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Sine Rule

In ABC, sides...

Question

In $△ A B C,$ sides opposite to angles $A, B, C$ are denoted by $a, b, c$ respectively. Then the value of $a cos A + b cos B + c cos C =$

2 a sin B \cdot sin C

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2 b sin C \cdot sin A

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2 c sin A \cdot sin B

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2 a b c sin A \cdot sin B \cdot sin C

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Solution

The correct option is C $2 c sin A \cdot sin B$
Using sine rule, we have : $\frac{a}{sin A} = \frac{b}{sin B} = \frac{c}{sin C} = 2 R$
We have,
$a cos A + b cos B + c cos C = 2 R sin A \cdot cos A + 2 R sin B cos B + 2 R sin C cos C = R (sin 2 A + sin 2 B + sin 2 C) = R \times (4 sin A sin B sin C) (∵ sin 2 A + sin 2 B + sin 2 C = 4 sin A sin B sin C) = 2 \cdot (2 R sin A) sin B sin C$
$= 2 a sin B sin C$
Similarly
$a cos A + b cos B + c cos C = 2 b sin C sin A$
Similarly
$a cos A + b cos B + c cos C = 2 c sin A sin B$

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