if-sin-A-sin-B-sin-D-sin-C-and-A-B-C-D-then-prove-that-A-B-180-C-D-180-

Question Number 78877 by M±th+et£s last updated on 21/Jan/20

i f \frac{s i n (A)}{s i n (B)} = \frac{s i n (D)}{s i n (C)}

a n d A + B = C + D

t h e n p r o v e t h a t

A + B = 180

C + D = 180

Commented by mr W last updated on 21/Jan/20

y o u c a n n o t p r o v e s i r, b e c a u s e t h e

s t a t e m e n t i s n o t c o r r e c t . i n f a c t t h e r e

a r e i n f i n i t e p o s s i b i l i t i e s!

e . g . A + B = C + D = n \times 180 °

e . g . A + B = C + D = a n y v a l u e i f A = D, B = C .

Commented by M±th+et£s last updated on 21/Jan/20

y o u a r e r i g h t s i r

Commented by M±th+et£s last updated on 21/Jan/20

i t s m y f a u l t s o r r y

Commented by M±th+et£s last updated on 21/Jan/20

i f \frac{s i n (A)}{s i n (B)} = \frac{s i n (D)}{s i n (C)}

a n d A + B = D + C A \neq c

p r o v e t h a t

A + B = 180 n n \in Z

Commented by M±th+et£s last updated on 21/Jan/20

i s i t r i g h t n o w s i r

Commented by mr W last updated on 21/Jan/20

n o!

a l l t h e s e f u l f i l l t h e e q n . :

A = 5 °, B = 6 ° \Rightarrow A + B = 11 °

C = 6 °, D = 5 ° \Rightarrow C + D = 11 °

A = 185 °, B = 6 ° \Rightarrow A + B = 191 °

C = 186 °, D = 5 ° \Rightarrow C + D = 191 °

Answered by $@ty@m123 last updated on 21/Jan/20

G i v e n \frac{s i n (A)}{s i n (B)} = \frac{s i n (D)}{s i n (C)}

\Rightarrow \frac{s i n A + \sin B}{s i n A - \sin B} = \frac{s i n (D) + \sin C}{s i n D - \sin C}

\Rightarrow \frac{2 s i n \frac{A + B}{2} \cos \frac{A - B}{2}}{2 \cos \frac{A + B}{2} \sin \frac{A - B}{2}} = \frac{2 s i n \frac{D + C}{2} \cos \frac{D - C}{2}}{2 \cos \frac{D + C}{2} \sin \frac{D - C}{2}}

\Rightarrow \frac{\cos \frac{A - B}{2}}{\sin \frac{A - B}{2}} = \frac{\cos \frac{D - C}{2}}{\sin \frac{D - C}{2}} {∵ A + B = C + D

\Rightarrow \cot (A - B) = \cot (D - C) \dots . (i)

\Rightarrow A - B = D - C \lor A - B = π + (D - C)

\Rightarrow [A = D \land B = C] \lor [A + C - (B + D) = π]

I t h i n k t h e r e i s s o m e t y p o i n q u e s t i o n .

P l . c h e c k .