x-2a-3-sin-y-2b-3-sin-and-z-2c-3-sin-where-a-b-and-c-are-sides-of-ABC-such-that-pi-3-A-pi-3-B-and-pi-3-C-Find-at-least-one-feasible-solutio

Question Number 14438 by ajfour last updated on 31/May/17

x = \frac{2 a}{\sqrt{3}} \sin θ, y = \frac{2 b}{\sqrt{3}} \sin ϕ, a n d

z = \frac{2 c}{\sqrt{3}} \sin ψ; w h e r e a, b, a n d c

a r e s i d e s o f △ A B C s u c h t h a t

ϕ - ψ + \frac{π}{3} = ∠ A,

ψ - θ + \frac{π}{3} = ∠ B, a n d

θ - ψ + \frac{π}{3} = ∠ C .

F i n d a t l e a s t o n e f e a s i b l e

s o l u t i o n s e t o f θ, ϕ, a n d ψ i n

t e r m s o f ∠ A, ∠ B, a n d ∠ C

s u c h t h a t a l l a n g l e s a n d s i d e s

a r e p o s i t i v e w i t h a \neq b \neq c,

a n d ∠ A \neq ∠ B \neq ∠ C \neq \frac{π}{2}

F i n d x, y, a n d z e v e n i f y o u

y o u p l e a s e . .

Commented by ajfour last updated on 31/May/17

t h i s i s a g a i n r e l a t e d t o y o u r

Q . 14157 t h i s i s h o w w e c a n

f i n d x, y, a n d z, g i v e n a, b, a n d c .

Commented by b.e.h.i.8.3.4.1.7@gmail.com last updated on 31/May/17

m r A j f o u r! i t i s a v e r y d i f f i c u l t Q .

Commented by ajfour last updated on 01/Jun/17

Answered by ajfour last updated on 01/Jun/17

\cos C = \frac{a^{2} + b^{2} - c^{2}}{2 a b},

s = x + y + z, t h e n

s^{2} = a^{2} + b^{2} + 2 a b \cos (C - \frac{π}{3})

x = \frac{2 a}{\sqrt{3}} \sin [\cos^{- 1} (\frac{a^{2} + s^{2} - c^{2}}{2 a s})] = \frac{2 a}{\sqrt{3}} \sin θ

y = \frac{2 b}{\sqrt{3}} \sin [\cos^{- 1} (\frac{b^{2} + s^{2} - a^{2}}{2 b s})] = \frac{2 b}{\sqrt{3}} \sin ϕ

z = \frac{2 c}{\sqrt{3}} \sin [\cos^{- 1} (\frac{c^{2} + s^{2} - b^{2}}{2 c s})] = \frac{2 c}{\sqrt{3}} \sin ψ .

\dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots . .

Commented by b.e.h.i.8.3.4.1.7@gmail.com last updated on 01/Jun/17

g r e a t j o b m r A j f o u r .

Commented by ajfour last updated on 01/Jun/17

t h a n k s s i r, b u t i w i s h t o s o l v e

i t b y p y t h a g o r a s t h e o r e m c o z

t h e r e i s a v e r t i c a l a n d h o r i z o n t a l

s h i f t f r o m c o m p l e t e s y m m e t r y

a s a, b, a n d c b e c o m e u n e q u a l . .

Commented by ajfour last updated on 01/Jun/17

s e e Q . 14502 m y s o l u t i o n i a m

d e l a y i n g t i l l^{'} n i g h t t i m e s a r e

a l l m y o w n^{'} . .

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