Use the Law of Sines to solve for all possible triangles that satisfy the given conditions.
(a) a = 20, b = 45, ∠A = 125◦
(b) a = 75, b = 100, ∠A = 30

To solve for all possible triangles using the Law of Sines, we'll examine each case:

### Case (a): \( a = 20 \), \( b = 45 \), \( \angle A = 125^\circ \)
Given \( \angle A = 125^\circ \), \( a = 20 \), and \( b = 45 \).

#### Step 1: Calculate \(\angle B\) using the Law of Sines
The Law of Sines formula is:
\[
\frac{a}{\sin A} = \frac{b}{\sin B}
\]
Substitute the known values:
\[
\frac{20}{\sin 125^\circ} = \frac{45}{\sin B}
\]
Calculate \(\sin 125^\circ\):
\[
\sin 125^\circ \approx 0.8192
\]
So,
\[
\frac{20}{0.8192} \approx \frac{45}{\sin B}
\]
\[
\sin B \approx \frac{45 \cdot 0.8192}{20} \approx 1.8432
\]
Since \(\sin B\) exceeds 1, no valid solution exists for \(\angle B\). Therefore, there is no possible triangle that satisfies these conditions.

### Case (b): \( a = 75 \), \( b = 100 \), \( \angle A = 30^\circ \)
Given \( \angle A = 30^\circ \), \( a = 75 \), and \( b = 100 \).

#### Step 1: Calculate \(\angle B\) using the Law of Sines
Again, using the Law of Sines:
\[
\frac{a}{\sin A} = \frac{b}{\sin B}
\]
Substitute the values:
\[
\frac{75}{\sin 30^\circ} = \frac{100}{\sin B}
\]
Calculate \(\sin 30^\circ = 0.5\):
\[
\frac{75}{0.5} = \frac{100}{\sin B}
\]
\[
150 = \frac{100}{\sin B}
\]
\[
\sin B = \frac{100}{150} \approx 0.6667
\]
Now, we can find \(\angle B\):
\[
\angle B \approx \arcsin(0.6667) \approx 41.8^\circ
\]

#### Step 2: Calculate \(\angle C\)
Since the sum of angles in a triangle is \(180^\circ\):
\[
\angle C = 180^\circ - \angle A - \angle B
\]
\[
\angle C \approx 180^\circ - 30^\circ - 41.8^\circ \approx 108.2^\circ
\]

#### Step 3: Find side \(c\) using the Law of Sines
\[
\frac{a}{\sin A} = \frac{c}{\sin C}
\]
\[
\frac{75}{0.5} = \frac{c}{\sin 108.2^\circ}
\]
Calculate \(\sin 108.2^\circ \approx 0.9511\):
\[
150 = \frac{c}{0.9511}
\]
\[
c \approx 150 \cdot 0.9511 \approx 142.7
\]

#### Summary of Solution for Case (b)
- \(\angle B \approx 41.8^\circ\)
- \(\angle C \approx 108.2^\circ\)
- \(c \approx 142.7\)

There is only one solution for Case (b) as it forms a valid triangle.

---

Would you like further details on any part of this process?

### Related Questions
1. How does the Law of Sines apply to obtuse triangles specifically?
2. What alternative methods can solve triangles when the Law of Sines fails?
3. How does the Law of Cosines compare when solving for missing angles and sides?
4. In what cases does the Law of Sines provide two possible triangles (ambiguous case)?
5. How can one verify the triangle inequality in such problems?

### Tip
Always check the range of the sine function when applying the Law of Sines to avoid impossible triangle scenarios.

Use the Law of Sines to solve for all possible triangles that satisfy the given conditions.
(a) a = 20, b = 45, ∠A = 125°
(b) a = 75, b = 100, ∠A = 30°

Learn how to use the Law of Sines to solve triangles with given side lengths and angles, including cases where no solution exists. This trigonometry problem is suitable for students in Grades 10-12 and covers key concepts such as angle calculation, the Law of Sines, and ambiguous cases.

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