Maths-
General
Easy

Question

In straight triangle A B C sum a cubed Sin space left parenthesis B minus C right parenthesis =

  1. 0
  2. 1
  3. 2
  4. 3

Hint:

In this question, we have to find the value of sum a cubed Sin space left parenthesis B minus C right parenthesis. For this we will use sine rule, fraction numerator sin space A over denominator a end fraction equals fraction numerator sin space B over denominator b end fraction equals fraction numerator sin space C over denominator c end fraction equals k and cosine rule, cos space A equals fraction numerator b squared plus c squared minus a squared over denominator 2 b c end fraction comma space cos space B equals fraction numerator c squared plus a squared minus b squared over denominator 2 a c end fraction comma space cos space C equals fraction numerator a squared plus b squared minus c squared over denominator 2 a b end fraction.

The correct answer is: 0


    B y space sin space r u l e comma space
fraction numerator sin space A over denominator a end fraction equals fraction numerator sin space B over denominator b end fraction equals fraction numerator sin space C over denominator c end fraction equals k
B y space cos i n e space r u l e comma
cos space A equals fraction numerator b squared plus c squared minus a squared over denominator 2 b c end fraction comma space cos space B equals fraction numerator c squared plus a squared minus b squared over denominator 2 a c end fraction comma space cos space C equals fraction numerator a squared plus b squared minus c squared over denominator 2 a b end fraction
sum a cubed space sin space left parenthesis B minus C right parenthesis
equals a cubed left parenthesis sin space B. cos space C minus cos space B. sin space C right parenthesis plus b cubed left parenthesis sin space C. cos space A minus cos space C. sin space A right parenthesis plus c cubed left parenthesis sin space A. cos space B minus cos space A. sin space B right parenthesis
equals a cubed open parentheses k b cross times fraction numerator a squared plus b squared minus c squared over denominator 2 a b end fraction minus k c cross times fraction numerator c squared plus a squared minus b squared over denominator 2 a c end fraction close parentheses plus b cubed open parentheses k c cross times fraction numerator b squared plus c squared minus a squared over denominator 2 b c end fraction minus k a cross times fraction numerator a squared plus b squared minus c squared over denominator 2 a b end fraction close parentheses plus c cubed open parentheses k a cross times fraction numerator c squared plus a squared minus b squared over denominator 2 a c end fraction minus k b cross times fraction numerator b squared plus c squared minus a squared over denominator 2 b c end fraction close parentheses
equals a cubed cross times fraction numerator 1 over denominator 2 a end fraction open parentheses a squared plus b squared minus c squared minus c squared minus a squared plus b squared close parentheses plus b cubed cross times fraction numerator 1 over denominator 2 b end fraction open parentheses b squared plus c squared minus a squared minus a squared minus b squared plus c squared close parentheses plus c cubed cross times fraction numerator 1 over denominator 2 c end fraction open parentheses c squared plus a squared minus b squared minus b squared minus c squared plus a squared close parentheses
equals a squared cross times 1 half cross times 2 open parentheses b squared minus c squared close parentheses plus b squared cross times 1 half cross times 2 open parentheses c squared minus a squared close parentheses plus c squared cross times 1 half cross times open parentheses a squared minus b squared close parentheses
equals a squared b squared minus a squared c squared plus b squared c squared minus a squared b squared plus a squared c squared minus b squared c squared
equals 0

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