n many cases the Law of Sines works perfectly well and returns the correct missing values in a non-right triangle. However, in some cases the Law of Sines returns two possible measurements.

Consider the diagram below, and assume that m∠B=61∘
, ¯¯¯¯¯¯¯¯AB=4.28 cm, and ¯¯¯¯¯¯¯¯AC=3.87

cm.

Using the Law of Sines, determine the value of m∠C

. You should notice that there are actually two possible values - list both of them (separated by a comma).

m∠C=
Incorrect °

If we assume the diagram is to scale, which value of m∠C

makes more sense? Enter the appropriate value.

m∠C=
Incorrect °

Using your answer to part (b), determine the length of BC

.

¯¯¯¯¯¯¯¯BC=
Incorrect cm

Let's solve this problem step by step using the Law of Sines.

### Given:
- \( m\angle B = 61^\circ \)
- \( AB = 4.28 \, \text{cm} \)
- \( AC = 3.87 \, \text{cm} \)

### Step 1: Use the Law of Sines to find \( m\angle C \)

The Law of Sines states:
\[
\frac{\sin A}{a} = \frac{\sin B}{b} = \frac{\sin C}{c}
\]
Where:
- \( A, B, C \) are the angles of the triangle
- \( a, b, c \) are the sides opposite to angles \( A, B, C \) respectively

For this triangle:
\[
\frac{\sin B}{AC} = \frac{\sin C}{AB}
\]

Substitute the known values:
\[
\frac{\sin 61^\circ}{3.87} = \frac{\sin C}{4.28}
\]

### Step 2: Solve for \( \sin C \)

\[
\sin C = \frac{4.28 \times \sin 61^\circ}{3.87}
\]
Let's calculate the value of \( \sin C \).

\[
\sin 61^\circ \approx 0.8746
\]
\[
\sin C = \frac{4.28 \times 0.8746}{3.87} \approx 0.9669
\]

### Step 3: Find possible values of \( m\angle C \)

Now, \( \sin C = 0.9669 \).

1. \( m\angle C \approx \arcsin(0.9669) \approx 75.32^\circ \)

However, since \( \sin \theta = \sin (180^\circ - \theta) \), the second possible value for \( m\angle C \) is:
2. \( m\angle C = 180^\circ - 75.32^\circ = 104.68^\circ \)

Thus, the two possible values for \( m\angle C \) are:
\[
m\angle C = 75.32^\circ, 104.68^\circ
\]

### Step 4: Determine the appropriate value of \( m\angle C \)

To decide which value makes more sense, consider that the sum of angles in a triangle must equal \( 180^\circ \).

For the first case (\( m\angle C = 75.32^\circ \)):
\[
m\angle A = 180^\circ - 61^\circ - 75.32^\circ = 43.68^\circ
\]

For the second case (\( m\angle C = 104.68^\circ \)):
\[
m\angle A = 180^\circ - 61^\circ - 104.68^\circ = 14.32^\circ
\]

Assuming the diagram is to scale and based on typical geometric considerations, \( m\angle A \approx 43.68^\circ \) seems more reasonable than \( 14.32^\circ \). So, the more appropriate value for \( m\angle C \) is \( 75.32^\circ \).

### Step 5: Find the length of \( BC \)

Using the Law of Sines again:
\[
\frac{BC}{\sin A} = \frac{AC}{\sin C}
\]
Substitute the values for \( AC \), \( \sin C \), and \( \sin A \):
\[
BC = \frac{AC \times \sin A}{\sin C} = \frac{3.87 \times \sin 43.68^\circ}{\sin 75.32^\circ}
\]

Let's calculate this.

### Final Answer
- \( m\angle C = 75.32^\circ \)
- \( BC \approx 2.79 \, \text{cm} \)

Would you like further details or any clarifications?

Here are 8 related questions:
1. What is the Law of Cosines, and when is it used?
2. How do you solve a triangle given two angles and one side (ASA case)?
3. What is the ambiguous case in the Law of Sines?
4. Can a triangle have two obtuse angles?
5. How does the Law of Sines handle obtuse triangles differently from acute triangles?
6. Why are there two possible angles for \( m\angle C \) in this scenario?
7. How can the Law of Sines be applied to right triangles?
8. What are the limitations of the Law of Sines?

**Tip:** Always check for the possibility of two solutions when applying the Law of Sines in ambiguous cases.

Learn how to solve a triangle problem using the Law of Sines to find angle C. Given angle B, sides AB and AC, determine the possible values of angle C and choose the appropriate one. Detailed step-by-step solution with calculations included.

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