On the right is a "right" triangle. It is "right" because it has one angle - the one with the small red box - that is exactly 90° (say: ninety degrees)*. You will also notice that H and L are the same lengths.
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The Beginner's Glimpse at "Tangents"
Parents and teachers and younger readers: This page is meant to be a glimpse into trigonometry's fun with numbers and shapes. While this subject is usually taught in high school, there are many aspects that are very simple and even very young children will see amazing things if guided through some neat paper-folding projects. Afterall, remember that very ancient and uncomplicated minds used trigonometry thousands of years ago - before modern mathemeticians added complexity and pushed the subject into high school.
On the right is a "right" triangle. It is "right" because it has one angle - the one with the small red box - that is exactly 90° (say: ninety degrees)*. You will also notice that H and L are the same lengths.
What is a tangent? A tangent is a number that is calculated by dividing the lengths of the opposite side ("H") of the triangle by that of the adjacent side ("L"), or H/L.
In this figure as shown, H/L = 1, because H = L. Thus the tangent of angle "a" = 1
How big is angle "a"? It is 45°, and is exactly half that of the right angle. (You can prove this to yourself by taking a perfectly square piece of paper (all sides are of equal length), and folding it diagonally so that opposite corners exactly touch, and the crease is made. Unfold and you will see that the crease exactly cuts two of the corners' right angles in half.
Okay, so the "tan" of angle "a" equals 1 ("tan a = 1").
What if H and L were not equal?
What would happen if H were longer than L?
What would happen if H were shorter than L?
More than 2,000 years ago the ancient Greeks and Egyptians noticed these relationships in triangles and we now call the study of triangles "trigonometry", which literally means measuring (metry) triangles (trigons).
Because these ancient people learned from even more ancient people in Mesopotamia - present day Iraq, one of the things they learned was that a triangle was defined to have 180° when you added up all the angles. Not just in some triangles, but in ALL triangles. Sort of like a universal law for triangles.
These ancient people busied themselves making charts and tables of what the tangents were to many different angles. The triangle shown on this page is a 45° right triangle in which a right angle was exactly divided in half (remember folding the square piece of paper). It's tangent = 1.0000. But what if the corner of the square piece of paper was divided into three equal sections? Or four? Or five? And so on? For each division of the 90° right angle at the corner, they calculated new tangents and so they made of table of angle versus calculated tangent. This table became very useful for architects as they made the fancy designs for windows, bridges, and roofs - all where diagonals are needed - and diagonals mean triangles!
So you see that the doodling and having fun with numbers and triangles turned out to be very useful in constructing buildings.
* "Degrees?" Very ancient people discovered that there were about 360 days in a year. "360" is an especially nice number because it can be divided evenly by many different numbers: 2, 3, 4, 5, 6, 8, 9, 10, 12, 13, ... , 130, 180. "Ah, ha!" they thought, "360" must be a divine number!" So, rather than using the decimal system based on the numbers of fingers we have, they based their numbering system on "360" and so today we have inherited the "360" degree system for angles. Look at the globe of the world and you see degrees written on it, and look at the magnetic compass and you see degrees. A circle has 360°, and a triangle exactly half of that. "Yup! Must be some divine magic in those relationships," said Imhotep, the ancient Egyptian architect.
So, teachers and parents, you see how a small paper-folding project can turn into multidisciplinary learning - mathematics, history, art, and.....FUN!
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