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To solve these problems, we will use either the Law of Sines or the Law of Cosines, as indicated.

### Problem 21
In triangle \(ABC\), we are given:
- \( \angle A = 35^\circ \)
- \( \angle C = 85^\circ \)
- Side \( AC = 3 \)

We need to find side \( x = AB \).

1. **Find \( \angle B \):**
   \[
   \angle B = 180^\circ - \angle A - \angle C = 180^\circ - 35^\circ - 85^\circ = 60^\circ
   \]

2. **Apply the Law of Sines:**
   \[
   \frac{x}{\sin C} = \frac{AC}{\sin B}
   \]
   Substituting the values:
   \[
   \frac{x}{\sin 85^\circ} = \frac{3}{\sin 60^\circ}
   \]
   Solving for \( x \):
   \[
   x = \frac{3 \cdot \sin 85^\circ}{\sin 60^\circ}
   \]
   Calculating this value:
   \[
   x \approx \frac{3 \cdot 0.9962}{0.8660} \approx 3.45
   \]

### Solution for Problem 21:
\[ x \approx 3.45 \]

---

### Problem 22
In triangle \(ABC\), we are given:
- \( \angle A = 40^\circ \)
- Side \( AB = 18 \)
- Side \( AC = 10 \)

We need to find side \( x = BC \).

Since we know two sides and the included angle, we use the Law of Cosines.

1. **Apply the Law of Cosines:**
   \[
   x^2 = AC^2 + AB^2 - 2 \cdot AC \cdot AB \cdot \cos A
   \]
   Substituting the values:
   \[
   x^2 = 10^2 + 18^2 - 2 \cdot 10 \cdot 18 \cdot \cos 40^\circ
   \]
   \[
   x^2 = 100 + 324 - 360 \cdot \cos 40^\circ
   \]
   Calculating \( \cos 40^\circ \approx 0.7660 \):
   \[
   x^2 = 100 + 324 - 360 \cdot 0.7660
   \]
   \[
   x^2 = 424 - 275.76 = 148.24
   \]
   Taking the square root:
   \[
   x \approx \sqrt{148.24} \approx 12.17
   \]

### Solution for Problem 22:
\[ x \approx 12.17 \]

---

Would you like further details or explanations on these solutions?

Here are five additional questions to consider:

1. How would the solutions change if the angles were altered slightly in each problem?
2. Could you use the Law of Sines in Problem 22 if a different angle were given?
3. What are the implications of rounding intermediate values in trigonometric calculations?
4. How would you verify these solutions using a different trigonometric approach?
5. Why is the Law of Cosines preferable in Problem 22 instead of the Law of Sines?

**Tip:** When using trigonometric functions, it's useful to keep as many decimal places as possible in intermediate steps to avoid rounding errors.

Encuentre el lado indicado x o el ángulo θ en los triángulos dados utilizando la Ley de Senos o la Ley de Cosenos, según sea apropiado.

Learn how to find the missing side or angle in triangles using the Law of Sines or Law of Cosines. The example problems cover solving for side lengths and angles in triangles with given angles and side lengths, ideal for high school trigonometry.

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