Question -
Answer -
(i) √3 sin x – cos x
Let f(x) = √3 sin x – cos x
Dividing and multiplying by √((√3)2 +12) i.e. by 2
f(x) = 2(√3/2 sin x – 1/2 cos x)
Sine expression:
f(x) = 2(cos π/6 sin x – sin π/6 cos x) (since, √3/2 =cos π/6 and 1/2 = sin π/6)
We know that, sin A cos B – cos A sin B = sin (A – B)
f(x) = 2 sin (x – π/6)
Again,
f(x) = 2(√3/2 sin x – 1/2 cos x)
Cosine expression:
f(x) = 2(sin π/3 sin x – cos π/3 cos x)
We know that, cos A cos B – sin A sin B = cos (A + B)
f(x) = -2 cos(π/3 + x)
(ii) cos x – sin x
Let f(x) = cos x – sin x
Dividing and multiplying by √(12 + 12)i.e. by √2,
f(x) = √2(1/√2 cos x – 1/√2 sin x)
Sine expression:
f(x) = √2(sin π/4 cos x – cos π/4 sin x) (since, 1/√2= sin π/4 and 1/√2 = cos π/4)
We know that sin A cos B – cos A sin B = sin (A – B)
f(x) = √2 sin (π/4 – x)
Again,
f(x) = √2(1/√2 cos x – 1/√2 sin x)
Cosine expression:
f(x) = 2(cos π/4 cos x – sin π/4 sin x)
We know that cos A cos B – sin A sin B = cos (A + B)
f(x) = √2 cos (π/4 + x)
(iii) 24 cos x + 7 sin x
Let f(x) = 24 cos x + 7 sin x
Dividing and multiplying by √((√24)2 +72) = √625 i.e. by 25,
f(x) = 25(24/25 cos x + 7/25 sin x)
Sine expression:
f(x) = 25(sin α cos x + cos α sin x) where, sin α= 24/25 and cos α = 7/25
We know that sin A cos B + cos A sin B = sin (A + B)
f(x) = 25 sin (α + x)
Cosine expression:
f(x) = 25(cos α cos x + sin α sin x) where, cos α= 24/25 and sin α = 7/25
We know that cos A cos B + sin A sin B = cos (A – B)
f(x) = 25 cos (α – x)