## Chapter 10 – Sine and Cosine Formulae and Their Applications Exercise Ex. 10.1

In any triangle ABC, prove the following:

b sinB – c sinC = a sin (B – C)

In any triangle ABC, prove the following:

a^{2}sin(B – C)= (b^{2} –c^{2})sinA

In any triangle ABC, prove the following:

a(sinB – sinC) + b (sinC – sinA) + c (sinA – sinB) = 0

In any triangle ABC, prove the following:

a^{2}(cos^{2}B – cos^{2}C) + b^{2}(cos^{2}C – cos^{2}A) + c^{2}(cos^{2}A –cos^{2}B) = 0

In any triangle ABC, prove the following:

b cosB + c cosC = a cos(B – C)

In any triangle ABC, prove the following:

a cosA + b cosB + c cosC= 2b sinA sinC= 2c sinA sinB

a(cos B cosC + cosA)= b(cos C cosA + cosB)= c(cos A cosB + cosC)

In ΔABC prove that, if Ө be any angle, then b cosӨ = c cos(A – Ө) + a cos(C + Ө)

In a ΔABC, if sin^{2}A + sin^{2}B = sin^{2}C, show that the triangle is right angled.

In any ΔABC, if a^{2}, b^{2}, c^{2} are in A.P., prove that cot A, cot B and cot C are also in A.P.

The upper part of a broken over by the wind makes an angle of 30^{0} with the ground and the distance from the root to the point where the top of the tree touches the ground is 15m. Using sine rule, find the height of the tree.

At the foot of a mountain the elevation of its summit is 45^{0}; after ascending 1000m towards the mountain up a slope of 30^{0} inclination, the elevation is found to be 60^{0}. Find the height of the mountain.

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## Chapter 10 – Sine and Cosine Formulae and Their Applications Exercise Ex. 10.2

b(c cos A – a cos C) = c^{2} –a^{2}

C (a cos B – b cos A) = a^{2} – b^{2}

2(bc cos A + ca cos B +ab cosC)= a^{2} + b^{2} + c^{2}

In any DABC, prove the following:

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

In 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

## Chapter 10 – Sine and Cosine Formulae and Their Applications Exercise Ex. 10VSAQ

In any triangle ABC, find the value of a sin (B -C) + b sin (C – A) + c sin (A – B)