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Chapter 8: Transformation Formulae – Exercise 8.2

Transformation Formulae – Exercise 8.2 – Q.1

Transformation Formulae – Exercise 8.2 – Q.1

= 2 sin 8θ cos 4θ

= 2 sin 2θ cos 3θ

= 2cos 10θ cos 2θ

(iv) cos 12θ - cos 4θ

= -2 sin 8θ sin 4θ

(v) sin 2θ + cos 4θ

= sin 2θ + sin (90 - 4θ)

 

Transformation Formulae – Exercise 8.2 – Q.2

sin 38° + sin 22° = sin 82°

LHS = sin 38°+ sin 22°

Transformation Formulae – Exercise 8.2 – Q.2

= 2 sin 30°cos 8°

= cos(90 - 82)°

= sin 82° = RHS            [∵ cos θ = sin (90 - θ)]

 

Transformation Formulae – Exercise 8.2 – Q.2(i)

cos 100° + cos 20° = cos 40°

Transformation Formulae – Exercise 8.2 – Q.2(i)

= 2cos 60°cos 40°

 

Transformation Formulae – Exercise 8.2 – Q.2(ii)

sin 50°+ sin 10° = cos 20°

Transformation Formulae – Exercise 8.2 – Q.2(ii)

= 2 sin 30°cos 20°

 

Transformation Formulae – Exercise 8.2 – Q.2(iii)

Sin 30° + sin 37° = cos 7°

LHS = sin 23° + sin 37°

Transformation Formulae – Exercise 8.2 – Q.2(iii)

= 2 sin (30°) cos (-7°)

= cos 7° = RHS

 

Transformation Formulae – Exercise 8.2 – Q.2(iv)

LHS = sin 105° + cos 105°

= sin 105° + cos (90° + 15°)

= sin 105° - sin 15°

Transformation Formulae – Exercise 8.2 – Q.2(iv)

= 2 sin 45°cos 60°

= cos 45°

 

Transformation Formulae – Exercise 8.2 – Q.3(iii)

cos 80° + cos 40° - cos 20° = 0

(cos 80° + cos 40°) - cos 20°

= 2cos 60°cos 20°- cos 20°

= cos 20° - cos 20°

= 0

= RHS

 

Transformation Formulae – Exercise 8.2 – Q.3(iv)

cos 20° + cos100° + cos140° = 0

⟹ (cos 20° + cos100°) + cos140°

= 2cos 60°cos (- 40°) + cos140°

= cos 40° + cos (180° - 40°)

= cos 40° - cos 40°

= 0

= RHS

 

Transformation Formulae – Exercise 8.2 – Q.3(v)

Transformation Formulae – Exercise 8.2 – Q.3(v)

= sin⁡ 50°- cos⁡ 80°

= sin⁡ 50°- sin⁡ 10°

= 2 sin 20° cos 30°

 

Transformation Formulae – Exercise 8.2 – Q.3(vi)

Transformation Formulae – Exercise 8.2 – Q.3(vi)

Multiplying and dividing by √2 on LHS

= RHS

 

Transformation Formulae – Exercise 8.2 – Q.3(vii)

sin 80° - cos 70° = cos 50°

LHS = sin 80° = cos 50° + cos 70°

Now,

RHS = cos 50° + cos 70°

= 2 cos 60°cos (-10°)

= cos 10°

= sin 80°

= LHS       [∵ cos θ = sin (90 - θ)]  

 

Transformation Formulae – Exercise 8.2 – Q.3(viii)

sin 51° + cos 81° = cos 21°

sin 51° = cos 21° - cos 81°

RHS = cos 21° – cos 81°

= sin 51°

LHS

 

Transformation Formulae – Exercise 8.2 – Q.4

We have,

Transformation Formulae – Exercise 8.2 – Q.4

= RHS


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