Q. 114.7( 3 Votes )

# For any Δ ABC show that -

a cos A + b cos B + c cos C = 2b sin A sin C

Answer :

**Note:** In any ΔABC we define ‘a’ as the length of the side opposite to ∠A, ‘b’ as the length of the side opposite to ∠B and ‘c’ as the length of the side opposite to ∠C.

**The key** **point to solve the problem:**

The idea of sine Formula:

•

As we have to prove:

a cos A + b cos B + c cos C = 2b sin A sin C

We can observe that we all the terms present in the equation to be proved are not showing any resemblance with known formula but the term is RHS side has sine terms, so there is a possibility that sine formula can solve our problem

∴ from sine formula we have-

∴ a = 2k sin A , b = 2k sin B , c = 2k sin C

As,

LHS = a cos A + b cos B + c cos C

= 2k sin A cos A + 2k sin B cos B + 2k sin C cos C

= k(2sin A cos A + 2sin B cos B + 2sin C cos C

LHS = k( sin 2A + sin 2B + sin 2C) {using 2 sin X cos X = sin 2X }

Using transformation formula – sin X + sin Y =

LHS = k ( 2sin(A + B) cos (A – B) + sin 2C)

∵ ∠ A + ∠ B + ∠ C = π

∴ A + B = π – C

∴ LHS = k { 2sin (π – C) cos (A – B) + 2 sin C cos C }

[as sin (π – θ) = sin θ]

LHS = k{ 2 sin C cos (A – B) + 2 sin C cos C }

LHS = 2k sin C { cos (A – B) + cos C }

Using transformation formula – cos X + cos Y =

LHS = 2k sin C { }

LHS = 4k sin C {∵∠ A + ∠ B + ∠ C = π }

LHS = 4k sin C sin B sin A

∵ 2k sin B = b

We have,

LHS = 2b sin A sin C = RHS **…..Hence proved.**

Rate this question :

In a ΔABC, if prove that the triangle is isosceles.

RD Sharma - MathematicsIn a Δ ABC, if ∠B = 60^{o}, prove that (a + b + c) (a – b + c) = 3ca

For any Δ ABC show that –

= (a+b+c)^{2}

Two ships leave a port at the same time. One goes 24 km/hr in the direction N 38^{o} E and other travels 32 km/hr in the direction S 52^{o} E. Find the distance between the ships at the end of 3 hrs.

In a Δ ABC cos^{2}A + cos^{2} B + cos^{2} C = 1, prove that the triangle is right angled.

In any Δ ABC, then prove that

RD Sharma - MathematicsIn a Δ ABC prove that

sin^{3} A cos (B – C) + sin^{3} B cos (C – A) + sin^{3} C cos (A – B) = 3 sin A sin B sin C

In a Δ ABC cos^{2}A + cos^{2} B + cos^{2} C = 1, prove that the triangle is right angled.

Two ships leave a port at the same time. One goes 24 km/hr in the direction N 38^{o} E and other travels 32 km/hr in the direction S 52^{o} E. Find the distance between the ships at the end of 3 hrs.

In any Δ ABC, then prove that

RD Sharma - Mathematics