Mastering Pythagoras and Trigonometry for GCSE Maths

Pythagoras' Theorem

Pythagoras' theorem relates the sides of a right-angled triangle. If the lengths of the two shorter sides are a and b, and the length of the longest side (hypotenuse) is c, then:

a² + b² = c²

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 c: a = 5 cm, b = 12 cm
2. Using Pythagoras: a² + b² = c²
3. 5² + 12² = c²
4. 25 + 144 = 169
5. c² = 169
6. Take the square root: c = √169 = 13 cm

Trigonometry

Trigonometry involves the study of ratios between the sides of right-angled triangles. The three main ratios are:

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

The phrase SOHCAHTOA can help you remember these ratios.

Worked Example

Problem: In a right-angled triangle, the angle opposite the 6 cm side is 30°. Find the lengths of the other two sides.

Solution:

1. Let the hypotenuse be c and the adjacent side be b.
2. sin 30° = 6/c (Opposite/Hypotenuse)
3. 0.5 = 6/c (The sine of 30° is 0.5)
4. c = 6/0.5 = 12 cm
5. cos 30° = b/12 (Adjacent/Hypotenuse)
6. √3/2 = b/12 (The cosine of 30° is √3/2)
7. b = 12 * √3/2 = 6√3 cm

Angles of Elevation/Depression

Trigonometry can be applied to find angles of elevation (upwards from horizontal) and depression (downwards from horizontal) by using the tangent ratio:

tan θ = Opposite / Adjacent

Higher Tier: Non-Right-Angled Triangles

The Sine Rule and Cosine Rule are used to find unknown sides and angles in non-right-angled triangles:

Sine Rule: a/sin A = b/sin B = c/sin C

Cosine Rule: c² = a² + b² - 2ab cos C

Area: Area = (1/2) ab sin C

These advanced trigonometric concepts are covered in the Higher Tier GCSE specification.