Right Triangle Properties and Trigonometry
A right triangle contains one 90-degree angle. The side opposite the right angle is the hypotenuse (longest side). The other two sides are called legs. Right triangles are the foundation of trigonometry.
The Pythagorean Theorem
a² + b² = c²
where c is the hypotenuse, and a and b are the two legs.
where c is the hypotenuse, and a and b are the two legs.
SOH-CAH-TOA
| Function | Mnemonic | Formula |
|---|---|---|
| Sine | SOH | sin(θ) = Opposite / Hypotenuse |
| Cosine | CAH | cos(θ) = Adjacent / Hypotenuse |
| Tangent | TOA | tan(θ) = Opposite / Adjacent |
Finding Angles from Sides
θ = arctan(a / b) = arcsin(a / c) = arccos(b / c)
The other acute angle = 90° - θ
The other acute angle = 90° - θ
Special Right Triangles
30-60-90 Triangle
Sides are in the ratio 1 : √3 : 2.
| Angle | Opposite Side Ratio |
|---|---|
| 30° | 1 (shortest leg) |
| 60° | √3 ≈ 1.732 |
| 90° | 2 (hypotenuse) |
45-45-90 Triangle
An isosceles right triangle. The two legs are equal and the hypotenuse is √2 times a leg.
| Angle | Opposite Side Ratio |
|---|---|
| 45° | 1 (each leg) |
| 90° | √2 ≈ 1.414 (hypotenuse) |
Area and Perimeter
Area = (1/2) × a × b
Perimeter = a + b + c
Perimeter = a + b + c
Applications
- Construction: The 3-4-5 triangle is used to verify right angles.
- Navigation: Calculating distances and bearings.
- Physics: Resolving force vectors into components.
- Surveying: Measuring heights indirectly.
- Engineering: Ramp angles, roof pitch, and structural design.