The three ratios
In a right-angled triangle, choose one acute angle θ. The side facing θ is the opposite, the side next to θ (not the hypotenuse) is the adjacent, and the longest side (facing the right angle) is the hypotenuse.
Key idea
sin θ = opposite ÷ hypotenuse; cos θ = adjacent ÷ hypotenuse; tan θ = opposite ÷ adjacent. (Remember SOH-CAH-TOA.)
Using a known triangle
Take a triangle with opposite = 3, adjacent = 4 and hypotenuse = 5. Then sin θ = 3/5 = 0.6, cos θ = 4/5 = 0.8, tan θ = 3/4 = 0.75. These fit because 3² + 4² = 9 + 16 = 25 = 5² (Pythagoras).
Worked example
A ladder leans on a wall. Its foot is 8 m from the wall (adjacent) and it reaches 15 m up (opposite). The hypotenuse = √(8² + 15²) = √289 = 17 m. So tan θ = 15/8 = 1.875 and sin θ = 15/17 ≈ 0.88.
Finding a side
If you know one angle's ratio and one side, multiply or divide. Knowing sin θ = 0.6 and hypotenuse = 20 cm gives opposite = 0.6 × 20 = 12 cm. If instead the opposite side were known and you wanted the hypotenuse, you would divide: hypotenuse = opposite ÷ sin θ = 12 ÷ 0.6 = 20 cm.
Choosing the ratio is simple once the three sides are labelled: pick the ratio that uses the two sides you actually have. If you have the opposite and adjacent, use tan θ; if one of them is the hypotenuse, use sin θ or cos θ.
Remember
- Every ratio needs the SAME angle θ.
- sin θ and cos θ are always between 0 and 1; tan θ can be more than 1.
- Label opposite, adjacent, hypotenuse before choosing a ratio.