A trigonometric ratio compares two sides of a right triangle. Because similar triangles have equal ratios, each ratio depends only on the angle, which is how sine, cosine and tangent were defined long before calculators existed. This tool works in both directions: enter any two sides of a right triangle to get all six ratios as simplified fractions and decimals along with both acute angles, and use the table of special angles below for the exact values worth memorizing.
How to use the trigonometric ratios calculator
- Choose which two sides you know: opposite and adjacent, opposite and hypotenuse, or adjacent and hypotenuse. “Opposite” and “adjacent” are measured from the angle θ you care about.
- Enter the two lengths in any unit; ratios have no units.
- Read the six ratios on the tape. Each shows the side fraction, its simplest form (radicals are rationalized, so 1/√2 becomes √2/2) and a decimal.
- Check the missing side, θ in degrees and radians, the other acute angle, and the labeled diagram. The work panel shows the Pythagorean theorem and each SOH-CAH-TOA step.
SOH-CAH-TOA and the reciprocal ratios
For an acute angle θ in a right triangle:
Flip each fraction to get the reciprocal ratios:
The missing side comes from the Pythagorean theorem, hyp² = opp² + adj², and the angle from θ = arctan(opp ÷ adj).
Worked example: adjacent 5, hypotenuse 13
1. Find the opposite side. opp = √(13² − 5²) = √144 = 12.
2. Write the ratios. sin θ = 12/13 ≈ 0.9230769231, cos θ = 5/13 ≈ 0.3846153846, tan θ = 12/5 = 2.4.
3. Flip them. csc θ = 13/12, sec θ = 13/5 = 2.6, cot θ = 5/12.
4. Find the angles. θ = arctan(2.4) ≈ 67.38°, and the other acute angle is 90° − 67.38° ≈ 22.62°.
Special angles table
These exact values come from two triangles: half of an equilateral triangle (30-60-90, sides 1, √3, 2) and half of a square (45-45-90, sides 1, 1, √2).
| θ | sin θ | cos θ | tan θ | csc θ | sec θ | cot θ |
|---|---|---|---|---|---|---|
| 0° | 0 | 1 | 0 | undefined | 1 | undefined |
| 30° | 1/2 | √3/2 | √3/3 | 2 | 2√3/3 | √3 |
| 45° | √2/2 | √2/2 | 1 | √2 | √2 | 1 |
| 60° | √3/2 | 1/2 | √3 | 2√3/3 | 2 | √3/3 |
| 90° | 1 | 0 | undefined | 1 | undefined | 0 |
Notice that each row for 60° is the 30° row with sine and cosine swapped. That is the cofunction rule at work.
Core trig identities
Every identity below follows from the ratio definitions and the Pythagorean theorem:
- Reciprocal: csc θ = 1/sin θ, sec θ = 1/cos θ, cot θ = 1/tan θ.
- Quotient: tan θ = sin θ ÷ cos θ and cot θ = cos θ ÷ sin θ.
- Pythagorean: sin²θ + cos²θ = 1, 1 + tan²θ = sec²θ, 1 + cot²θ = csc²θ. Divide the first by cos²θ or sin²θ to get the other two.
- Cofunction: sin(90° − θ) = cos θ, tan(90° − θ) = cot θ, sec(90° − θ) = csc θ. The two acute angles of a right triangle are complements, so one angle’s opposite side is the other’s adjacent side.
Choosing the right ratio
Pick the ratio that uses the two quantities you know and the one you want. Know the hypotenuse and need the side across from θ? That is sine: opp = hyp × sin θ. Standing 40 feet from a tree and measuring a 52° angle of elevation to its top? Tangent links the distance (adjacent) to the height (opposite): 40 × tan 52° ≈ 51.2 feet. For solving a whole right triangle from one side and one angle, use the right triangle calculator; for non-right triangles, the triangle calculator applies the laws of sines and cosines.
Frequently asked questions
What does SOH-CAH-TOA stand for?
Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, Tangent = Opposite/Adjacent. The sides are named relative to the angle θ you are working with, and the hypotenuse is always the side across from the right angle.
Which side is opposite and which is adjacent?
The opposite side is across the triangle from θ and does not touch it. The adjacent side touches θ but is not the hypotenuse. If you switch to the other acute angle, the opposite and adjacent sides swap.
Do the ratios depend on the size of the triangle?
No. A 3-4-5 triangle and a 30-40-50 triangle have the same angles, so every ratio matches: sin θ = 3/5 = 30/50 = 0.6. That is what makes the ratios useful for measuring things you cannot reach.
Why do I get an error when the hypotenuse is shorter than a leg?
In a right triangle the hypotenuse is always the longest side, because hyp² = opp² + adj². If you enter a leg equal to or longer than the hypotenuse, no right triangle has those sides.