Understanding Pythagoras' Theorem and Trigonometry in GCSE Maths

Pythagoras' Theorem

Pythagoras' Theorem is a fundamental concept in GCSE Maths that relates the sides of a right-angled triangle. It states that in a right-angled triangle, the square 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 we denote the lengths of the sides as a, b, and c (where c is the hypotenuse), Pythagoras' Theorem can be written as:

c² = a² + b²

Worked Example

Problem: In a right-angled triangle, two sides have lengths 5 cm and 12 cm. Find the length of the hypotenuse.

Solution:

1. Let a = 5 cm and b = 12 cm
2. Using Pythagoras' Theorem: c² = a² + b²
3. c² = 5² + 12²
4. c² = 25 + 144
5. c² = 169
6. Taking the square root of both sides: c = √169
7. c = 13 cm

Trigonometry

Trigonometry is the study of the relationships between the sides and angles of triangles. In GCSE Maths, you will learn about the three main trigonometric ratios: sine, cosine, and tangent, often referred to as SOHCAHTOA.

• SOH: Sine ratio = Opposite side / Hypotenuse
• CAH: Cosine ratio = Adjacent side / Hypotenuse
• TOA: Tangent ratio = Opposite side / Adjacent side

These ratios allow you to find missing angles or sides in right-angled triangles, as well as solve problems involving angles of elevation and depression.

Worked Example

Problem: In a right-angled triangle, one angle is 30°, and the hypotenuse is 10 cm. Find the lengths of the other two sides.

Solution:

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

At the higher tier, you will also learn the Sine and Cosine Rules for solving non-right-angled triangles, as well as the Area Rule for finding the area of any triangle given two sides and the included angle.