WildTrig5: Applications of rational trigonometry

[footstep002]

I was confused about the cross law a little bit. So After reviewing this and previous videos several times, and I realized that due to the cyclic property of the cross law and the cross law being quadratic equation gives the formula able to manipulate its internal signs, practically there exists 6 different forms of cross laws.

Am I right?

[njwildberger]

@footstep002 There are three separate Cross laws if we consider the three fold symmetry in a triangle: ie there are three different spreads that can be involved. As for rewriting one of these laws, that can be done in any number of ways. Remember that (a+b-c)^2 is the same as (c-a-b)^2, and then it is clearer.

[horstmueller1000]

Why should I allow the maximum to be not a triangle - too much for me, right now. So, you want to make me rethink my view? Okay, I still have to work through only the first ten of your lectures. (Which I do slowly.... but I still do.... ).

[horstmueller1000]

The solution of the quadratic equation in that kite calculation has a "mistake" . The Root of 2 become root of 20 and the numbers in front of the root change. Only, that one may have the impression, they are 1*8 and 3*6 - which they are not. There is no "mistake", it is just 1,8 and 3,6 or as anlgosachsen world writes it 1.8 and 3.6

I achieved to get 18 *( 1+/- (1/5)^(1/2)) . The comment about the solution "cut" and giving 18 *(1 - Root (1/5)) as the unique solution was helpful.

[horstmueller1000]

Which is, by the way, exactly the wording that follows, after the "zero" solution is suggested. (Just that I (me , myself and I ) consider it as "not existing" so to say ... in this particular case.).

[horstmueller1000]

That mini gap continues at 5:05 where it is said that 1 solution of the spread equation would be zero.

Of course, I disagree and reply in the sense, that this is the collinear case (spread zero) and the triple quad formular still holds.

[horstmueller1000]

There is a mini gap.

At 4:45 the idea is to use the spread formular /law in the simpler case of two equal spreads.

The same idea applies at min 3:10 or so, where the quadrance is searched for three collinear points and two of them have the same distance to the third, which is then the mid point.

You can use the tripple quad formular in the same manner as in that spread case and get the factor 1/4.

[dreznik]

Interesting that, as you show here, square roots are unavoidable in calculating quadrances... similarly, on th e previous video I was interested in finding a simple formula for the spread of a bisected spread. from the half-angle identity, Sin[x/2]^2 = (1 - Cos[x])/2, so this seems it will require a square root as well (Cos[x] is not squared).

[noobenstein]

I like it.

I kinda found it harder to understand once you divorced it from the real world. The flag example was good.

[Cieros]

this teory is so good, too bad the world will never accept it.

[isilder]

Have you considered calling this "right angle analysis", because it sure isnt trigonometry.... :)

[kojelja0]

Well, we dont need to use spreads at all, because spread(alfa)=sin^2(alfa). For example: if we have sides of a triangle with anlge 90 degrees. a=5 and c=20 we could simply say sin(alfa)=5/20, which is a rational trigonometricy also. We could use spread(alfa)= (5/20)^2 instead sin(alfa) as notation, but it doesnt make calculation easier. We still prefer to speak in degrees cuz its more natural to people I guess. Ty, good video anyway, gave me few ideas, and helped, keep good job ;)

[porschev]

I watched the first 5 videos. My impression is that this method is no easier. It is more accurate. I would put the effort into learning your method, but it still leaves me with more work to do since all my modules deal with Cos Sin and Tan and converting the angles.

[iambowman]

At time 6.25, how do you determine which solution is correct, besides identifying the figure as "close" to an equilateral triangle?

[njwildberger]

It is a reasonable question. The other solution would have Q roughly 26, but this is too much, since then the top of the kite would form a triangle with quadrances 4,4,26. If two quadrances are 4, then the maximum possible quadrance for the third side is 4x4=16.

[horstmueller1000]

@njwildberger 16 is not the maximum. The maximum is less than but not equal to 16, i.e. the 16 cannot be reached in a triangle. For 16 there is no third and no second side. ( Let me see if I am still so precise at WTxx, with xx > 16 .... :-)) )

[njwildberger]

@horstmueller1000 So what is the maximum according to your thinking?

[sixbillionmorons]

I think there's sign error at 7:15 in this movie. The first 4 in the line below the "In ABC" should be positive, not negative. If I'm right, you could note it in the video description text, or else if you still have the original for editing, you could superimpose text to alert watchers to the typo. (I realize you probably have other stuff to do.)

[njwildberger]

I don't see that there is an error. The signs are correct. Remember that (A+B-C)^2 is the same as (C-A-B)^2, so the Cross law is written in two slightly different ways sometimes, depending on the situation.

[sixbillionmorons]

You're right. I should have been watching closer before I posted this as an "error." Sorry.

[Scientisticsoviet]

This great stuff, a nice break from some of youtube's irrational garbage. I really enjoyed this review of some of these euclidean notions. It just wonderful that someone put some pure geometry on youtube rather than one of their "opinions" (I mean to say their ideological positions that have no evidence to back them up).

Keep it coming!

[GAZA118118]

whats this about? we always use SOH CAH TOA