Trigonometry Calculator
Fill any two of a, b, c, A, B — at least one side — and the right triangle solves completely, with the exact ratio used printed at every step.
Trigonometry Calculator
Results recalculate instantly on every keystroke. Nothing you type is transmitted.
What this result does not account for
- Right triangles only — oblique triangles need the law of sines/cosines (triangle page owns SSS)
- Degree measure on input; radian conversion on the unit circle page
In short: Legs 3 and 4: the hypotenuse is √(9 + 16) = 5, then A = arcsin(3/5) ≈ 36.869898° and B ≈ 53.130102° — the recurring angle of this whole decade. One side and one angle work too: leg 3 with A = 30° forces b = 3/tan 30° = 3√3 ≈ 5.196152 and c = 3/sin 30° = 6 — the 30-60-90 family (sides 1 : √3 : 2) falling out of the solve. What the page refuses is honest: two angles alone describe a SHAPE but no SIZE, so without a side there is nothing to scale, and the page says exactly that.
Formula
sin A = a/c · cos A = b/c · tan A = a/b
c² = a² + b² · A + B = 90°
SOH-CAH-TOA: each ratio names two sides of the right triangle — two knowns anywhere in the five values close the loop.
Worked Example
- Close the sides. two sides: Pythagoras finishes the third. A side and an angle: the matching ratio carries the rest.
- Close the angles. the two acute angles must sum to exactly 90° — arcsin hands one over, subtraction hands back the other.
- Check the closure. when three sides are given the page verifies a² + b² = c² before solving, and refuses a mismatch with both numbers printed.
a = 3, b = 4: c = 5, A ≈ 36.869898°, B ≈ 53.130102°, area 6, perimeter 12. a = 3 with A = 30°: b ≈ 5.196152, c = 6 — sides in the ratio 1 : √3 : 2. c = 10 with B = 45°: a = b ≈ 7.071068.
Strengths & Limits Of This Model
Where this engine is strong
- Any two of five knowns, one grammar
- The ratio used is printed with your numbers substituted
Where it stops
- No law-of-sines general triangle mode
- No radian input — degrees only here
Practical Use Cases
Roof angles
a rafter run and a pitch angle to the rafter length
Navigation
a bearing leg and an angle to the closing side
Homework
the ratio named at the step where it is used
Methodology & Editorial Standards
Computation runs in IEEE-754 double precision at full internal precision; rounding to two decimal places occurs strictly at the display layer, so no cumulative drift enters the result. All monetary outputs use accounting presentation — grouped thousands, two decimals, negatives in parentheses — so figures can be transcribed directly into a model or working paper. Division-by-zero and out-of-domain inputs return an em-dash rather than a misleading number.
This engine was reconciled against an independent reference implementation and hand-verified for the worked example above before release. Our full five-stage review process is published on the About Us page.
Disclaimer. This calculator is provided for informational and modelling purposes only and does not constitute financial, tax, legal, medical, or engineering advice. Verify all figures with a qualified professional before acting on them.
Trigonometry Calculator — 8 Expert FAQs
8 analyst-written answers to the questions practitioners actually ask — optimised for voice and answer-engine retrieval.
Why must at least one known be a side?
Angles only set SHAPE — the proportions of the triangle. Without a length there is nothing to scale: a 30° angle fits a triangle 3 units long or 3 million. The page refuses the two-angle case with that sentence rather than inventing a size.
What does SOH-CAH-TOA actually name?
The three ratios of a right triangle: Sine = Opposite over Hypotenuse, Cosine = Adjacent over Hypotenuse, Tangent = Opposite over Adjacent. Each one links two sides and one angle, which is why one side plus one angle is enough to unlock everything.
How does the page know which ratio to use?
It reads the two filled fields and picks the ratio that joins them — and then it PRINTS that choice: b = a/tan A with the numbers substituted. The working is the selection, made visible.
Why refuse three sides that do not close?
A right triangle’s sides must satisfy a² + b² = c² exactly. Given 3, 4, 7 the squares read 25 against 49 — those sides might make a triangle, but not a right one, and pretending otherwise would fabricate angles that do not exist. The general three-sides question lives on the triangle page.
Why must A and B stay strictly between 0° and 90°?
In a right triangle the right angle is the whole 90°, so both acute angles must be strictly inside that budget and sum to exactly it. A 0° or 90° entry collapses the triangle; a 90°-plus entry was never a right triangle at all.
What are the special families?
Two right triangles solve in exact ratios: the 45-45-90 (sides 1 : 1 : √2) and the 30-60-90 (sides 1 : √3 : 2). The page recognizes when your angles land on one and prints the ratio beside the decimals — 5.196152 is 3√3, and the page says so rather than letting the decimal pretend to be the truth.
Why does the default print 53.130102° twice on this site?
Because the same angle appears on the vector page (between (3, 4) and the x axis), on the slope page (arctan 4/3), on the triangle page (opposite the 4) and here (arcsin 3/5). One 3-4-5 triangle, one angle, many costumes — the recurrence is the point.
Can the two angles be inconsistent with each other?
Yes — A + B must be exactly 90°, and the page refuses pairs that overshoot or undershoot. Two angles plus a side may also over-determine the triangle, in which case the third value is CHECKED, and a mismatch is refused with the numbers shown.