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Law of Sin by HP - acoto - 08-03-2018 05:46 PM
Hi
On this HP application:
https://www.hpcalc.org/details/4167
The diagram is a bit odd. Are not A, B and C supposed to be the angles; X, Y and Z the side lenghts?
RE: Law of Sin by HP - Albert Chan - 08-03-2018 05:59 PM
I just peek at your link, A, B, C are sides, X, Y, Z angles.
C^2 = A^2 + B^2 − 2 A B cos(Z)
So, for right angle Z, A^2 + B^2 = C^2, as expected
Edit: typo fixed
RE: Law of Sin by HP - Dieter - 08-03-2018 06:08 PM
(08-03-2018 05:59 PM)Albert Chan Wrote: I just peek at your link, A, B, C are sides, X, Y, Z angles.
A^2 + B^2 = C^2 - 2 A B cos(Z)
I'd say A² + B² = C² + 2AB cos(Z)
Or C² = A² + B² – 2AB cos(Z)
(08-03-2018 05:59 PM)Albert Chan Wrote: So, for right angle Z, A^2 + B^2 = C^2, as expected
That's also true for the correct formula. ;-)
Regarding the screenshot on the linked site:
The formulas refer both to the law of cosines (the three equations at the top) and the law of sines (smaller, right of the triangle), but here sides and angles have been swapped. It should read sin(X)/A = sin(Y)/B = sin(Z)/C.
So this looks like an error.
Dieter
RE: Law of Sin by HP - Albert Chan - 08-03-2018 06:22 PM
My head get the correct formula, but when I type, somehow I switched the order ...
The screen shot was right in front of me ! Sorry ...
It should be this:
C^2 = A^2 + B^2 - 2 A B cos(Z)
So, if Z is right triangle, we get C^2 = A^2 + B^2
If Z is tiny, cos(Z) ~ 1,
C^2 ~ A^2 + B^2 - 2 A B ~ (A - B)^2
C ~ abs(A - B)
If Z is almost 180 degree, coz(Z) ~ -1:
C^2 ~ A^2 + B^2 + 2 A B ~ (A + B)^2
C ~ A + B
RE: Law of Sin by HP - acoto - 08-03-2018 07:40 PM
Comparing with Wolfram Alpha.
http://www.wolframalpha.com/input/?i=law+of+sines
I think labels on HP diagram do not correspond to formulas, or viceversa.
RE: Law of Sin by HP - Albert Chan - 08-03-2018 08:32 PM
Dieter just answer your question earlier ...
(08-03-2018 06:08 PM)Dieter Wrote: It should read sin(X)/A = sin(Y)/B = sin(Z)/C.
While we are on the subject on triangle, Law of Cosine formula is not suited for needle-like triangle.
Prof. Kahan had the answer (see example 5) : Miscalculating Area and Angles of a Needle-like Triangle
c² = a² + b² - 2ab*cosC = (a-b)² + 4ab*(sin(C/2))² = (a-b)² + 4Δ tan(C/2)
Triangle area, Δ = ½ ab sinC
RE: Law of Sin by HP - acoto - 08-03-2018 08:36 PM
Thank you Albert, Dieter!