In a ABC the expression sin 2 A -sin 2 B -sin 2 C -sin A -sin B -sin C - k sin A -2sin B -2sin C -2- then the value of k is

We know that nbsp; A+B+C= π sin2A+sin2B+sin2C nbsp;= nbsp;4sin Asin Bsin C nbsp; sin A+sin B+sin C =2sinA+B2cosA B2+2sinC2cosC2 (∵sin(A+B2) = sin(π2 ...

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Byju's Answer

Standard XII

Mathematics

Conditional Identities

In a ABC the ...

Question

In a $△ A B C$ the expression $\frac{sin 2 A + sin 2 B + sin 2 C}{sin A + sin B + sin C} = k sin \frac{A}{2} sin \frac{B}{2} sin \frac{C}{2}$ , then the value of $k$ is

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Solution

We know that
$A + B + C = π sin 2 A + sin 2 B + sin 2 C = 4 sin A sin B sin C$

$sin A + sin B + sin C = 2 sin \frac{A + B}{2} cos \frac{A - B}{2} + 2 sin \frac{C}{2} cos \frac{C}{2} (∵ sin (\frac{A + B}{2}) = sin (\frac{π}{2} - \frac{C}{2}) = cos \frac{C}{2}) = 2 cos \frac{C}{2} [cos \frac{A - B}{2} + sin \frac{C}{2}] = 2 cos \frac{C}{2} [cos \frac{A - B}{2} + cos \frac{A + B}{2}] (∵ sin (\frac{C}{2}) = sin (\frac{π}{2} - \frac{A + B}{2}) = cos \frac{A + B}{2}) = 4 cos \frac{A}{2} cos \frac{B}{2} cos \frac{C}{2}$

Now,
$\frac{sin 2 A + sin 2 B + sin 2 C}{sin A + sin B + sin C} = \frac{4 sin A sin B sin C}{4 cos \frac{A}{2} cos \frac{B}{2} cos \frac{C}{2}} = \frac{8 sin \frac{A}{2} cos \frac{A}{2} sin \frac{B}{2} cos \frac{B}{2} sin \frac{C}{2} cos \frac{C}{2}}{cos \frac{A}{2} cos \frac{B}{2} cos \frac{C}{2}} = 8 sin \frac{A}{2} sin \frac{B}{2} sin \frac{C}{2} ∴ k = 8$

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Similar questions

Q. If

A + B + C = 180^{o}, \frac{sin 2 A + sin 2 B + sin 2 C}{sin A + sin B + sin C} = k sin \frac{A}{2} sin \frac{B}{2} sin \frac{C}{2}

, then the value of

k

is equal to

Q. Prove that:

\frac{sin 2 A + sin 2 B + sin 2 C}{sin A + sin B + sin C} = 8 sin (\frac{A}{2}) sin (\frac{B}{2}) sin (\frac{C}{2})

Q. In

△ A B C, \frac{sin 2 A + sin 2 B + sin 2 C}{sin A + sin B + sin C}

is equal to :-

Q.
lf

A + B + C = 180^{\circ}

. Match the following
List -I List -II

1.

cos A + cos B + cos C

2 + 2 cos A cos B cos C

{sin}^{2} A + {sin}^{2} B + {sin}^{2} C

4 cos \frac{A}{2} cos \frac{B}{2} cos \frac{C}{2}

sin 2 A + sin 2 B + sin 2 C

1 + 4 sin \frac{A}{2} sin \frac{B}{2} sin \frac{C}{2}

sin A + sin B + sin C

4 sin A sin B sin C

Q. If the line

x = y = z

intersect line in

sin A + sin B + sin C = {2 d}^{2}

sin 2 A + sin 2 B + sin 2 C = d^{2}

Then

sin \frac{A}{2} sin \frac{B}{2} \frac{C}{2}

is equal to

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In a △ABC the expression sin2A+sin2B+sin2CsinA+sinB+sinC=ksinA2sinB2sinC2 , then the value of k is

In a $△ A B C$ the expression $\frac{sin 2 A + sin 2 B + sin 2 C}{sin A + sin B + sin C} = k sin \frac{A}{2} sin \frac{B}{2} sin \frac{C}{2}$ , then the value of $k$ is