Pythagoras and Trigonometry in Right-Angled Triangles
The most useful triangle
The right-angled triangle is one of the most useful shapes in all of mathematics, and this unit gathers several powerful tools for working with it: the angle facts that constrain it, the ideas of scale and similarity that relate triangles of the same shape, Pythagoras' theorem for its sides, and trigonometry for linking its sides to its angles. Together these let you find an unknown length or angle in a huge range of practical problems, from measuring a building's height to checking that a corner is truly square.
Angle facts and complementary angles
Start with the angle facts, because they are the simplest constraint. The three angles of any triangle add to one hundred and eighty degrees. In a right-angled triangle one angle is already ninety degrees, so the remaining two must add to ninety; they are said to be complementary. This means that knowing one of the acute angles immediately gives the other, a small fact that saves a great deal of work later.
Scale and similar triangles
Scale and similarity come next. Two triangles are similar when they have the same shape but possibly different sizes, which happens exactly when their angles match. Corresponding sides of similar triangles are then in a fixed ratio called the scale factor. A three, four, five triangle enlarged by a scale factor of two becomes a six, eight, ten triangle, and every pair of corresponding sides shares the ratio two, while the angles stay exactly the same. This is the engine behind indirect measurement: if a two metre stick casts a three metre shadow while a tree casts an eighteen metre shadow, the triangles are similar, the scale factor is six, and the tree must be twelve metres tall.
Pythagoras' theorem for the sides
Pythagoras' theorem connects the three sides of a right-angled triangle through their squares. It states that the square of the hypotenuse, the longest side opposite the right angle, equals the sum of the squares of the other two sides. So if the two shorter sides are three and four, the hypotenuse squared is nine plus sixteen, which is twenty-five, making the hypotenuse five. Certain whole-number combinations like three, four, five and five, twelve, thirteen recur so often they are worth memorising. The theorem also runs backwards to find a shorter side: if the hypotenuse is thirteen and one side is five, the missing side squared is one hundred and sixty-nine minus twenty-five, which is one hundred and forty-four, so that side is twelve.
When the hypotenuse is irrational
Most triangles do not have whole-number sides, and here the earlier work on real numbers matters. A right-angled triangle with both short sides equal to one has a hypotenuse whose square is two, so the hypotenuse is the square root of two, an irrational number approximately one point four one but never exactly that decimal. Leaving such an answer as the square root of two, or simplifying a surd like the square root of eight to two root two, keeps it exact; the decimal is only an approximation and should not be written with an equals sign.
Trigonometry: SOH CAH TOA
Trigonometry is the final and most powerful tool, linking the angles of a right-angled triangle to the ratios of its sides. For a chosen acute angle, the side touching it is the adjacent, the side facing it is the opposite, and the longest is the hypotenuse. The three ratios are sine, equal to opposite over hypotenuse, cosine, equal to adjacent over hypotenuse, and tangent, equal to opposite over adjacent, captured by the memory aid SOH CAH TOA. A few special values are exact: the sine of thirty degrees is exactly one half, and the tangent of forty-five degrees is exactly one. Others, like the sine of forty-five degrees at about nought point seven zero seven, are irrational.
Finding sides and angles
These ratios solve two kinds of problem. To find a side, multiply by the ratio: in a triangle with a thirty degree angle and a hypotenuse of ten, the opposite side is ten times the sine of thirty, which is five. To find an angle, use the inverse functions: if the opposite is three and the adjacent is four, the tangent of the angle is nought point seven five, so the angle is the inverse tangent of nought point seven five, about thirty-seven degrees. Choosing the right ratio depends only on which two sides you know and which one you want.