
In \[\Delta ABC\] find \[a\sin \left( {B - C} \right) + b\sin \left( {C - A} \right) + c\sin \left( {A - B} \right)\]
A. 0
B. \[a + b + c\]
C. \[{a^2} + {b^2} + {c^2}\]
D. \[2\left( {{a^2} + {b^2} + {c^2}} \right)\]
Answer
161.4k+ views
Hint: Using sine law, we will find the value \[\sin A\], \[\sin B\], and \[\sin C\]. Then using difference formula of sin we will find the value of of \[\sin \left( {B - C} \right)\], \[\sin \left( {C - A} \right)\], \[\sin \left( {A - B} \right)\] and substitute in the given expression. After simply the expression we will get the required solution.
Formula used:
Sine law:
\[\dfrac{a}{{\sin A}} = \dfrac{b}{{\sin B}} = \dfrac{c}{{\sin C}}\]
Difference of sine function formula:
\[\sin \left( {a - b} \right) = \sin a\cos b - \cos a\sin b\]
Complete step by step solution:
Given expression is
\[a\sin \left( {B - C} \right) + b\sin \left( {C - A} \right) + c\sin \left( {A - B} \right)\]
We know that, \[\dfrac{{\sin A}}{a} = \dfrac{{\sin B}}{b} = \dfrac{{\sin C}}{c} = k(say)\]
Now calculating the value of \[\sin A\], \[\sin B\], and \[\sin C\].
\[\sin A = ak\],\[\sin B = bk\], \[\sin C = ck\]
Then find \[\sin \left( {B - C} \right)\], \[\sin \left( {C - A} \right)\], \[\sin \left( {A - B} \right)\] using the formula \[\sin \left( {a - b} \right) = \sin a\cos b - \cos a\sin b\]
\[\sin \left( {B - C} \right) = \sin B\cos C - \cos B\sin C\]
\[ \Rightarrow \sin \left( {B - C} \right) = bk\cos C - ck\cos B\]
\[\sin \left( {C - A} \right) = \sin C\cos A - \cos C\sin A\]
\[ \Rightarrow \sin \left( {C - A} \right) = ck\cos A - ak\cos C\]
\[\sin \left( {A - B} \right) = \sin A\cos B - \cos A\sin B\]
\[ \Rightarrow \sin \left( {A - B} \right) = ak\cos B - bk\cos A\]
Now putting the value of \[\sin \left( {B - C} \right)\], \[\sin \left( {C - A} \right)\], \[\sin \left( {A - B} \right)\]in the given expression
\[ = a\left( {bk\cos C - ck\cos B} \right) + b\left( {ck\cos A - ak\cos C} \right) + c\left( {ak\cos B - bk\cos A} \right)\]
Simplify the above equation
\[ = abk\cos C - ack\cos B + bck\cos A - abk\cos C + cak\cos B - bck\cos A\]
=0
Hence option A is the correct option
Note: If we calculate the value of a, b, c from the sine law and substitute it in the given expression, then we are unable to reach the correct answer. So from the sine law we will find the \[\sin A\], \[\sin B\], and \[\sin C\].
Formula used:
Sine law:
\[\dfrac{a}{{\sin A}} = \dfrac{b}{{\sin B}} = \dfrac{c}{{\sin C}}\]
Difference of sine function formula:
\[\sin \left( {a - b} \right) = \sin a\cos b - \cos a\sin b\]
Complete step by step solution:
Given expression is
\[a\sin \left( {B - C} \right) + b\sin \left( {C - A} \right) + c\sin \left( {A - B} \right)\]
We know that, \[\dfrac{{\sin A}}{a} = \dfrac{{\sin B}}{b} = \dfrac{{\sin C}}{c} = k(say)\]
Now calculating the value of \[\sin A\], \[\sin B\], and \[\sin C\].
\[\sin A = ak\],\[\sin B = bk\], \[\sin C = ck\]
Then find \[\sin \left( {B - C} \right)\], \[\sin \left( {C - A} \right)\], \[\sin \left( {A - B} \right)\] using the formula \[\sin \left( {a - b} \right) = \sin a\cos b - \cos a\sin b\]
\[\sin \left( {B - C} \right) = \sin B\cos C - \cos B\sin C\]
\[ \Rightarrow \sin \left( {B - C} \right) = bk\cos C - ck\cos B\]
\[\sin \left( {C - A} \right) = \sin C\cos A - \cos C\sin A\]
\[ \Rightarrow \sin \left( {C - A} \right) = ck\cos A - ak\cos C\]
\[\sin \left( {A - B} \right) = \sin A\cos B - \cos A\sin B\]
\[ \Rightarrow \sin \left( {A - B} \right) = ak\cos B - bk\cos A\]
Now putting the value of \[\sin \left( {B - C} \right)\], \[\sin \left( {C - A} \right)\], \[\sin \left( {A - B} \right)\]in the given expression
\[ = a\left( {bk\cos C - ck\cos B} \right) + b\left( {ck\cos A - ak\cos C} \right) + c\left( {ak\cos B - bk\cos A} \right)\]
Simplify the above equation
\[ = abk\cos C - ack\cos B + bck\cos A - abk\cos C + cak\cos B - bck\cos A\]
=0
Hence option A is the correct option
Note: If we calculate the value of a, b, c from the sine law and substitute it in the given expression, then we are unable to reach the correct answer. So from the sine law we will find the \[\sin A\], \[\sin B\], and \[\sin C\].
Recently Updated Pages
If there are 25 railway stations on a railway line class 11 maths JEE_Main

Minimum area of the circle which touches the parabolas class 11 maths JEE_Main

Which of the following is the empty set A x x is a class 11 maths JEE_Main

The number of ways of selecting two squares on chessboard class 11 maths JEE_Main

Find the points common to the hyperbola 25x2 9y2 2-class-11-maths-JEE_Main

A box contains 6 balls which may be all of different class 11 maths JEE_Main

Trending doubts
JEE Main 2025 Session 2: Application Form (Out), Exam Dates (Released), Eligibility, & More

JEE Main 2025: Derivation of Equation of Trajectory in Physics

Displacement-Time Graph and Velocity-Time Graph for JEE

Electric Field Due to Uniformly Charged Ring for JEE Main 2025 - Formula and Derivation

JoSAA JEE Main & Advanced 2025 Counselling: Registration Dates, Documents, Fees, Seat Allotment & Cut‑offs

NIT Cutoff Percentile for 2025

Other Pages
JEE Advanced Marks vs Ranks 2025: Understanding Category-wise Qualifying Marks and Previous Year Cut-offs

JEE Advanced Weightage 2025 Chapter-Wise for Physics, Maths and Chemistry

NCERT Solutions for Class 11 Maths Chapter 4 Complex Numbers and Quadratic Equations

JEE Advanced 2025: Dates, Registration, Syllabus, Eligibility Criteria and More

Degree of Dissociation and Its Formula With Solved Example for JEE

Free Radical Substitution Mechanism of Alkanes for JEE Main 2025
