a/sin A = b/sin B = c/sin C is usually taught as three equal ratios and left there, as if it were a coincidence worth memorising. It is not a coincidence. All three equal 2R — the diameter of the one circle that passes through all three corners. Move a corner: the triangle changes completely, the three ratios stay locked together, and the number they equal is the width of the circle you can see.
Give a triangle two sides and the angle between them and there is exactly one answer. Give it two sides and an angle not between them — the SSA case — and there can be two, one, or none. This is the single most common way a sine-rule answer comes out wrong: the calculator returns one angle, and its supplement fits just as well.
A degree is a convention: somebody divided a circle into 360 parts, probably because 360 has a great many factors and a year has roughly that many days. Nothing in mathematics prefers it. A radian is not a convention at all — it is the arc length divided by the radius, so it is a ratio of two lengths and therefore a pure number with no units attached. Sweep out an arc as long as the radius and you have turned through one radian, which is 57.2958°, and a full circle is 2π of them because the circumference is 2πr.
That distinction is the whole reason the rest of mathematics insists on them. sin x ≈ x is only true in radians — in degrees sin 1° is 0.01745, not 1. The derivative of sin x is cos x only in radians; in degrees it picks up a stray factor of π/180 and every equation downstream inherits it. Series expansions, differential equations, Fourier analysis, the complex exponential — all of them treat the angle as a number, and only radians are one.
Which leaves one practical rule. Angles you measure are in degrees; angles you calculate with are in radians. A surveyor, a machinist and a navigator all work in degrees because a degree is a readable size. The moment that angle enters a formula it wants converting, and the moment it comes back out for a human it wants converting again. Nearly every wrong trig answer in the world is that conversion missed once — which is why the mode selector on a calculator is the first thing to check when a result is off by a factor nobody can account for.
180° = π is the only one you need; everything else is a fraction of it. 90° = π/2, 60° = π/3, 45° = π/4, 30° = π/6. To go from degrees to radians multiply by π/180; the other way, by 180/π. And one radian is 57.3°, near enough, which is a useful sanity check: if an angle in radians looks like a plausible number of degrees, it is almost certainly in the wrong one.
Pendulums, lenses, small oscillations, the paraxial approximation, half of structural analysis: all of them replace sin x with x and tan x with x, and none of them is exactly true. The useful question is not whether it holds but up to what angle, for an error you can live with.
The thresholds worth carrying. sin x is within 0.1% of x up to 4.4°, within 1% up to 14.1°, and within 5% up to 31.6°. tan x drifts the other way and twice as fast — 1% at 9.9°. A pendulum swung to 20° runs about 0.76% slow against the small-angle formula, which is eleven minutes a day: ignorable in a physics question and hopeless in a clock.
The two legs and the hypotenuse of any right triangle: a² + b² = c². The fastest way to find a missing side.
sin = opposite/hypotenuse, cos = adjacent/hypotenuse, tan = opposite/adjacent: one triangle, three ratios.
A point at angle θ on a radius-1 circle has coordinates (cos θ, sin θ); unwrapped, that is the sine wave.
These ratios need one angle of exactly 90°. Any other triangle needs the Law of Sines or Cosines, and the angle unit is the commonest wrong answer.
sin = Opposite/Hypotenuse, cos = Adjacent/Hypotenuse, tan = Opposite/Adjacent: the three ratios that connect an angle to a right triangle’s side lengths.
a² + b² = c² for the two legs and hypotenuse of any right triangle: one of the oldest results in mathematics, and still the fastest way to find a missing side.
A radian is the angle where the arc length equals the radius, so a full circle is 2π radians (≈ 6.283) or 360°. Most programming languages’ trig functions expect radians, not degrees.
tan θ = sin θ / cos θ, and cos(90°) = 0. Dividing by zero means the tangent shoots off to infinity exactly where the unit-circle point sits directly above the centre.
Results carry ordinary floating-point rounding, so an exact value like sin 60° = √3/2 displays as 0.8660254. That is not an error in the maths.
Two sides and a non-included angle can describe two different triangles, one, or none: the ambiguous case, which no calculator can resolve without more information.
A point moving around a circle of radius 1 has coordinates (cos θ, sin θ) by definition — the triangle formed by that point, the centre, and the x-axis is a right triangle with hypotenuse 1, so "opposite over hypotenuse" is just the y-coordinate, and "adjacent over hypotenuse" is just the x-coordinate.
It's the same rotation, just unwrapped — plot the circle's y-coordinate against the angle travelled instead of against the x-coordinate, and the smooth up-down oscillation of a wave is exactly what falls out. This is literally how the wave gets traced in the animation above.
A set of three whole numbers that satisfy a² + b² = c² exactly — 3-4-5 is the smallest and most famous, but there are infinitely many (5-12-13, 8-15-17, 7-24-25...), and any multiple of a triple (6-8-10, 9-12-15) is also a valid triple.
Not directly — a general triangle needs the Law of Sines or Law of Cosines instead, since SOH-CAH-TOA and a²+b²=c² both specifically rely on one angle being exactly 90°.
A right triangle in a flat plane, with one angle exactly 90 degrees — the side ratios here do not hold otherwise. Results carry ordinary floating-point rounding, so an exact value like sin 60° = √3⁄2 displays as 0.8660254; that is a display artefact, not an error in the maths. Angles are read in whichever unit you have selected, which is the single most common source of a wrong answer here.
Use the Law of Sines, a / sin A = b / sin B = c / sin C, or the Law of Cosines, c² = a² + b² − 2ab cos C. Two sides and a non-included angle can describe two different triangles, one or none (the ambiguous case), which the general triangle solver on this page flags.
Releases in which this page changed, newest last. Derived from the archived copy of every release, not from notes written afterwards — so it reflects what actually shipped. Site-wide passes are left out; they are in the full changelog.