Back again. To begin this exercise I am going to take the diagram we ended with and turn it 90 degrees and we will have this:
And then next I am going to duplicate on the left what we have drawn on the right and we will get this configuration.
Okay at this point a very quick lesson on trigonometry and incorporating what you learned about before. Right angle triangles have a second guiding principle and that is that if the sides are in ratio to each other the angles will always be the same and the tan or the ratio between side A and side B will be the same as well. For example we are going to use side 19 and side 12. The ratio between 12 and 19 is 12/19 or 0.631578947368. the neat thing about right angled triangles is that any ratio that agrees with this for example 24 / 38 or 48 / 76 or even 18 / 28.5 will yield the ratio of 0.631578947368 and this we call "the tan " of the angle. A "tan" of 0.631578947368 can and does yield only 1 possible angle. So even though all these ratio have different numbers the ratio remains the same and then so does the angle. So in the old days we would have to either figure out the angle or consult a book but now most calculators have the function built in so we would simply key in 12 / 19 up would come 0.63157894736842105263157894736842 you would then press inverse tan and we would get an angle of 32.2756443 degrees. It is that simple.
Now another thing we have to know is that all triangles when their interior angles are added together ALWAYS EQUAL 180 DEGREES So in the triangle of 12, 19 and 22.47220505 we have an angle of 90 (by definition of right angled triangle) and 32.2756443 so this would mean the other missing angle HAS TO EQUAL 180 - 90 - 32.2756443 or 57.724355685422368135731924931312. Again forgive the long number but it is imperative to prove a point. Since 90 degrees is always a given it is quicker and the norm to simply take the one angle from 90 degrees for example 90 - 32.2756443 = 57.7243556... So to sum up a tan (or cotangent) is simply the ratio of the small side into the medium side (or reverse) Every angle has a tan and an inverse or cotangent and they are always different except for the only exception and that is 45 degrees where they are the same.
So now let's solve for our solar system diagram and calculate the angles involved in the diagram. I am first going to label the angles as so:
Angle A would simply be tan of 19 /12 or 1.5833333 or angle of 57.724355685422368135731924931312
Angle C is 90 - 57.724355685422368135731924931312 or 32.275644314577631864268075068688
Okay now let's solve for B. Angle B would be tan of 19 / 34.472205054244231864598140445491 or 0.55116868706548588761042844449951 and we find that this relates to angle 28.862177842711184067865962465656
Angle D is 90 - 28.862177842711184067865962465656 or 61.137822157288815932134037534344
On the surface this doesn't seem overly important but it is showing us an amazing thing. The two angles off of the main line of 34.7422 are EXACTLY in the ratio of 1 to 2. That is that angle 57.724355685422368135731924931312 is EXACTLY double the angle of 28.862177842711184067865962465656 2 x 28.862177842711184067865962465656 = 57.724355685422368135731924931312
THEY ARE EXACTLY DOUBLE OR 1/2 OF EACH OTHER. SO IN OUR LITTLE DIAGRAM OF OUR SOLAR SYSTEM WE HAVE MANAGED TO FIND A WAY OF SHOWING HOW TO DRAW ONE HALF OF ANOTHER ANGLE.
But I then found this ratio and diagram in the most unusual of places and that will be for our next posting.