Chapter 10: Sine and Cosine Formulae and Their Applications – Exercise 10.2
Sine and Cosine Formulae and Their Applications – Exercise 10.2 – Q.1
The area of a triangle ABC is given by
Sine and Cosine Formulae and Their Applications – Exercise 10.2 – Q.2
The area of a triangle ABC is given by
Therefore,
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Sine and Cosine Formulae and Their Applications – Exercise 10.2 – Q.3
We have, a = 4, b = 6 and c = 8
Sine and Cosine Formulae and Their Applications – Exercise 10.2 – Q.4
In any ∆ABC, we have
we have,
a = 18, b = 24, c = 30
Therefore,
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Sine and Cosine Formulae and Their Applications – Exercise 10.2 – Q.5
b(c cos A – a cos C) = c2 - a2
RHS
= c2 - a2
= k2 sin2C - k2 sin2 A
= k2(sin2 C - sin2 A)
= k2sin(C + A). sin(C - A)
= k2 sin(π - B).sin(C - A)
= k2 sin B.sin(C - A)
= k sin B.k sin(C - A)
= bk sin (C-A)
= bk(sin C.cos A - sin A.con C)
= b(k sin C.cos A - k sin A. cos C)
= b(b cos A - a cos C) = LHS
Sine and Cosine Formulae and Their Applications – Exercise 10.2 – Q.6
c(a cos B - b cos A)
= ac.cos B - bc cos A
Sine and Cosine Formulae and Their Applications – Exercise 10.2 – Q.7
2(bc cos A + ca cos B + ab cos C) = a2 + b2 + c2
LHS
= 2bc cos A + 2ca cos B + 2ab cos C
= b2 + c2 - a2 + a2 + c2 - b2 + a2 + b2 - c2
= a2 + b2 + c2 = RHS
Sine and Cosine Formulae and Their Applications – Exercise 10.2 – Q.8
For any ∆ABC, We have
therefore,
.png "Sine and Cosine Formulae and Their Applications – Exercise 10.2 – Q.8(i)")
Also,
.png "Sine and Cosine Formulae and Their Applications – Exercise 10.2 – Q.8(ii)")
Now,
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Hence proved.
Sine and Cosine Formulae and Their Applications – Exercise 10.2 – Q.10
In any ∆ABC, we have
a = b cos C +c cos B
b = c cos A + a cos C
c = a cos B + b cos A
Therefore,
L.H.S = a(cos B + cos C - 1) + b(cos C + cos A - 1) + c(cos A + cos B - 1)
= a cos B + a cos C - a + b cos C + b cos A - b + c cos A + c cos B - c
= c - b cos A + a cos C - a + a - c cos B + b cos A - b + b - a cos C + c cos B - c
= 0
= RHS
Hence proved.
Sine and Cosine Formulae and Their Applications – Exercise 10.2 – Q.11
By sine rule we have
K sin A = a, K sin B = b, k sin C = c
a cos A + b cos B + cos C = k sin A cos A + k sin B cos B + k sin C cos C
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=k[sin(A+B) cos (A – B) + sin C cos C]
=k [sin C cos (A – B) – sin C cos (A + B)]
=k [sin C(cos (A – B) – cos(A + B))]
= k sin C [2 sin A sin B]
= 2 sin C (k sin A) sin B
Sine and Cosine Formulae and Their Applications – Exercise 10.2 – Q.12
We know that by cosine rule
⇒2bc cos A = b2 + c2 - a2
⇒ a2 = b2+c2 - 2bc cosA
.png "Sine and Cosine Formulae and Their Applications – Exercise 10.2 – Q.12(i)")
Sine and Cosine Formulae and Their Applications – Exercise 10.2 – Q.13
LHS,
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= 2bc – b2 - c2 + a2 + 2ca – a2 - c2 + b2 + 2ab – b2 - a2 + c2
= a2 + b2 + c2+ 2ab + 2bc + 2ca
= (a + b + c)2 = RHS
Sine and Cosine Formulae and Their Applications – Exercise 10.2 – Q.14
sin3 A cos(B - C) + sin3B.cos(C - A) + sin3C.cos(A - B)
= sin2A sin A cos (B - C) + sin2B.sin B cos(C - A) + sin2 C. sin C. cos (A - B)
= sin2A sin (π -(B + C))cos(B - C) + sin2B. sin(π - (A + C)).cos(C - A)
+ sin2 C. sin (π - (A + B)). cos (A - B)
= sin2 A sin(B + C) cos(B - C) + sin2B. sin(C + A). cos(C - A)
+ sin2 C. sin(A + B).cos(A - B)
= sin2A. (sin2B + sin2C) + sin2B. (sin2C + sin2A) + sin2C.(sin2A + sin2B)
= sin2A.(2sinB.cosB + 2sinCcosC) + 2sin2B.(2sin C cos C + 2 sin A cos A)
+ sin2C. (2 sin A cos A + 2 sin B cos B)
= sin2A. (2sin B cos B + 2 sin C cos C) + sin2B. (2sin C cos C + 2 sin A cos A)
+ sin2C. (2sin A cos A + 2 sin B cos B)
= sin2A. 2 sin B cos B + sin2A. 2 sin C cos C + sin2B. 2 sin C cos S
+ sin2B. 2sin A cos A + sin2C. 2 sin A cos A + sin2 C. 2sin B cos B
= k2a2 2kb cos B + k2a2.2kc cos C + k2b2.2ka cos C
+ k2b2.2 ka cos A + k2c2.2ka cos A + k2c2.2kb cos B
= k3ab(a cos B + b cos A) + k3ac (a cos C + c cos A) + k3bc(c cos B + b cos C)
= k3abc + k3acb + k3bca
= k33abc
= 3(k sin A k sin B k sin C)
= 3abc = RHS
Sine and Cosine Formulae and Their Applications – Exercise 10.2 – Q.15
b + c = 12λ, c + a = 13λ, a + b = 15λ
(b + c + c + a + a + b) = 12λ + 13λ + 15λ
2(a + b + c) = 40λ
a + b + c = 20λ
b + c = 12λ and a + b + c = 20λ ⟹ a = 8λ
c + a = 13λ and a + b + c = 20λ ⟹ b = 7λ
a + b = 15λ and a + b + c = 20λ ⟹ c = 5λ
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