In a right-angled triangle, the square of the hypotenuse equals the sum of squares of the other two sides.
a² + b² = c²
a, b = legs (shorter sides)
c = hypotenuse (longest side, opposite right angle)
Given: legs a = 3, b = 4
Find: hypotenuse c
c² = 3² + 4² = 9 + 16 = 25
c = √25 = 5
Given: leg a = 5, hypotenuse c = 13
Find: leg b
5² + b² = 13²
25 + b² = 169
b² = 144, so b = 12
Trigonometry uses ratios in right triangles to find unknown sides and angles.
Sine
sin θ = opposite/hypotenuse
SOH
Cosine
cos θ = adjacent/hypotenuse
CAH
Tangent
tan θ = opposite/adjacent
TOA
This helps remember which ratio to use for each trigonometric function!
Use trigonometry to find missing sides or angles in right triangles.
Given: angle = 30°, hypotenuse = 10
Find: opposite side
sin 30° = opposite/10
opposite = 10 × sin 30° = 10 × 0.5 = 5
Given: opposite = 7, adjacent = 7
Find: angle θ
tan θ = 7/7 = 1
θ = tan⁻¹(1) = 45°
Some right triangles have special angle and side relationships.
Sides in ratio 1 : 1 : √2
If legs = 1, hypotenuse = √2
Sides in ratio 1 : √3 : 2
Short leg : long leg : hypotenuse
1. In a right triangle with legs 6 and 8, find the hypotenuse
2. In a right triangle, if the hypotenuse is 5 and one leg is 3, find the other leg