Library / Foundations: Maths / FMA-13
Foundations: Maths
Right-angled triangle trigonometry
Opposite and adjacent sides from the hypotenuse and an angle.
- Ref
- FMA-13
- Rev
- v1
- Field
- Foundations: Maths
- Result
- opp [m]
Relative to angle θ: opposite is the side facing θ, adjacent is the side next to θ (not the hypotenuse). Angles in degrees.
Trigonometric ratios link an angle to the ratio of two sides, so a known angle and one side give the other sides.
Calculator
- Adjacent side8.192 m
- tan θ0.7002
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- The mnemonic SOH-CAH-TOA matches each ratio to its sides.
- Height from distance and angle uses tan θ = height/distance.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (m) | Local step | Change in result |
|---|---|---|---|---|---|---|
| hyp Hypotenuse | 1 to 100 m | 0.5736 | 57.36 | 56.78 | +10% | +10 % |
| θ Angle | 5 to 85 ° | 0.8716 | 9.962 | 9.09 | +10% | +8.53 % |
Sweep of Angle
| Angle (°) | Opposite side (m) |
|---|---|
| 5 | 0.8716 |
| 15 | 2.588 |
| 25 | 4.226 |
| 35 | 5.736 |
| 45 | 7.071 |
| 55 | 8.192 |
| 65 | 9.063 |
| 75 | 9.659 |
| 85 | 9.962 |
Sweep one variable
Variables and how each one contributes
Opposite side
What the equation gives you. Everything else feeds it.
Hypotenuse
The longest side. Directly proportional.
Angle
Angle between the hypotenuse and the adjacent side. Steeper angles increase the opposite side and reduce the adjacent one.
Worked example with the values above
- sin 35° = 0.5736.
- opp = hyp × sin θ = 10 × 0.5736.
- Opposite side 5.736 m.
Assumptions and limits
- A right-angled triangle.
- Calculator in degree mode.
Common mistakes
- Calculator in radian mode.
- Using the wrong side as adjacent.
Statistics and uncertainty
Near 90° small angle errors have little effect on sin θ, but near 0° they matter a lot proportionally: δ(opp)/opp ≈ cot θ · δθ.
Where it is used
- Heights and ranges from angles of elevation.
- Resolving a force or velocity into components.
Maths you need
- Right triangles: opposite, adjacent, hypotenuse
- Ratios: sin, cos and tan