https://file.aiquickdraw.com/mathbot/file/1726494541130ohu2ixil.jpg

Let's work through each triangle step by step.

### Problem (a):
We are given a triangle with:
- \( \angle ABC = 52^\circ \),
- \( AC = 6 \, \text{cm} \),
- \( AE = 12 \, \text{cm} \),
- A right angle at \( C \), and
- The unknown side \( y = BC \).

Using the **sine rule** in the right-angled triangle \( \triangle ABC \):
\[
\sin(52^\circ) = \frac{AC}{BC} = \frac{6}{y}
\]
\[
y = \frac{6}{\sin(52^\circ)}
\]
Let's calculate \( y \).

### Problem (b):
We are given a right-angled triangle with:
- \( AB = 3 \, \text{m} \),
- \( AD = 15 \, \text{m} \),
- \( CD = 9 \, \text{m} \),
- \( \angle ABE = y^\circ \) (unknown angle), and
- Right angles at \( B \) and \( C \).

We can use the **tangent rule** to find the unknown angle \( y \):
\[
\tan(y) = \frac{AB}{BD} = \frac{3}{9}
\]
\[
y = \tan^{-1}\left(\frac{3}{9}\right)
\]
Let's calculate \( y \).

### Problem (c):
We have a circle and the triangle \( PQR \), where:
- \( PQ = 3 \, \text{cm} \),
- \( PR = 5 \, \text{cm} \),
- \( QR = 6 \, \text{cm} \), and
- \( \angle PRQ = x^\circ \) (unknown).

Using the **cosine rule**:
\[
\cos(x) = \frac{PR^2 + PQ^2 - QR^2}{2 \cdot PR \cdot PQ}
\]
Let's calculate \( x \).

### Problem (d):
We have:
- \( BE = 8.6 \, \text{cm} \),
- \( AE = 5.6 \, \text{cm} \),
- \( \angle D = 75^\circ \),
- \( \angle E = 45^\circ \), and
- \( x = BD \) is the unknown side.

We can use the **sine rule** to find \( x \):
\[
\frac{BE}{\sin(45^\circ)} = \frac{x}{\sin(75^\circ)}
\]
\[
x = \frac{BE \cdot \sin(75^\circ)}{\sin(45^\circ)}
\]
Let's calculate \( x \).

Now, I will calculate all the unknowns.Here are the calculated results for each triangle:

### (a)
The length of side \( BC \) (denoted \( y \)) is approximately **7.61 cm**.

### (b)
The unknown angle \( y^\circ \) is approximately **18.4°**.

### (c)
The angle \( x^\circ \) in triangle \( PQR \) is approximately **93.8°**.

### (d)
The length of side \( BD \) (denoted \( x \)) is approximately **11.75 cm**.

Would you like further details on these solutions, or have any specific questions?

Here are 5 related questions for further practice:
1. How would you solve triangle (a) using the cosine rule instead?
2. Can you use the sine rule to find the missing angles in triangle (b)?
3. How would the solution change in problem (c) if one side length were altered?
4. What is the general formula for solving triangles using the tangent rule?
5. How can the sine rule be applied to solve non-right-angled triangles?

**Tip**: The sine and cosine rules are most useful when you have non-right-angled triangles or when you know two angles or two sides.

For each of the following triangles find the length of the unknown lettered side (correct to two decimal places) or the size of the lettered angle (correct to one decimal place).

Learn how to solve triangles with unknown sides or angles using trigonometric formulas such as the sine rule, cosine rule, and tangent rule. This problem set covers both right-angled and non-right-angled triangles. Suitable for high school students in Grades 9-12.

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