PROVING QUESTION

Asked by RACHIT_MAHAAN | 19th Sep, 2010, 02:37: PM

Expert Answer:

Dear Student
Following is the solution of your question

Let cos A + cos  B + cos C = a

And sin A + sin B + sin C = b

To prove a =b = 0

We can prov that a2 + b2 = 0.

Now (cos A + cos B + cos C)2

= cos2 A + cos2 B + cos2 C + 2 (cos A cos B + cos B cos C + cos C cos A)      …(i)

and (sin A + sin B + sin C)2 

= sin2 A + sin 2 B  + sin2 C + 2 (sin A sin B + sin B sin C + sin C sin A).

Given cos (B-C) + cos (A-B) + cos (C-A) = -3/2                                         …(ii)

i.e., cos B cos C + sin B sin C + cos A cos B + sin A sin B + cos C cos A

+ sin C sin A = -3/2                                                                               …(iii)

Adding (i) and (ii)

(cos2 A + sin2 A) + (cos2 B + sin2 B) + (sin2 C + cos2 C) +  

2(cos B cos C + sin B sin C + cos A cos B + sin A sin B + cos C cos A + sin C sin A)

3 + 2 x -3/2 = o      (using (iii))

So a2 + b2 = 0

Þ a = 0 and b = 0.

i.e., cos A + cos B + cos C = sin A + sin B + sin C = 0.

Regards

Team Topperlearning

Answered by  | 23rd Sep, 2010, 09:15: AM

Queries asked on Sunday & after 7pm from Monday to Saturday will be answered after 12pm the next working day.