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To solve for \( x \) in this triangle, where we have:

- Side \( 2.1 \) km adjacent to the angle \( 105^\circ \),
- Side \( 3.3 \) km opposite to angle \( 105^\circ \),
- Unknown side \( x \) opposite the angle that isn’t given.

### Solution Approach
This is an oblique triangle, and we can apply the **Law of Cosines** to solve for \( x \):
\[
x^2 = 2.1^2 + 3.3^2 - 2 \times 2.1 \times 3.3 \times \cos(105^\circ)
\]

1. **Square the given side lengths**:
   - \( 2.1^2 = 4.41 \)
   - \( 3.3^2 = 10.89 \)

2. **Calculate \( \cos(105^\circ) \)** (approximately \( -0.2588 \)).

3. **Plug in the values**:
   \[
   x^2 = 4.41 + 10.89 - 2 \times 2.1 \times 3.3 \times (-0.2588)
   \]

4. **Simplify and solve for \( x \)**.

Would you like the detailed calculations for each step, or a direct answer? 

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### Related Questions

1. How does the Law of Cosines help in finding unknown sides in a triangle?
2. What are the conditions required to use the Law of Sines instead?
3. How can we verify if the triangle is obtuse, acute, or right-angled?
4. How do changes in angle affect the opposite side length in such a triangle?
5. What other real-world applications use triangle side-angle relationships?

### Tip
Always verify if the angle is obtuse or acute, as this affects the cosine value and thus the calculations.

Find the length of the unknown side x in a triangle with sides 2.1 km and 3.3 km, and an angle of 105° between them.

Learn how to find the length of an unknown side x in a triangle given sides 2.1 km and 3.3 km with a 105° angle between them. This solution utilizes the Law of Cosines, a key trigonometric concept for solving oblique triangles. Suitable for high school students in Grades 10-12.

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