The Right Triangle

A right angle An angle of exactly 90 degrees, or π/2 radians, which is a quarter of a full turn. Two lines meeting at a right angle are perpendicular. It is conventionally drawn as a small square rather than an arc, and a triangle containing one is a right triangle, the shape all six trigonometric functions are defined from. is a quarter turn. If you face north and turn to face east, you have turned through one right angle. Measured in degrees One 360th of a full turn, written with the symbol °. The number 360 is inherited from Babylonian base-60 arithmetic and has no geometric justification, unlike the radian. Degrees remain the practical unit for navigation, surveying and everyday description. that is 90 degrees, because a full turn was divided into 360 parts by Babylonian astronomers and we kept their number.

A right triangle A triangle containing one right angle. Because a triangle's angles sum to 180 degrees, its other two angles are acute and sum to 90 degrees, so they are complementary. The side opposite the right angle, the hypotenuse, is always the longest. is a triangle with one right angle in it. That single constraint is doing a surprising amount of work. The three angles of any triangle add up to 180 degrees, so if one of them is 90 degrees, the other two must add up to 90 degrees between them. Both of the others are therefore smaller than a right angle, which makes them acute An angle smaller than 90 degrees. Both non-right angles of a right triangle are acute, which is why the triangle definitions of sine, cosine and tangent only cover angles from 0 to 90 degrees and why the unit circle is needed beyond that..

Two consequences follow, and both matter later. The two acute angles are locked together: fix one and the other is decided. And the side opposite the right angle is always the longest side, because the largest angle in any triangle faces the longest side.

Everything in trigonometry is built on this shape. The functions are defined by picking one of the two acute angles and comparing the lengths of the sides. That is the whole idea, and the next five steps are the consequences of it.

Pick one of the two acute angles. It is traditional to call it theta, written as the Greek letter θ. Once you have picked it, the three sides get names, and the names are relative to that choice.

The hypotenuse The side of a right triangle opposite the right angle, and therefore its longest side. It does not depend on which acute angle you are working from, unlike the opposite and adjacent sides. On the unit circle it is the rotating radius and has length 1, which is what reduces sine and cosine to plain coordinates. is the side opposite the right angle, which is the longest side. It never depends on which acute angle you picked, because there is only one right angle to be opposite.

The opposite The side of a right triangle across from the chosen angle theta, touching neither end of it. It is the numerator of sine (opposite over hypotenuse) and of tangent (opposite over adjacent). Choosing the other acute angle makes this same edge the adjacent side. side is the side across the triangle from theta, the side that does not touch theta at all. The adjacent The side of a right triangle running from the chosen angle theta to the right angle, the non-hypotenuse side that touches theta. It is the numerator of cosine and the denominator of tangent. Choosing the other acute angle makes this same edge the opposite side. side is the remaining one, the side that runs from theta to the right angle. Adjacent means next to, and this side is next to theta.

Now the point people miss. Choose the other acute angle instead, and opposite and adjacent trade places. The same physical edge is the opposite side for one angle and the adjacent side for the other. The hypotenuse does not move.

This is not a technicality. It is the reason there are pairs of functions rather than single ones. Sine and cosine, tangent and cotangent, secant and cosecant are each the same construction seen from the other acute angle. Page 3 makes that precise under the name cofunction The function of the complementary angle. cos θ = sin(90° − θ), cot θ = tan(90° − θ) and csc θ = sec(90° − θ), which is what the "co" prefix in cosine, cotangent and cosecant records. Geometrically it is the same construction measured from the other acute angle of the same right triangle..

The Pythagorean theorem For any right triangle, a² + b² = c², where c is the hypotenuse. Read as areas: the squares built on the two shorter sides have the same total area as the square built on the hypotenuse. It appears as Proposition 47 of Book I of Euclid's Elements and was known to Babylonian scribes far earlier. Applied to the unit circle triangle it becomes x² + y² = 1, and hence sin²θ + cos²θ = 1, which is where all three Pythagorean identities come from. says that in a right triangle, a² + b² = c², where c is the hypotenuse and a and b are the other two sides.

Written like that it looks like a formula to plug numbers into. It is a statement about area. Draw a square on each side of the triangle, using that side as the square’s edge. The square on side a has area a². The square on side b has area b². The square on the hypotenuse has area c². The theorem says the first two squares, added together, have exactly the same area as the third. Not approximately, exactly, for every right triangle that exists.

The classic demonstration is a rearrangement. Take four copies of the right triangle and place them inside a large square whose side is a + b. Arrange them one way and the uncovered space is two squares, of areas a² and b². Arrange the same four triangles the other way and the uncovered space is a single square of area c². The four triangles cover the same amount both times, so the uncovered areas must be equal. That is the proof, and it needs no algebra at all.

This appears as Proposition 47 of Book I of Euclid’s Elements, around 300 BCE, though the relationship was known to Babylonian scribes over a thousand years earlier. Clay tablet Plimpton 322 lists triples of whole numbers that satisfy it.

Hold on to the area picture rather than the letters. On page 2 this theorem becomes the equation of a circle, and on page 6 it becomes all three of the Pythagorean identities. It is the same fact each time, rescaled.

Here is the fact that makes trigonometry possible, and it is usually skipped.

Two triangles are similar Triangles with the same three angles, and therefore the same shape at different sizes. Euclid, Elements VI.4, proves their corresponding sides are in proportion. This is the fact that makes trigonometry possible: scaling a right triangle multiplies every side by the same factor, so any ratio of two sides is unchanged and depends on the angle alone. if they have the same three angles. Similar triangles are the same shape at different sizes. Euclid proves in Book VI, Proposition 4 of the Elements that equiangular triangles have their corresponding sides in proportion. If one triangle is twice the size of another, then every side is twice as long, without exception.

Now do the division. Take the opposite side divided by the hypotenuse in the small triangle. In the doubled triangle both of those lengths are twice as big, so the division gives 2a divided by 2c, and the 2s cancel. The ratio One quantity divided by another, expressing relative size rather than absolute size. The six trigonometric functions are all ratios of two sides of a right triangle, which is why they are pure numbers with no units attached and why they are unaffected by the size of the triangle. is identical. This holds for any scale factor and for any pair of sides you choose.

So a ratio of two sides in a right triangle does not depend on how big the triangle is. It depends only on the angle. That is the entire foundation. It is why sin(30 degrees) is a single fixed number, exactly 0.5, rather than something you would have to recompute for every triangle you happen to draw. Everybody who draws a right triangle with a 30 degree angle, at any size, on any paper, gets the same value when they divide the opposite side by the hypotenuse.

This is also why a table of values was worth compiling. If the ratios changed with size, a table would be useless. Because they do not, one table serves every right triangle forever.

Three of the possible ratios get the primary names.

Sine For an acute angle in a right triangle, the opposite side divided by the hypotenuse. On the unit circle it is the y coordinate of the point at angle theta, which extends the definition to every angle. It is odd, periodic with period 2π, and its range is [-1, 1]. The name is a translation accident: Sanskrit jya became Arabic jb, was read as jayb meaning bay or fold, and was rendered in Latin as sinus. of theta is the opposite divided by the hypotenuse. Cosine For an acute angle in a right triangle, the adjacent side divided by the hypotenuse. On the unit circle it is the x coordinate of the point at angle theta. It equals the sine of the complementary angle, cos θ = sin(90° − θ), which is what the "co" prefix means. It is even, periodic with period 2π, and its range is [-1, 1]. of theta is the adjacent divided by the hypotenuse. tangent For an acute angle in a right triangle, the opposite side divided by the adjacent side, equal to sin θ / cos θ and to the gradient of a line at that angle. On the unit circle it is the length cut off on the vertical tangent line at (1, 0) by the extended radius, which is where the name comes from (Latin tangere, to touch). Period π, range all real numbers, undefined wherever cosine is zero. of theta is the opposite divided by the adjacent. Written short: sin θ, cos θ, tan θ.

The standard mnemonic is SOH CAH TOA. Sine equals Opposite over Hypotenuse. Cosine equals Adjacent over Hypotenuse. Tangent equals Opposite over Adjacent. It is worth memorising as a keyboard shortcut, but it is not the reason any of it is true. The reason is the previous step: the ratio depends only on the angle.

Two properties fall out immediately from the hypotenuse being the longest side. Sine and cosine both have the hypotenuse on the bottom of the fraction, and the top is always shorter than the bottom, so both of them are always between 0 and 1 for an acute angle. Tangent has no such limit. Its bottom is the adjacent side, which can be very short, so tangent can be as large as you like. That difference shows up as a very visible difference between the graphs on page 4.

There are six ratios in total, not three, because you can also divide each of these the other way up. Those three get their own page.

These ratios were not invented as algebra. They were compiled as tables by astronomers who needed to predict where things in the sky would be. Hipparchus, in the second century BCE, is credited with the first such table, and Ptolemy’s Almagest, around 150 CE, contains a table of chord lengths in steps of half a degree. A chord A straight line segment joining two points on a circle. It stops at those two points, unlike the whole line through them, which carries on in both directions and is a secant line. Early trigonometric tables, including Ptolemy's in the Almagest around 150 CE, tabulated chord lengths rather than sines. The sine is the half chord, which is the form Indian mathematicians adopted and the one that survived. is the straight line segment joining two points on a circle, and for a long time the chord, rather than the sine, was the tabulated quantity. Indian mathematicians, including Aryabhata around 500 CE, switched to tabulating the half chord, which is what we now call the sine.

The name is a translation accident. The Sanskrit for that half chord was jya, transliterated into Arabic as jiba and written without vowels as jb. Twelfth century Latin translators read those consonants as the ordinary Arabic word jayb, meaning a bay, a bosom or a fold in a garment, and translated that word rather than the technical one, giving Latin sinus, which is where the English "sine" comes from.

"Cosine" is shorter to explain. It is the sine of the complementary angle, from complementi sinus, and page 3 makes the relationship exact.