Formula and Method for Triangle Ratio (Trigonometric Ratios)
The "triangle ratios" are the trigonometric ratios of a right triangle: sine, cosine, and tangent, defined as ratios between the two legs and the hypotenuse relative to a reference angle θ. Given the lengths of the side opposite θ and the side adjacent to θ, this calculator finds the hypotenuse, the angle θ itself, and all six trigonometric ratios (sine, cosine, tangent, and their reciprocals cosecant, secant, cotangent).
How the calculation works
First, the hypotenuse is found with the Pythagorean theorem: c = √(a² + b²), where a is the opposite side and b is the adjacent side. The reference angle is then θ = arctan(a / b) (the inverse tangent of opposite over adjacent). From there the three primary ratios follow directly: sin θ = a/c (SOH: opposite over hypotenuse), cos θ = b/c (CAH: adjacent over hypotenuse), and tan θ = a/b (TOA: opposite over adjacent). The three reciprocal ratios are csc θ = c/a, sec θ = c/b, and cot θ = b/a.
Common mistakes
- Mixing up opposite and adjacent: "opposite" and "adjacent" are always relative to the angle you're solving for — swapping them flips sine and cosine and changes the angle you compute.
- Degrees vs. radians: arctan() in most calculators and programming languages returns radians by default; convert to degrees by multiplying by 180/π before comparing to a protractor reading.
- Non-right triangles: these ratios only apply to right triangles. For a triangle with no 90° angle, use the Law of Sines or Law of Cosines instead.
Real-world applications
- Surveying and navigation use these ratios to find distances and heights that can't be measured directly (e.g., a building's height from a known distance and angle of elevation)
- Construction and carpentry use them to calculate roof pitch, stair rise/run angles, and ramp slopes
- Physics uses sine and cosine constantly to resolve vectors (forces, velocities) into perpendicular components
- Engineering and architecture use trigonometric ratios for structural load angles and design geometry
Frequently Asked Questions
What are the trigonometric ratios of a right triangle?
For a reference angle θ in a right triangle, sine is the opposite side divided by the hypotenuse (sin θ = opposite/hypotenuse), cosine is the adjacent side divided by the hypotenuse (cos θ = adjacent/hypotenuse), and tangent is the opposite side divided by the adjacent side (tan θ = opposite/adjacent). These are commonly remembered with the mnemonic SOH-CAH-TOA.
How do I find the angle from two side lengths?
If you know the side opposite and adjacent to the angle, use the inverse tangent: θ = arctan(opposite/adjacent). For example, with opposite = 3 and adjacent = 4, θ = arctan(3/4) ≈ 36.87°.
What are cosecant, secant, and cotangent?
They are the reciprocals of the three primary ratios: cosecant is 1/sin θ (hypotenuse/opposite), secant is 1/cos θ (hypotenuse/adjacent), and cotangent is 1/tan θ (adjacent/opposite).
Do these ratios only work for right triangles?
Yes. Sine, cosine, and tangent as opposite/adjacent/hypotenuse ratios are defined for right triangles. For triangles without a 90° angle, use the Law of Sines or Law of Cosines instead.
Common Tip Percentages
| Service | Typical Tip Range | Notes |
|---|
| Restaurant (sit-down) | 15–20% | 20%+ for exceptional service |
| Food delivery | 10–20% | Consider extra for long distances |
| Taxi / rideshare | 10–20% | More for assistance with luggage |
| Hotel housekeeping | $2–5/night | Leave daily, not just at checkout |
| Barber / hair salon | 15–20% | On pre-discount total |
| Pizza delivery | 10–15% or $3–5 minimum | Higher in bad weather |
General Household Budget Ratios (50/30/20 Rule)
| Category | % of Net Income | Examples |
|---|
| Needs | 50% | Rent, groceries, utilities, insurance |
| Wants | 30% | Dining out, entertainment, hobbies |
| Savings / Debt | 20% | Emergency fund, retirement, extra payments |
Common Measurement Conversions for Everyday Use
| Quantity | US / Imperial | Metric Equivalent |
|---|
| 1 cup | 8 fl oz | 237 mL |
| 1 pint | 16 fl oz | 473 mL |
| 1 quart | 32 fl oz | 946 mL |
| 1 gallon | 128 fl oz | 3.785 L |
| 1 lb | 16 oz | 453.6 g |
| 1 ft | 12 in | 30.48 cm |