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The image contains the following trigonometric equation:

\[
\text{tg} 82^\circ + 3 \cdot \text{csc} 30^\circ = 3x \cdot \text{ctg} 37^\circ + 3\sqrt{2} \cdot \text{csc} 45^\circ
\]

We need to solve for \( x \).

### Step-by-step solution:

1. **Calculate trigonometric values**:
   - \( \text{tg} 82^\circ \) (tangent of 82 degrees).
   - \( \text{csc} 30^\circ = \frac{1}{\sin 30^\circ} = \frac{1}{0.5} = 2 \).
   - \( \text{ctg} 37^\circ = \frac{1}{\tan 37^\circ} \).
   - \( \text{csc} 45^\circ = \frac{1}{\sin 45^\circ} = \frac{1}{\frac{\sqrt{2}}{2}} = \sqrt{2} \).

2. **Substitute the known values** into the equation:

\[
\text{tg} 82^\circ + 3 \times 2 = 3x \times \text{ctg} 37^\circ + 3\sqrt{2} \times \sqrt{2}
\]

Simplify further:

\[
\text{tg} 82^\circ + 6 = 3x \times \text{ctg} 37^\circ + 6
\]

3. **Isolate \( x \)** by moving terms around:

\[
\text{tg} 82^\circ = 3x \cdot \text{ctg} 37^\circ
\]

4. **Solve for \( x \)**:

\[
x = \frac{\text{tg} 82^\circ}{3 \cdot \text{ctg} 37^\circ}
\]

We can now compute the values of \( \text{tg} 82^\circ \) and \( \text{ctg} 37^\circ \) using a calculator:

- \( \text{tg} 82^\circ \approx 7.115 \)
- \( \text{ctg} 37^\circ = \frac{1}{\tan 37^\circ} \approx 1.327 \)

Thus:

\[
x = \frac{7.115}{3 \times 1.327} = \frac{7.115}{3.981} \approx 1.787
\]

So, \( x \approx 1.787 \).

Would you like further details or clarifications on any step?

Here are 5 related questions:
1. What is the difference between the cotangent and tangent functions?
2. How do you calculate the cosecant of an angle?
3. What are common trigonometric identities that can simplify such equations?
4. Can this method be applied to angles in radians instead of degrees?
5. How do inverse trigonometric functions help in solving equations involving trigonometry?

**Tip**: Always ensure your calculator is set to the correct mode (degrees or radians) when solving trigonometric problems!

Determine the value of x in: tg 82° + 3 csc 30° = 3x ctg 37° + 3√2 csc 45°.

Learn how to determine the value of x in the trigonometric equation involving tangent, cosecant, and cotangent functions: tg 82° + 3 csc 30° = 3x ctg 37° + 3√2 csc 45°. This step-by-step solution covers essential trigonometric concepts and key formulas for students in Grades 9-12.

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