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To work out the size of angle \( \angle BCD \), we need to apply trigonometric relationships, particularly using the triangle formed. Here's a step-by-step breakdown:

### Step 1: Identify the Triangle
- The triangle \( ABD \) is a right-angled triangle with:
  - \( AB = 15 \, \text{m} \)
  - \( BD = 15 \, \text{m} \) (as it’s a vertical side).
  - Angle \( \angle BAD = 62^\circ \)

Since triangle \( BCD \) has angle \( BCD \) as the target, we'll first solve triangle \( ABD \), then focus on the larger triangle.

### Step 2: Use the Right Angle Trigonometry in Triangle ABD
From triangle \( ABD \), we can use the tangent function to find \( BD \), the height of the triangle.
\[
\tan(\angle BAD) = \frac{BD}{AD}
\]

Let me calculate the required angle from here.The length of \( BD \) is approximately \( 28.2 \, \text{m} \).

### Step 3: Use the Cosine Rule in Triangle BCD
Now, we move to the larger triangle \( BCD \). We know:
- \( BC = 24 \, \text{m} \)
- \( BD = 28.2 \, \text{m} \)
- \( CD = 15 \, \text{m} \)

To find the angle \( \angle BCD \), we will use the cosine rule:

\[
\cos(\angle BCD) = \frac{BD^2 + BC^2 - CD^2}{2 \cdot BD \cdot BC}
\]

Let me compute the angle.The size of \( \angle BCD \) is approximately \( 32.1^\circ \), correct to one decimal place.

Would you like further clarification or additional details?

Here are some related questions you could explore:
1. How is the cosine rule derived and when is it applied?
2. Can the sine rule be applied to this problem, and how would the result differ?
3. What are the conditions for using trigonometry in non-right-angled triangles?
4. How does the accuracy of measurements affect trigonometric calculations?
5. What are some real-life applications of the cosine rule?

**Tip:** When working with angles and side lengths in trigonometry, it's helpful to double-check the triangle type to use the correct rules (sine, cosine, or tangent).

AB = 15 m, BC = 24 m, angle BAD = 62°. Work out the size of angle BCD. Give your answer correct to 1 decimal place.

Solve a trigonometry problem where AB = 15 m, BC = 24 m, and angle BAD = 62°. Learn how to find the size of angle BCD using the tangent function and cosine rule. Suitable for students in Grades 10-12.

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