Pythagoras' theorem
In any right-angled triangle, the square of the hypotenuse (the side opposite the right angle) equals the sum of the squares of the other two sides.
Key idea
c² = a² + b², where c is the hypotenuse. To find a shorter side, rearrange: a² = c² − b².
Right-angled trigonometry (SOH CAH TOA)
For an acute angle θ in a right-angled triangle, name the sides opposite, adjacent and hypotenuse. Then:
- sin θ = opposite / hypotenuse
- cos θ = adjacent / hypotenuse
- tan θ = opposite / adjacent
Worked example
A right-angled triangle has legs 6 cm and 8 cm. Hypotenuse = √(6² + 8²) = √(36 + 64) = √100 = 10 cm. If instead the hypotenuse is 10 cm and one angle is 30°, the opposite side = 10 × sin 30° = 10 × 0.5 = 5 cm.
Remember
- Pythagoras finds sides; trig ratios link sides to angles.
- To find an angle, use the inverse: θ = sin⁻¹(opp/hyp), etc.
Angles of elevation and depression
Many real problems use the angle of elevation (looking up) or depression (looking down) measured from the horizontal. Draw the right-angled triangle, label the sides opposite, adjacent and hypotenuse relative to that angle, then pick sin, cos or tan. If two sides are known instead of the angle, use the inverse function to find the angle.