceed with the next inequalities or have a detailed explanation for any part of the process? Here are some follow-up questions:

1. What happens when \(\tan x = \frac{1}{\sqrt{3}}\) at the critical points?
2. Can we represent the general solution for \(\tan x > \frac{1}{\sqrt{3}}\) graphically?
3. How do periodic functions like \(\tan x\) behave over intervals?
4. How can we verify the solution using specific values for \(x\)?
5. What are other ways to approach trigonometric inequalities?

### Tip:
Always remember that solving trigonometric inequalities involves checking for periodicity and analyzing the principal solutions before generalizing for all solutions.

Solve the inequality √3 sin x - cos x > 0.

Explore how to solve the trigonometric inequality √3 sin x - cos x > 0 with a detailed explanation of rearranging the equation, dividing by cos x, and analyzing the tangent function. Suitable for high school students learning advanced trigonometry.

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