Transcendental Algebraic Equations
A long-time obsession of mine has been transcendental algebraic equations, which despite sounding a long way away from the real world, threw up a real gem of a function the other night. I have a function plotter that will map out phenomenally complex equations of the single variant variety. It looks like this (click on image to download 384K zip file) :-
Trying out a few randomisations, I began to recognise patterns and groups of curves and asymptotes. Then a level 9 function produced this :-
The amazing thing about this curve is that it seems to go along smoothly, then flips out completely, but finitely, and then continues along smoothly in the same direction and from the same area as where the flip-out started. I call this the first portrayal of a mathematically rigorous representation of a quantum jump. For example, consider an electron orbiting a nucleus. Prior to the point of flip-out (or supposed asymptote), the electron is in a steadily-changing energy field which is drawing the electron out of its current orbit, then a sudden plummeting followed by an even more sudden rise, as the electron joins its new orbit, which continues on smoothly. Many other representations may be inferred from this graph, but it's just one of many that can be generated. This is my "pet" application and it was ported from a programmable Casio calculator. Here is another one portraying Electron Shell Mappings :-
Trigonometrical Addition Theorems and Repercussions
Once you have accepted that a2 + b2 = c2 in the case of a right-angled triangle where c is the hypotenuse, it becomes quite interesting how things turn out for the relationships between the angles and sides of a triangle. The following diagram is representative of a couple of simple, but sweet theorems, which lead nicely on to the next section on sine and cosine laws.
Using Pythagoras we see that
c2 + d2 = 1
Using definitions of sine and cosine we get
1) c = cos B
2) d = sin B
Hence
sin2 B + cos2 B = 1
Continuing, we get
3) a + b = c cos A = cos B cos A from (1) and
4) b = d sin A = sin B sin A from (2).
We also know
cos (A + B) = a = (a + b) - b
Hence from (3) and (4), we get
cos (A + B) = cos A cos B - sin A sin B
Similarly,
sin (A + B) = e + f = d cos A + c sin A = sin A cos B + sin B cos A
Sine and Cosine Laws for any Triangle
The next diagram shows how to derive the sine and cosine laws for triangles.
1) A + B = 180 - C
2) cos (180 - C) = - cos C
3) f = b sin A
4) f = a sin B
5) d = b cos A
6) e = a cos B
(3) and (4) yield the sine law in
7) a / sin A = b / sin B
Again, using Pythagoras, we can see from the diagram that
8) d2 + f2 = b2
9) e2 + f2 = a2
Adding (8) and (9), we get
a2 + b2 = d2 + e2 + 2 f2
a2 + b2 = (d + e)2 - 2 d e + 2 f2
Using (4), (5), and (6), we get
a2 + b2 = c2 - 2 a b cos A cos B + 2 a2 sin2 B
Using a rearranged (7), so that a sin B = b sin A, we get
a2 + b2 = c2 - 2 a b (cos A cos B - sin A sin B)
Now, we use the Addition Theorem above for cosine to simply the bracket contents and rearrange to get
c2 = a2 + b2 + 2 a b cos (A + B)
Putting (1) and (2) together, we get cos (A + B) = - cos C, giving the final equation :-
c2 = a2 + b2 - 2 a b cos C
This looks like a crooked variety of Pythagoras, bent by the fact that there may not be a right angle. When C is 90 degrees, cos 90 is zero, so it is Pythagoras.
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