Solve the triangle using the Law of Sines. (Assume
b = 7, ∠A = 30°, and ∠C = 120°. Round the lengths to two decimal places.)

To solve this triangle using the Law of Sines, we need to find all sides and angles. We're given:

- \( b = 7 \)
- \( \angle A = 30^\circ \)
- \( \angle C = 120^\circ \)

### Step 1: Find \(\angle B\)
The sum of the angles in any triangle is always \(180^\circ\). Therefore, we can find \(\angle B\) as follows:

\[
\angle B = 180^\circ - \angle A - \angle C = 180^\circ - 30^\circ - 120^\circ = 30^\circ
\]

### Step 2: Use the Law of Sines
The Law of Sines states:

\[
\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}
\]

We can start by finding side \(a\) using \(b = 7\), \(\angle B = 30^\circ\), and \(\angle A = 30^\circ\).

\[
\frac{a}{\sin 30^\circ} = \frac{7}{\sin 30^\circ}
\]

Since \(\sin 30^\circ = 0.5\), we simplify the equation:

\[
\frac{a}{0.5} = \frac{7}{0.5} \implies a = 7
\]

### Step 3: Find side \(c\)
Now, using the Law of Sines again to find \(c\):

\[
\frac{c}{\sin 120^\circ} = \frac{7}{\sin 30^\circ}
\]

We know that \(\sin 120^\circ = \sin 60^\circ = \frac{\sqrt{3}}{2} \approx 0.866\), so:

\[
\frac{c}{0.866} = \frac{7}{0.5}
\]

Simplifying:

\[
c = \frac{7 \times 0.866}{0.5} = 12.12
\]

### Final Answer:
- \( a = 7 \)
- \( b = 7 \) (given)
- \( c = 12.12 \)
- \( \angle A = 30^\circ \), \( \angle B = 30^\circ \), \( \angle C = 120^\circ \)

Let me know if you'd like more details or have any questions! 

Here are 5 related questions:
1. How does the Law of Sines apply to non-right triangles?
2. What are the possible cases when the Law of Sines leads to an ambiguous triangle?
3. How would you solve the triangle if only two sides and an angle were given?
4. Can the Law of Sines be used to solve a right triangle? Why or why not?
5. What is the relationship between the Law of Sines and the Law of Cosines?

**Tip:** Always remember to check if you are dealing with an ambiguous case when applying the Law of Sines, especially when two angles or sides are similar.

Solve the triangle using the Law of Sines. (Assume b = 7, ∠A = 30°, and ∠C = 120°. Round the lengths to two decimal places.)

Learn how to solve a triangle using the Law of Sines, given b = 7, ∠A = 30°, and ∠C = 120°. This detailed solution covers key trigonometric concepts, the Law of Sines, and step-by-step calculations. Suitable for students in Grades 9-12.

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