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GCSE trigonometry skills are essential for students to solve the three types of right-angled triangle problems presented, common in GCSE Mathematics and crucial for those aiming for grades 4 or 5. Here’s a summary of the key skills required for each problem, along with tailored advice for students and teachers, and probing questions to assess comprehension.

**Skill**: Understanding and applying the trigonometric ratios (Sine, Cosine, and Tangent) to find angles.

**Advice**: Remember that for any right-angled triangle, SOH-CAH-TOA represents the ratios of sides: Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, and Tangent = Opposite/Adjacent. Identify which sides are given and which ratio to use accordingly.

**Key Questions**:

- Which trigonometric ratio would you use to find angle θ in this triangle?
- How do you know if you should use the sine, cosine, or tangent function?

**Skill**: Using trigonometric ratios to find unknown sides in right-angled triangles.

**Advice**: Students should focus on identifying the right ratio to use based on the sides given and the side they need to find. Drawing and labeling the triangle can help in visualising the problem.

**Key Questions**:

- Given angle θ and side PR, which trigonometric function helps us find QR?”
- “What is the relationship between side QR and side PR in this context?

**Skill**: Demonstrating the Pythagorean theorem and calculating the area of irregular shapes by decomposing them into basic shapes.

**Advice**: Be sure to review the Pythagorean theorem (a² + b² = c²) where ‘c’ is the hypotenuse. For areas, students should break down complex shapes into simpler shapes like triangles or rectangles whose areas they can calculate.

**Key Questions**:

- How can we show that triangle DCA is a right-angled triangle?
- Can you decompose quadrilateral ABCD into shapes with areas you know how to calculate?

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