Pythagoras and right-angled trigonometry

Pythagoras' theorem

In a right-angled triangle, the square of the hypotenuse equals the sum of the squares of the other two sides.

Formula: a² + b² = c²

Where c is the hypotenuse (the side opposite the right angle — always the longest side).

Finding the hypotenuse: c = √(a² + b²). Example: legs are 6 cm and 8 cm. c = √(36 + 64) = √100 = 10 cm.

Finding a shorter side: a = √(c² − b²). Example: hypotenuse is 13 cm, one leg is 5 cm. Other leg = √(169 − 25) = √144 = 12 cm.

Pythagorean triples (common sets to memorise): 3-4-5; 6-8-10; 5-12-13; 8-15-17; 7-24-25.

3D Pythagoras (Higher): the space diagonal of a cuboid is d = √(l² + w² + h²).

SOHCAHTOA — right-angled trigonometry

In a right-angled triangle with reference angle θ:

• Opposite = side directly opposite to θ.
• Adjacent = side next to θ (not the hypotenuse).
• Hypotenuse = longest side (opposite the right angle).

The three ratios:

• sin θ = Opposite / Hypotenuse (SOH)
• cos θ = Adjacent / Hypotenuse (CAH)
• tan θ = Opposite / Adjacent (TOA)

Finding a side: identify which two sides are involved (given and required), choose the correct ratio, rearrange.

Example: In a right-angled triangle, angle = 40°, hypotenuse = 12 cm. Find the opposite side. sin 40° = opp/12 → opp = 12 × sin 40° ≈ 7.71 cm.

Finding an angle: use inverse trig. Example: opposite = 7, adjacent = 24. Find θ. tan θ = 7/24 → θ = tan⁻¹(7/24) ≈ 16.3°.

Angles of elevation and depression

• Angle of elevation: measured upward from horizontal.
• Angle of depression: measured downward from horizontal.

These appear frequently in CCEA context questions (e.g. "a lighthouse, a cliff, a shadow").