Mastering Pythagoras and Trigonometry for GCSE Maths

Introduction to Pythagoras and Trigonometry

Pythagoras' theorem and trigonometry are fundamental concepts in geometry and essential components of the GCSE Mathematics curriculum. These topics equip students with the necessary tools to solve problems involving right-angled triangles and provide a foundation for further study in mathematics and related fields.

Pythagoras' Theorem

Pythagoras' theorem states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. Mathematically, it is expressed as:

a² + b² = c²

Where 'a' and 'b' are the lengths of the shorter sides, and 'c' is the length of the hypotenuse.

Worked Example

Problem: Find the missing side length in a right-angled triangle with sides 5 cm and 12 cm.

Solution:

1. Let the missing side be 'x'
2. Using Pythagoras' theorem: 5² + 12² = x²
3. 25 + 144 = x²
4. 169 = x²
5. √169 = x
6. x = 13 cm

Trigonometry and SOHCAHTOA

Trigonometry deals with the relationships between the sides and angles of triangles. The acronym SOHCAHTOA (Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, Tangent = Opposite/Adjacent) helps us remember the trigonometric ratios for right-angled triangles.

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. Let the opposite side be 'x' and the adjacent side be 'y'
2. Using SOHCAHTOA: sin(30°) = x/10 and cos(30°) = y/10
3. sin(30°) = 0.5, cos(30°) = 0.866
4. 0.5 = x/10, x = 5 cm
5. 0.866 = y/10, y = 8.66 cm

At the higher tier, students will also learn the Sine and Cosine Rules for non-right-angled triangles, as well as the Area Rule for finding the area of any triangle using trigonometry.

Additional Resources

• AQA GCSE Mathematics specification: https://www.aqa.org.uk/subjects/mathematics/gcse/mathematics-8300
• BBC Bitesize GCSE Maths guide: https://www.bbc.co.uk/bitesize/topics/zcmtmp3
• Trigonometry examples on TRH Learning: https://www.trh-learning.com/blog/index.html#trigonometryExample