Solving Every Side and Angle: Right Triangle Computation
Right triangle computation combines the Pythagorean theorem with trigonometric ratios to find all unknown sides and angles of a right triangle when given just a few measurements. This process — called "solving" a triangle — is used in surveying, navigation, construction, and any field where indirect measurement is needed.
Components of Right Triangle Computation
This section covers the strategies for fully solving a right triangle:
- Given Two Sides: Use the Pythagorean theorem to find the third side, then inverse trig to find the angles.
- Given One Side and One Acute Angle: Use sine, cosine, or tangent to find the other sides, and subtract from 90° to find the other acute angle.
- Special Right Triangles: The 45-45-90 triangle (sides in ratio 1:1:√2) and the 30-60-90 triangle (sides in ratio 1:√3:2) have exact values.
- Applications: Solving real-world problems involving heights, distances, angles of elevation, and angles of depression.
Examples of Right Triangle Computation
Given Two Sides Examples
- Legs are 5 and 12. Hypotenuse = √(25 + 144) = √169 = 13. The smaller angle = tan⁻¹(5/12) ≈ 22.6°, and the larger = 90 - 22.6 = 67.4°.
- Hypotenuse is 17, one leg is 8. Other leg = √(289 - 64) = √225 = 15. Angle opposite the 8 side = sin⁻¹(8/17) ≈ 28.1°.
- Legs are 7 and 7. Hypotenuse = √(49 + 49) = √98 ≈ 9.90. Both acute angles are 45° (isosceles right triangle).
Given One Side and One Angle Examples
- Angle = 35°, hypotenuse = 20. Adjacent side = 20 × cos(35°) ≈ 16.38. Opposite side = 20 × sin(35°) ≈ 11.47. Other angle = 55°.
- Angle = 50°, opposite side = 10. Hypotenuse = 10 / sin(50°) ≈ 13.05. Adjacent side = 10 / tan(50°) ≈ 8.39. Other angle = 40°.
- Angle = 60°, adjacent side = 6. Opposite side = 6 × tan(60°) = 6√3 ≈ 10.39. Hypotenuse = 6 / cos(60°) = 12. Other angle = 30°.
Special Right Triangle Examples
- A 45-45-90 triangle with legs of 8: Hypotenuse = 8√2 ≈ 11.31.
- A 30-60-90 triangle with shortest side 5: The side opposite 60° is 5√3 ≈ 8.66, and the hypotenuse is 10.
- A square with diagonal 14 cm: Each side is 14/√2 = 7√2 ≈ 9.90 cm (the square's diagonal creates two 45-45-90 triangles).
Applications Examples
- A person stands 30 meters from a building and looks up at a 55° angle. The building height above eye level is 30 × tan(55°) ≈ 42.84 meters.
- A pilot at 5,000 feet altitude spots a runway at a 12° angle of depression. The ground distance to the runway is 5,000 / tan(12°) ≈ 23,517 feet.
- A 20-foot ramp must rise 4 feet. The angle of incline is sin⁻¹(4/20) = sin⁻¹(0.2) ≈ 11.54°.