ASKSAGE: Sage Q&A Forum - Latest question feedhttps://ask.sagemath.org/questions/Q&A Forum for SageenCopyright Sage, 2010. Some rights reserved under creative commons license.Wed, 07 Feb 2018 08:45:18 -0600How do you find all three angles ?Are there any proof of how to claim all three?https://ask.sagemath.org/question/41011/how-do-you-find-all-three-angles-are-there-any-proof-of-how-to-claim-all-three/
For a triangle of $\angle A B C$ the sides of $ a,b,c$ are written in a way of $a=\frac{\sin A}{\sin C}$, $b=\frac{\sin B}{\sin C}$, $c=\frac{\sin C}{\sin C}$ and the heights $h_a,h_b,h_c$ are written in a form $\frac {h_c}{h_a}=$,$\frac{h_c}{h_b}=$,$\frac{h_c}{h_c}$ to give us $ a$ and $b$. and the base $c=1$.
If I have a triangle with sides $5,5,4 $ and their altitudes are $\sqrt {21},\sqrt {13.44},\sqrt {13.44}$ , why do they simplify to give us a special kind of triangle where $a=1.25,b=1.25,c=1$
>$\sqrt\frac {21}{13.44}=1.25$
Angles
$3=0.16+0.16+0.68^2+0.84+0.84+(1-0.68^2)$
Laws of Cosine ,when we have all $3$ lengths are:
>$a^2=b^2+c^2-2bc\cos(A)$
>$b^2=a^2+c^2-2ac\cos(B)$
>$c^2=a^2+b^2-2ab\cos(C)$
Here we have sides $1.25,1.25 and 1$,a simplest version of the triangle measuring $5,5,4$.
And for three sides of a triangle $a,b,c$,and $\angle ABC$. The legs of the heights $h_a,h_b,h_c$ are situated on three sides of the triangle.
For one side $a$ I have $\frac{b-\cos(A)}{\cos(C)}=a$ and for the second side $b$ I have $\frac{a-\cos(B)}{\cos(C)}=b$ and the third side which is $c$ as the base of the triangle equals to $1$.
>$\sqrt{\sin^2(B)+(a-\cos(B))^2}=b$
>$\sqrt{\sin^2(A)+(b-\cos(A))^2}=a$
>$\sqrt{\sin^2(A)+(\cos(A))^2}=c$
>by using consecutive or non consecutive numbers:
>$a<b<c $ we're able to define $\theta$ without conversion
>$\frac{a}{c}=\cos(C)$ $(1-\frac{a}{c})\times\sqrt\frac{(a+c)}{(c-a)}=\sin(C)$
>$\sqrt\frac{(c-b)}{c}=\cos(B)$ $\sqrt\frac{b}{c}=\sin(B)$
Find
>$\cos(A)$ & $\sin(A)$ which we already know.
> Question is: why $\frac{h_c}{h_b}$; $\frac{h_c}{h_a}$;$\frac{h_c}{h_c}$ and. $\frac{\sin A}{\sin C}$; $\frac{\sin B}{\sin C}$ $\frac{\sin C}{\sin C}$equal to the sides $a,b,c$?LarrousseWed, 07 Feb 2018 08:45:18 -0600https://ask.sagemath.org/question/41011/