Sine, cosine, and tangent as geometric ratios

In brief

Two ladders lean against a wall at the same angle from the ground, one short, one long. Despite being different sizes, the ratio between how high each ladder reaches up the wall and how long the ladder itself is turns out to be identical for both, because the angle, not the size, is what fixes that ratio. This single fact, that in similar right triangles the same angle always produces the same ratio of sides, however large or small the triangle, is the entire foundation of sine, cosine, and tangent: they are simply names for the three most useful such ratios, tabulated once and for all, angle by angle, so that a single measured angle can stand in for an entire triangle's proportions.

The full treatment

First look: two ladders, one angle

Lean a two meter ladder against a wall so it makes an angle of thirty degrees with the ground, and separately lean a four meter ladder against a different wall at that same thirty degree angle. The four meter ladder reaches twice as high, and its base sits twice as far from the wall, as the two meter ladder's does, doubled in every length, because doubling every side of a triangle while keeping its angles fixed simply produces a bigger copy of the same shape, a similar triangle. But now divide height reached by ladder length for each: both divisions give the same number, since both the height and the length doubled together. That constant ratio, fixed by the angle alone and indifferent to the ladder's actual size, is what trigonometry names and tabulates.

Building the idea: similarity forces fixed ratios

This rests entirely on the geometry of similar triangles, triangles with the same angles but possibly different sizes. Two triangles are similar exactly when their corresponding angles match; when that happens, every pair of corresponding sides is in the same proportion. A right triangle with one additional angle, call it theta, fixed, has all three angles fixed, since the third angle is whatever remains from a half turn after removing the right angle and theta. So any two right triangles sharing that angle theta are similar, and every ratio of two sides in one such triangle equals the same ratio of corresponding sides in the other, no matter their absolute size. This is the entire justification for defining ratios that depend only on the angle: similarity guarantees the ratio is well defined, the same answer regardless of which particular right triangle you drew to compute it.

The formal definitions: naming the three ratios

Label a right triangle's angle of interest theta, not the right angle itself. Relative to theta, one leg is adjacent, touching theta and the right angle; the other leg is opposite, across from theta; and the hypotenuse is the side opposite the right angle, the longest side. Three ratios are named. Sine of theta, written sin(theta), is the length of the opposite side divided by the length of the hypotenuse. Cosine of theta, written cos(theta), is the length of the adjacent side divided by the length of the hypotenuse. Tangent of theta, written tan(theta), is the length of the opposite side divided by the length of the adjacent side.

Because of the similarity argument above, each of these three numbers depends only on the angle theta, never on the triangle's size. Note also that tangent is not an independent idea: since opposite divided by adjacent equals opposite divided by hypotenuse, all divided by adjacent divided by hypotenuse, tangent of theta always equals sine of theta divided by cosine of theta, a relationship, not a coincidence, following directly from the two definitions.

Extending the ratios with rotation

The right triangle definition above only makes sense for angles between zero and ninety degrees, since a right triangle cannot contain a second angle of ninety degrees or more. To extend sine and cosine to any angle, including negative angles and angles beyond a full turn, place a right triangle inside a circle of radius one centered at the origin, with theta measured, in radians, as the angle of rotation from the positive horizontal direction. The point where the rotated radius meets the circle has a horizontal coordinate and a vertical coordinate; define cosine of theta as that horizontal coordinate and sine of theta as that vertical coordinate. For angles between zero and ninety degrees this matches the right triangle definitions exactly, because the radius, the horizontal coordinate, and the vertical coordinate literally form a right triangle with hypotenuse length one, so opposite over hypotenuse becomes just the vertical coordinate and adjacent over hypotenuse becomes just the horizontal coordinate. But the circle definition keeps making sense past ninety degrees, where the horizontal or vertical coordinate turns negative, giving sine and cosine values for any angle whatsoever, including full and repeated rotations.

Lineage

Systematic tables of chord lengths, close relatives of sine, appear in Hipparchus's astronomical work in the second century BCE and were developed further by Ptolemy in the Almagest. Indian mathematicians and astronomers, notably in the Surya Siddhanta and in Aryabhata's fifth century work, defined something very close to the modern half chord, the direct ancestor of sine, and the word sine itself descends, through a chain of Arabic and Latin translation and a mistranslation of the Sanskrit word for a bowstring, jya, into the Latin sinus. Islamic mathematicians including al-Battani and al-Biruni extended tables to cosine and tangent and applied them extensively to astronomy and geography. European trigonometry, systematized by figures such as Regiomontanus in the fifteenth century, brought the ratios into the algebraic, symbol based form used today, and the extension to the unit circle and all real angles matured alongside the development of calculus in the seventeenth and eighteenth centuries.

The strongest case for it

The ratios are exact consequences of similarity, not empirical approximations, so a value of sine or cosine looked up or computed for a given angle applies to every right triangle containing that angle, of any size whatsoever, which is precisely what makes indirect measurement possible: a single angle, measured with an instrument like a theodolite or even a protractor, combined with one known length, yields every other length in the triangle. This is the working principle behind surveying inaccessible heights and distances, behind navigation by bearing and distance, and behind resolving any force, velocity, or displacement into perpendicular components, since a slanted quantity is itself the hypotenuse of a right triangle whose legs are found using exactly these ratios. The unit circle extension reaches further still, describing any periodic, back and forth phenomenon, the swing of a pendulum, the alternation of an electric current, the vibration of a wave, because all repeat with the same rotational structure a circle has.

The strongest case against it

The ratios are defined for, and make full geometric sense as stated only for, flat, Euclidean triangles; on a curved surface the relationships between angle and side ratio change, and spherical trigonometry, used for instance in long distance navigation and astronomy, requires different formulas. A common misconception is applying the plain right triangle ratios directly to a triangle that is not right angled at all; sine and cosine of an angle still exist for such a triangle, by the circle definition, but they no longer equal opposite over hypotenuse in that triangle, and using them that way silently gives a wrong length. Another common error is confusing degree and radian mode when evaluating these ratios on a calculator, since sin(30) means something entirely different depending on whether 30 is read as degrees or as radians, and the calculator will not warn you. Finally, the ratios alone cannot recover an angle uniquely from a computed sine or cosine value without care, because the same sine value corresponds to more than one angle within a full rotation, an ambiguity that must be resolved using other information about the situation.

Where it stands now

The right triangle ratios and their unit circle extension are complete, settled mathematics, unchanged since their systematization and used without modification across science and engineering. What continues to develop are computational methods for evaluating them efficiently and to high precision, and their extension into complex analysis, where sine and cosine connect to exponential functions, an extension with real depth still taught and used but resting entirely on the same geometric ratios defined here.

Test yourself

You are standing a measured one hundred meters from the base of a tall building, and you measure the angle from the ground up to the building's top as thirty five degrees, using an instrument that reads angle only, not distance. Using the ratios developed above, derive the building's height, showing which of the three ratios you chose and why the other two would not directly give you the answer from the information available, then explain what would go wrong in your calculation if your reported angle were actually measured from the top of your five foot instrument stand rather than from true ground level.