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.