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To solve these problems, we'll apply the **Sine Rule**, which is useful when we know two angles and one side, or two sides and one non-included angle.

### Part 1: Calculating two possible values for angle \( ACB \)

We are given:
- \( AB = 8.1 \, \text{cm} \)
- \( BC = 4.9 \, \text{cm} \)
- \( \angle BAC = 31^\circ \)

Using the **Sine Rule**:

\[
\frac{AB}{\sin(\angle ACB)} = \frac{BC}{\sin(\angle BAC)}
\]

Let's solve for \( \angle ACB \).

### Step 1: Rearrange the Sine Rule to solve for \( \sin(\angle ACB) \):

\[
\sin(\angle ACB) = \frac{AB \cdot \sin(\angle BAC)}{BC}
\]

Substitute the given values:

\[
\sin(\angle ACB) = \frac{8.1 \cdot \sin(31^\circ)}{4.9}
\]

Now, I'll compute this to get the value of \( \sin(\angle ACB) \) and then find the corresponding angles \( ACB \).From the calculations:

- \( \sin(\angle ACB) = 0.851 \)
- The **acute** angle \( ACB \) is approximately \( 58^\circ \)
- The **obtuse** angle \( ACB \) is approximately \( 122^\circ \)

Thus, the two possible values for \( \angle ACB \) are \( 58^\circ \) and \( 122^\circ \).

### Part 2: Finding the obtuse angle \( x \) in the lower triangle

For the lower triangle, we are given:
- \( \angle A = 29^\circ \)
- \( BC = 6.6 \, \text{cm} \)
- \( AB = 4.3 \, \text{cm} \)

We can again use the **Sine Rule** to calculate \( x \). Let's apply the same process here:

\[
\frac{AB}{\sin(x)} = \frac{BC}{\sin(29^\circ)}
\]

Now, I will calculate \( x \).For the second triangle:

- The value of \( \sin(x) = 0.316 \)
- The corresponding angle \( x \) is approximately \( 18^\circ \).

### Summary of results:
1. The two possible values for \( \angle ACB \) are:
   - \( 58^\circ \) (acute)
   - \( 122^\circ \) (obtuse)
   
2. The obtuse angle \( x \) is approximately \( 18^\circ \).

Would you like more details on the steps or have any questions?

Here are 5 related questions to expand on this:
1. How does the sine rule apply to non-right-angled triangles?
2. Why are there two possible angles for \( \angle ACB \) in the ambiguous case?
3. How can we verify which of the two solutions is more likely in a real-world scenario?
4. How do we determine whether the given angle is obtuse or acute?
5. Can the cosine rule be used for these triangles as well?

**Tip:** Always ensure that your calculator is set to the correct mode (degrees or radians) when solving trigonometric problems!

Triangle ABC is such that AB = 8.1 cm, BC = 4.9 cm and angle BAC = 31°. Work out the values of these two answers for angle ACB, including the obtuse angle x in the lower triangle.

Learn how to solve for two possible angles ACB in triangle ABC using the Sine Rule, with sides AB = 8.1 cm, BC = 4.9 cm, and angle BAC = 31°. Also, find the obtuse angle x using the Sine Rule for another triangle with sides BC = 6.6 cm, AB = 4.3 cm, and angle A = 29°.

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