Trigonometry Calculator
Enter any 2 known values of a right triangle — 2 sides, or 1 side plus 1 acute angle — to solve every side, every angle, the area and the perimeter, with the SOH-CAH-TOA working shown step by step in both degrees and radians. Also includes a sin/cos/tan evaluator with exact values at the 30°, 45° and 60° reference angles.
Solve a right triangle
c = 10
Hypotenuse (one leg + angle A (a, A))
| Sides | Angles (degrees) | Angles (radians) |
|---|---|---|
| a = 5 | A = 30° | A = 0.5236 rad |
| b = 8.6603 | B = 60° | B = 1.0472 rad |
| c = 10 | C = 90° | C = 1.5708 rad |
Area: 21.6506 sq units · Perimeter: 23.6603 units
Show the working (SOH-CAH-TOA, step by step)
Evaluate sin, cos, tan (and inverses)
0.5
sin(30°) — normalised to 30° on the unit circle · exact value: 1/2
Right-triangle solving assumes the right angle is at C (90°). How we calculate →
SOH-CAH-TOA: the rule behind every right-triangle calculation
SOH-CAH-TOA is a memory aid for the three basic trigonometric ratios in a right-angled triangle: Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, Tangent = Opposite/Adjacent. "Opposite" and "adjacent" are always relative to whichever angle you're working with — the same side can be "opposite" for one angle and "adjacent" for the other.
Worked example: a right triangle with legs a = 3 and b = 4 has hypotenuse c = √(a² + b²) = 5 (the Pythagorean theorem), and angle A = atan(a/b) = 36.8699° using the TOA ratio.
Solving a right triangle from just 2 known values
You only need 2 pieces of information to solve an entire right triangle — either 2 sides, or 1 side plus 1 acute angle — because the right angle (90°) is already fixed and the two acute angles always add up to 90°. This calculator accepts all 6 valid combinations: (a,b), (a,c), (b,c), (a,A), (b,A) and (c,A).
Worked example: given a = 5 (opposite angle A) and A = 30°, SOH gives the hypotenuse c = a / sin(A) = 5 / sin(30°) = 10, and TOA gives the other leg b = a / tan(A) = 8.6603. Angle B is simply 90° − 30° = 60°.
Exact values at 30°, 45° and 60° — and why they matter
A handful of angles have exact trigonometric values expressed as fractions and square roots, rather than endless decimals — these come directly from two special triangles: the 45-45-90 (an isosceles right triangle) and the 30-60-90 (half of an equilateral triangle).
sin(30°) = 1/2 (= 0.5000), cos(45°) = √2/2 (= 0.7071), tan(60°) = √3 (= 1.7321). These exact forms are useful in exams and proofs where a decimal approximation isn't precise enough or isn't allowed.
Degrees vs. radians: converting between the two
Degrees split a full turn into 360 equal parts; radians measure the angle by the arc length it sweeps out on a unit circle, so a full turn is 2π radians. The conversion is straightforward: radians = degrees × (π/180), and degrees = radians × (180/π). A right angle is 90° or π/2 radians; a full turn is 360° or 2π radians.
Most calculators and this tool default to degrees for everyday geometry problems, but radians are the standard unit in calculus and physics — always check which mode your calculator (or this tool) is set to before reading off a result.
Frequently asked questions
What is SOH-CAH-TOA?
It's a mnemonic for the three basic trig ratios in a right triangle: Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, Tangent = Opposite/Adjacent. "Opposite" and "adjacent" are defined relative to the angle you're solving for.
How many values do I need to solve a right triangle?
Just 2 — either 2 sides, or 1 side plus 1 acute angle. The right angle (90°) is already known, and the two acute angles always sum to 90°, so 2 additional pieces of information fully determine every side and angle.
What is the exact value of sin(30°), cos(45°) and tan(60°)?
sin(30°) = 1/2, cos(45°) = √2/2 (also written 1/√2), and tan(60°) = √3. These come from the 30-60-90 and 45-45-90 special right triangles and are exact fractions/roots rather than rounded decimals.
How do I convert between degrees and radians?
Multiply degrees by π/180 to get radians, or multiply radians by 180/π to get degrees. For example 90° × (π/180) = π/2 radians, and π radians × (180/π) = 180°.
What's the difference between sine, cosine and tangent?
All three compare two sides of a right triangle relative to a chosen angle: sine compares the opposite side to the hypotenuse, cosine compares the adjacent side to the hypotenuse, and tangent compares the opposite side to the adjacent side (equivalently, tangent = sine ÷ cosine).
Why is tan(90°) undefined?
Because tan(θ) = sin(θ)/cos(θ), and cos(90°) = 0 — dividing by zero is undefined. Geometrically, at 90° the "adjacent" side of the reference triangle shrinks to zero, so the opposite/adjacent ratio has no finite value.
Can this calculator work in radians instead of degrees?
Yes — the sin/cos/tan evaluator accepts either degrees or radians as the input unit; switch the unit selector and the calculation updates immediately, still checking for exact values at the standard reference angles.
Researched & verified by the Calcuris Data & Research Team. How we build and check our tools →