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Use the Law of Sines to solve the triangle.  Show all parts for the solution to the triangle.  
SHOW THE WORK FOR EACH PART!
1)	Solve the missing parts – write all values together including known/shown values as solution.  ABC-abc

Let's solve the triangle using the **Law of Sines**! The given triangle has:

- \( a = 10 \) (side opposite angle \( A \))  
- \( b = 7 \) (side opposite angle \( B \))  
- \( \angle A = 42^\circ \)  

We need to find the following:
1. Angle \( B \)  
2. Angle \( C \)  
3. Side \( c \)  

The **Law of Sines** states:

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

---

### Step 1: Find \( \angle B \) using the Law of Sines
\[
\frac{a}{\sin A} = \frac{b}{\sin B}
\]

Substitute the known values:

\[
\frac{10}{\sin 42^\circ} = \frac{7}{\sin B}
\]

First, calculate \( \sin 42^\circ \):

\[
\sin 42^\circ \approx 0.6691
\]

Now, solve for \( \sin B \):

\[
\frac{7}{10 / 0.6691} = \sin B
\]

\[
\sin B = \frac{7 \times 0.6691}{10} \approx 0.467
\]

Now, find \( \angle B \) by taking the inverse sine:

\[
B = \sin^{-1}(0.467) \approx 27.9^\circ
\]

So, \( \angle B \approx 27.9^\circ \).

---

### Step 2: Find \( \angle C \)
Since the sum of the angles in a triangle is \( 180^\circ \):

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

\[
\angle C = 180^\circ - 42^\circ - 27.9^\circ \approx 110.1^\circ
\]

So, \( \angle C \approx 110.1^\circ \).

---

### Step 3: Find Side \( c \) using the Law of Sines
We now use the Law of Sines again to find side \( c \):

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

Substitute the known values:

\[
\frac{10}{\sin 42^\circ} = \frac{c}{\sin 110.1^\circ}
\]

First, calculate \( \sin 110.1^\circ \):

\[
\sin 110.1^\circ \approx 0.936
\]

Now, solve for \( c \):

\[
c = \frac{10 \times 0.936}{0.6691} \approx 14.0
\]

So, \( c \approx 14.0 \).

---

### Final Solution
The solution for the triangle is:

- \( \angle A = 42^\circ \)  
- \( \angle B \approx 27.9^\circ \)  
- \( \angle C \approx 110.1^\circ \)  
- \( a = 10 \)  
- \( b = 7 \)  
- \( c \approx 14.0 \)

---

Do you need any more steps or further explanation?

---

### Related Questions:
1. How can the Law of Sines be applied in non-obtuse triangles?
2. What happens if the calculated sine value gives no valid angle (ambiguous case)?
3. How would the Law of Cosines help in this triangle?
4. Can we solve the triangle if only one side and two angles are known?
5. Why is it important to use degrees consistently when using trigonometric functions?

**Tip:** Always double-check angles to ensure the sum of the triangle's angles is exactly \(180^\circ\).

Use the Law of Sines to solve the triangle ABC. Given: side a = 10, side b = 7, angle A = 42°. Solve for the missing angles B, C, and side c.

Learn how to solve triangle ABC using the Law of Sines with a step-by-step solution. Given side a = 10, side b = 7, and angle A = 42°, this problem covers trigonometric concepts and the Law of Sines to find missing angles and sides. Suitable for Grades 9-12.

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