Reduce each of the following expressions to the Sine And Cosine of A s

1.	Reduce each of the following expressions to the Sine And Cosine of A single expression:\(\sqrt3\)sin x - cos x
Answer» Let f(x) = √3 sin x – cos x Dividing and multiplying by √(3 + 1) = 2, ⇒ f(x) = 2\((\frac{\sqrt3}2sinx - \frac{1}2cosx)\) Sine of expression: ⇒ f(x) = 2\((cos\frac{π }6sinx-sin\frac{π }6cosx)\) We know that sinA cosB – cosA sinB = sin(A –B) ∴ f(x) = 2sin\((x-\frac{π }6)\) Cosine of the expression: ⇒ f(x) = 2\((sin\frac{π }6sinx-cos\frac{π }6cosx)\) We know that cosA cosB – sinA sinB = cos(A +B) ∴ f(x) = -2cos\((\frac{π }3+x)\)

Reduce each of the following expressions to the Sine And Cosine of A single expression:\(\sqrt3\)sin x - cos x

Answer»

Let f(x) = √3 sin x – cos x

Dividing and multiplying by √(3 + 1) = 2,

⇒ f(x) = 2\((\frac{\sqrt3}2sinx - \frac{1}2cosx)\)

Sine of expression:

⇒ f(x) = 2\((cos\frac{π }6sinx-sin\frac{π }6cosx)\)

We know that sinA cosB – cosA sinB = sin(A –B)

∴ f(x) = 2sin\((x-\frac{π }6)\)

Cosine of the expression:

⇒ f(x) = 2\((sin\frac{π }6sinx-cos\frac{π }6cosx)\)

We know that cosA cosB – sinA sinB = cos(A +B)

∴ f(x) = -2cos\((\frac{π }3+x)\)