Mastering Pythagoras' Theorem and Trigonometry for GCSE Maths

Pythagoras' Theorem

Pythagoras' theorem is a fundamental concept in geometry that relates the sides of a right-angled triangle. It states that in any right-angled triangle, the square of the length of the hypotenuse (the longest side opposite the right angle) is equal to the sum of the squares of the other two sides.

Mathematically, if the sides of the right-angled triangle are a, b, and c (where c is the hypotenuse), then:

a² + b² = c²

Example

Problem: Find the length of the hypotenuse of a right-angled triangle with sides 3 cm and 4 cm.

Solution:

1. Let a = 3 cm and b = 4 cm
2. Using Pythagoras' theorem: a² + b² = c²
3. Substitute the values: 3² + 4² = c²
4. Evaluate: 9 + 16 = 25
5. Take the square root: c = √25 = 5 cm

Trigonometry

Trigonometry is the study of relationships between the sides and angles of triangles. In a right-angled triangle, the trigonometric ratios sine (sin), cosine (cos), and tangent (tan) are defined in terms of the opposite, adjacent, and hypotenuse sides.

• Sine (sin): Opposite / Hypotenuse
• Cosine (cos): Adjacent / Hypotenuse
• Tangent (tan): Opposite / Adjacent

The acronym SOHCAHTOA (Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, Tangent = Opposite/Adjacent) helps in remembering these ratios.

Example

Problem: In a right-angled triangle, the angle is 30°, and the hypotenuse is 10 cm. Find the opposite and adjacent sides.

Solution:

1. sin(30°) = Opposite / Hypotenuse
2. Opposite = sin(30°) × Hypotenuse
3. Opposite = 0.5 × 10 = 5 cm
4. cos(30°) = Adjacent / Hypotenuse
5. Adjacent = cos(30°) × Hypotenuse
6. Adjacent = 0.866 × 10 = 8.66 cm

The AQA GCSE Mathematics specification also includes the Sine and Cosine Rules for finding angles and sides in non-right-angled triangles, as well as the Area Rule for calculating the area of a triangle using trigonometry.