in-AB-C-a-3-b-6-c-7-find-the-value-of-E-a-b-cos-C-b-c-cos-A-a-c-cos-B-

Question Number 188515 by mnjuly1970 last updated on 02/Mar/23

i n A \overset{Δ}{B C} : a = 3, b = 6, c = 7

f i n d t h e v a l u e o f :

E = (a + b) c o s (C) + (b + c) c o s (A) + (a + c) c o s (B) = ?

Answered by mr W last updated on 02/Mar/23

E = (a + b + c) (\cos A + \cos B + \cos C) - (a \cos A + b \cos B + c \cos C)

= (a + b + c) (1 + 4 \sin \frac{A}{2} \sin \frac{B}{2} \sin \frac{C}{2}) - R (\sin 2 A + \sin 2 B + \sin 2 C)

= (a + b + c) + 4 (a + b + c) \sin \frac{A}{2} \sin \frac{B}{2} \sin \frac{C}{2} - 4 R \sin A \sin B \sin C

= (a + b + c) + 8 R (\sin A + \sin B + \sin C) \sin \frac{A}{2} \sin \frac{B}{2} \sin \frac{C}{2} - 4 R \sin A \sin B \sin C

= (a + b + c) + 8 R \times 4 \cos \frac{A}{2} \cos \frac{B}{2} \cos \frac{C}{2} \sin \frac{A}{2} \sin \frac{B}{2} \sin \frac{C}{2} - 4 R \sin A \sin B \sin C

= (a + b + c) + 4 R \sin A \sin B \sin C - 4 R \sin A \sin B \sin C

= a + b + c

= 3 + 6 + 7 = 16 ✓

Commented by manxsol last updated on 03/Mar/23

p e r p e n d i c u l a r p r o j e c t i o n

Commented by manxsol last updated on 03/Mar/23

Commented by mnjuly1970 last updated on 03/Mar/23

t h a n k s a l o t s i r W