How do I find the sine, cosine and tangent of an angle?

In a right triangle, sine is the opposite side over the hypotenuse, cosine is the adjacent over the hypotenuse, and tangent is the opposite over the adjacent — the mnemonic SOH-CAH-TOA. For angles beyond a right triangle the unit circle takes over: the point at angle θ on a circle of radius 1 has coordinates (cos θ, sin θ), and tangent is the ratio of the two. That is why sin 30° is exactly 1/2, or 0.5, and why cos 150° is −√3/2, or about −0.866025.

Why does this say tan 90° is undefined when my calculator gives a huge number?

Because undefined is the right answer and the huge number is an artefact. Tangent is sine divided by cosine, and cos 90° is exactly zero, so tan 90° is a division by zero — it does not exist. A calculator working in radians has to represent 90° as π/2, which is irrational and cannot be stored exactly, so its cosine comes out as roughly 6.1×10⁻¹⁷ rather than 0, and dividing by that gives 16331239353195370. This page looks up every multiple of 15° instead of computing it, so cosine is exactly zero and the answer is simply undefined.

What is the ambiguous case, and how do I know when I have one?

It arises with SSA — two sides and an angle that is not between them. Knowing a, b and A does not pin the triangle down, because side a can often swing to meet the base at two different points. Take a = 5, b = 8, A = 30°: angle B comes out as either 53.1301° or 126.8699°, and both give a valid triangle, with side c at 9.9282 or 3.9282 respectively. The tell is that the two values for B are supplementary — they add to 180°. Every other combination, SSS, SAS, ASA and AAS, gives exactly one triangle or none.

How can I tell whether an SSA problem has two triangles, one, or none?

Compare side a against the height h = b·sin A, which is the shortest distance from the far vertex to the base. If a is less than h it cannot reach and there is no triangle. If a equals h it just touches, giving exactly one right triangle. If a lies between h and b, it reaches in two places and you get two triangles. If a is at least b, only one of those two positions is still valid. With b = 8 and A = 30° the height is 4, so a = 3 gives nothing, a = 4 gives one triangle with B = 90°, a = 5 gives two, and a = 10 gives one.

When do I use the law of sines and when the law of cosines?

Use the law of cosines, a² = b² + c² − 2bc·cos A, whenever you know three sides or two sides with the angle between them, since those are the cases where the sine rule cannot get started. Use the law of sines, a/sin A = b/sin B = c/sin C, when you have an angle paired with its opposite side. One caution worth knowing: the sine rule cannot tell an angle from its supplement, because sin 53.13° and sin 126.87° are identical, which is precisely what creates the ambiguous case. This page finds the remaining angles with the cosine rule wherever it can, for that reason.

What is a reference angle and why does it matter?

The reference angle is the acute angle between your angle and the horizontal axis, always between 0° and 90°. It matters because every trigonometric value of any angle is the value of its reference angle, with a sign decided by the quadrant. The reference angle of 150° is 30°, and so is that of 210° and 330° — which is why sin 30°, sin 150°, sin 210° and sin 330° are all 0.5 in magnitude, differing only in sign. Learn the first quadrant and the rest is bookkeeping.

How do degrees, radians and gradians relate?

A full turn is 360 degrees, 2π radians or 400 gradians. Degrees are the everyday unit, radians are what mathematics and every programming language actually use because they make calculus behave, and gradians are a metric attempt that survives mostly in surveying. So 30° is π/6, which is 0.52359878 radians, and 33.3333 gradians; 180° is exactly π, or 3.14159265 radians. This calculator accepts any of the three and shows all three at once, and if you enter a radian value that lands on a familiar angle it recognises it — π/6 is treated as exactly 30°, not as 29.999999999999996.

Which exact values are worth memorising?

The first quadrant at multiples of 30° and 45°, because everything else follows from them by reference angle and sign. Sine runs 0, 1/2, √2/2, √3/2, 1 at 0°, 30°, 45°, 60° and 90°; cosine is the same list backwards. Tangent gives 0, √3/3, 1, √3, undefined. The pattern is easier to hold if you write the sines as √0/2, √1/2, √2/2, √3/2, √4/2 — the numerators simply count up. This page shows the surd alongside the decimal at every multiple of 15°, including the less familiar ones like sin 15° = (√6 − √2)/4.

Can any three lengths make a triangle, and what are csc, sec and cot?

No — the two shorter lengths must add to more than the longest, which is the triangle inequality. Lengths of 1, 2 and 3 fail, because 1 + 2 exactly equals 3 and the shape collapses into a straight line; 1, 2 and 10 fail more obviously. As for the other three functions, they are simply reciprocals: cosecant is 1/sin, secant is 1/cos, and cotangent is 1/tan. They exist because certain formulas are far tidier written with them, and each is undefined wherever the function it inverts is zero — so csc is undefined at 0° and 180°, and sec at 90° and 270°.

Trigonometry Calculator

One engine for the whole of school and undergraduate trigonometry. Give it an angle and it returns all six functions with the exact surd where one exists, the quadrant, the reference angle and the same angle in degrees, radians and gradians, while a unit circle sweeps round to show you where the val...

ANGLE

UNIT

COMMON ANGLES

SIN 30°

0.500000

EXACT

1/2

30.0000° = 0.523599 rad = 33.3333 grad · quadrant 1 · reference angle 30.00°

Created with❤️byeaglecalculator.com

HOW TO USE

1
Pick a mode. FUNCTIONS evaluates all six trigonometric functions at an angle, RIGHT TRIANGLE solves a triangle with a 90° corner from any two measurements, and ANY TRIANGLE handles the general case from whatever combination you happen to know.
2
In FUNCTIONS, type an angle and choose degrees, radians or gradians. The tiles below give all six values at once — tap any of them to make it the headline, and where an exact surd exists it is shown beside the decimal. The unit circle sweeps to your angle as you type.
3
In ANY TRIANGLE, first say what you know: SSS, SAS, ASA, AAS or SSA. The three input boxes relabel themselves to match, and the note underneath tells you whether that combination can be ambiguous before you enter anything.
4
Read the diagram as well as the numbers. It is drawn to scale, the travelling dot runs A → B → C so you can see which side faces which angle, and when two triangles fit your measurements both are drawn together with the second one dashed.

THE FORMULAS

Sinesin A = opposite ÷ hypotenuse

Cosinecos A = adjacent ÷ hypotenuse

Tangenttan A = opposite ÷ adjacent

Reciprocalscsc = 1/sin · sec = 1/cos · cot = 1/tan

Pythagorean identitysin²θ + cos²θ = 1

Law of sinesa/sin A = b/sin B = c/sin C

Law of cosinesa² = b² + c² − 2bc·cos A

Area from two sidesarea = ½·a·b·sin C

Heron's formulaarea = √(s(s−a)(s−b)(s−c))

CircumradiusR = a ÷ (2·sin A)

Inradiusr = area ÷ s

Angle sumA + B + C = 180°

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Last updated: July 28, 2026 · Formula verified · Every multiple of 15° is returned exactly, so tan 90° reads undefined rather than 1.6×10¹⁶ · Eagle-eyed accuracy for every calculation.