WildTrig4: Five main laws of rational trigonometry

We derive from first principles the main laws of rational trigonometry, using the concepts of quadrance and spread to replace the usual distance and angle. Most everything works out much simpler.

Category:

Science & Technology

Tags:

• rational
• trigonometry
• Pythagoras'
• theorem
• mathematics
• Triple
• quad
• formula
• spread
• law
• cross
• quadrance

[footstep002]

This may be a silly question, but as I heard from the previous video triple quad formula is (internal large interval + internal small interval - whole interval) squared equal 4 times (internal large interval times internal small interval), and the triple quad formula from this video seems different. Instead, the triple quad formula in this video says (internal large interval + whole interval - internal small interval), which is p_3 - q_2 - p_1. Am I missing somthing here?

[njwildberger]

@footstep002 I think your question is answered amongst the earlier questions.

[horstmueller1000]

What is not "simple" or "natural" to me - at least, it does not come naturally to me - is when the hight is outside the triangle. The same difficulty with me, when I look at a spread and make it open more than the perpendicular...

[njwildberger]

@horstmueller1000 It does take some getting used to after all our experience with angles. Keep in mind that the spread is defined between two lines, not between two rays. It might be instructive to look at the spread protractor posted by M. Ossmann online.

[yuraaa1990]

Q1+Q2-Q3 have to be equal to zero , nope ?!

[njwildberger]

@yuraaa1990 Only if the triangle has perpendicular sides, ie Pythagoras' thm applies.

[horstmueller1000]

the question naturally arises , why the triple quad formular can be used the way as described (+a + b - c)^2 = 4ab. This is a great excercise, given the hints in wildtrig 2 (?) where the simple case is proven - but the same ideas apply, when all other options are written down. It is a lovely exercise.

[varungargg8]

P3<Q2 how come sister formula is valid in cross law

[QuantumMaths]

Holy crap I just realized you posted that comment three years ago...

[QuantumMaths]

@sixbillionmorons

In other words, P3, Q2, and P1 are *symmetric* and those totally interchangeable amongst one another as long as the general form is followed. Although Wildberger gave the negative sign to the largest quadrance in his initial derivation, he didn't have to.