To calculate the angle \( \angle ABC \) using the information given, you can use the **Sine Rule**. Here’s a step-by-step guide on how to do it:

### Step 1: Understand the Sine Rule
The Sine Rule states that in any triangle:
\[
\frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)}
\]
where \( a \), \( b \), and \( c \) are the sides opposite to angles \( A \), \( B \), and \( C \), respectively.

### Step 2: Apply the Sine Rule to Triangle ABC
For triangle \( ABC \):
- \( a = AC = 310 \, \text{m} \) (opposite to angle \( \angle ABC \))
- \( b = BC \approx 652.15 \, \text{m} \) (opposite to angle \( \angle BAC = 135^\circ \))

You already calculated \( BC \) using the Cosine Rule.

### Step 3: Calculate \( \sin(\angle BAC) \)
Convert \( \angle BAC \) to radians and find \( \sin(\angle BAC) \):
\[
\sin(135^\circ) = \sin(180^\circ - 45^\circ) = \sin(45^\circ) = \frac{1}{\sqrt{2}} \approx 0.707
\]

### Step 4: Use the Sine Rule to Find \( \angle ABC \)
Now, apply the Sine Rule to find \( \sin(\angle ABC) \):
\[
\frac{\sin(\angle ABC)}{310} = \frac{\sin(135^\circ)}{652.15}
\]
\[
\sin(\angle ABC) = \frac{310 \times 0.707}{652.15} \approx 0.336
\]

### Step 5: Calculate \( \angle ABC \)
Finally, find \( \angle ABC \) by taking the inverse sine:
\[
\angle ABC = \arcsin(0.336) \approx 19.64^\circ
\]

### Summary
Thus, \( \angle ABC \) is approximately \( 19.64^\circ \).

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Would you like to explore further details or have any questions? Here are some related questions you might find interesting:

1. How can you verify the calculated bearing of \( C \) from \( B \)?
2. How do you apply the Cosine Rule to find unknown angles in a triangle?
3. What are the implications of rounding intermediate steps in trigonometric calculations?
4. How would the solution change if the bearings were different?
5. How can the Sine Rule be applied to non-right triangles in different scenarios?

**Tip:** When solving trigonometric problems involving bearings, always draw a clear diagram to visualize the angles and sides involved. This helps avoid errors and ensures you use the correct angles for calculations.

Learn how to calculate angle ABC in triangle ABC using the Sine Rule. Step-by-step guide with detailed trigonometric calculations and example values provided. Suitable for Grades 10-12 students and anyone studying trigonometry.

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