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Question

$I n space triangle A B C comma sum a left parenthesis sin invisible function application B minus sin invisible function application C right parenthesis equals$

0
1
1/2
2R

hint Hint:

First we will rewrite the expression as follows $a left parenthesis sin open parentheses B close parentheses minus sin open parentheses C close parentheses plus b left parenthesis sin open parentheses C close parentheses minus sin open parentheses A close parentheses right parenthesis plus c left parenthesis sin open parentheses A close parentheses minus sin open parentheses B close parentheses right parenthesis$ and then we will use the Sine Rule to find the value of the expression.

The correct answer is: 0

In this question we have to find the sum of $sum a left parenthesis sin B minus sin C right parenthesis$
Step1: Rewriting the expression
The given expression can be rewritten as follows
$a left parenthesis sin open parentheses B close parentheses minus sin open parentheses C close parentheses plus b left parenthesis sin open parentheses C close parentheses minus sin open parentheses A close parentheses right parenthesis plus c left parenthesis sin open parentheses A close parentheses minus sin open parentheses B close parentheses right parenthesis$
Step2: Using Sine Rule
$fraction numerator sin open parentheses A close parentheses over denominator a end fraction equals fraction numerator sin open parentheses B close parentheses over denominator b end fraction equals fraction numerator sin open parentheses C close parentheses over denominator c end fraction equals lambda$
$sin open parentheses A close parentheses equals a lambda comma space sin open parentheses B close parentheses equals b lambda comma sin open parentheses C close parentheses equals c lambda$
By putting these values in the above expression we get,
$a left parenthesis b lambda minus c lambda right parenthesis plus b left parenthesis c lambda minus a lambda right parenthesis plus c left parenthesis a lambda minus b lambda right parenthesis$
=> $lambda left parenthesis a b minus a c plus b c minus a b plus a c minus b c right parenthesis$
=> $0$
So, the value of the expression is $0$

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