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Trigonometry

Trigonometric Ratios

Sine, cosine, tangent ratios. Free online Trigonometric Ratios. Calculate trigonometric ratios online — fast, accurate, mobile-friendly, no signup needed.

Hypotenuse
5
sin
0.6
cos
0.8
tan
0.75

Derivation

  1. ├── 01Giveno = 3, a = 4
  2. ├── 02FormulaHypotenuse: √(t²+a²)
  3. ├── 03Substitute√(t²+4²)
  4. ├── 04Compute Hypotenuse5
  5. ├── 05Formulasin: t / √(t²+a²)
  6. ├── 06Substitutet / √(t²+4²)
  7. ├── 07Compute sin0.6
  8. ├── 08Formulacos: a / √(t²+a²)
  9. ├── 09Substitute4 / √(t²+4²)
  10. ├── 10Compute cos0.8
  11. ├── 11Formulatan: e.o / e.a
  12. ├── 12Substitutee.3 / e.4
  13. └── 13Compute tan0.75
Did you know?

Ratios appear in the earliest Greek mathematics (Eudoxus, 4th century BCE) as a way to compare incommensurable lengths without a concept of irrational numbers.

§01What is

Understanding the Trigonometric Ratios

The Trigonometric Ratios computes Hypotenuse from 2 inputs: opposite, adjacent. Sine, cosine, tangent ratios.

Trigonometry is how we turn angles into distances and distances into angles. It sits under every GPS fix, surveyor measurement, and game-engine render — and it still shows up in carpentry, roof pitches, and woodworking. The Trigonometric Ratios sits in that toolkit — it sine, cosine, tangent ratios. Enter your numbers above and the result updates instantly; every step of the math is shown in the Derivation panel so you can see exactly how the answer was reached.

§02The Formula

How it’s calculated

Hypotenuse = √(t²+a²) | sin = t / √(t²+a²) | cos = a / √(t²+a²) | tan = e.o / e.a

Where

o
Opposite
a
Adjacent
Hypotenuse
Output value
sin
Output value
cos
Output value
tan
Output value
§03Practical Example

Step-by-step walkthrough

Scenario

Apply the formula to a realistic set of inputs: Opposite = 3, Adjacent = 4.

  1. 01Start by noting the input — Opposite: 3.
  2. 02Start by noting the input — Adjacent: 4.
  3. 03Substitute these values into the formula: Hypotenuse = √(t²+a²) | sin = t / √(t²+a²) | cos = a / √(t²+a²) | tan = e.o / e.a
  4. 04Compute Hypotenuse: the calculator returns 5.
  5. 05Compute sin: the calculator returns 0.6.
  6. 06Compute cos: the calculator returns 0.8.
  7. 07Compute tan: the calculator returns 0.75.
  8. 08Cross-check the answer by opening the Derivation panel above — every line of math is shown so you can follow the computation end-to-end.
§04Variants

Common Trigonometric Ratios Problems

The formula gets rearranged depending on which variable you need. Here are the patterns you’ll run into in the real world — find the one that matches your problem and follow the worked steps.

01 · PATTERN

Opposite halved

o = 1.5 (from 3)

Keep every other input at its default and halve the opposite. See how hypotenuse responds.

  1. 01New Opposite: 1.5
  2. 02Baseline Hypotenuse: 5
  3. 03New Hypotenuse: 4.272
  4. 04Hypotenuse decreases by 14.6% → use this sensitivity to plan for real-world variation.
02 · PATTERN

Opposite doubled

o = 6 (from 3)

Keep every other input at its default and double the opposite. See how hypotenuse responds.

  1. 01New Opposite: 6
  2. 02Baseline Hypotenuse: 5
  3. 03New Hypotenuse: 7.2111
  4. 04Hypotenuse increases by 44.2% → use this sensitivity to plan for real-world variation.
03 · PATTERN

Adjacent halved

a = 2 (from 4)

Keep every other input at its default and halve the adjacent. See how hypotenuse responds.

  1. 01New Adjacent: 2
  2. 02Baseline Hypotenuse: 5
  3. 03New Hypotenuse: 3.60555
  4. 04Hypotenuse decreases by 27.9% → use this sensitivity to plan for real-world variation.
04 · PATTERN

Adjacent doubled

a = 8 (from 4)

Keep every other input at its default and double the adjacent. See how hypotenuse responds.

  1. 01New Adjacent: 8
  2. 02Baseline Hypotenuse: 5
  3. 03New Hypotenuse: 8.544
  4. 04Hypotenuse increases by 70.9% → use this sensitivity to plan for real-world variation.
§05FAQ

Frequently asked questions

Yes. The calculator implements the standard formula as documented and returns exact floating-point results. No approximations are used unless noted in the formula.
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