Rewrite 4sin(x)-5cos(x) as Asin(x+phi)

Question

 Rewrite 4sin(x)-5cos(x) as Asin(x+phi)

Answer 1

The trick is to multiply and divide by the square root of the sum of the squares of the coefficients, distribute, and then use a trigonometric identity.

 

Divide and multiply by √(4² + 5²) = √41

 

4sin(x) - 5cos(x) = √41 [(4/√41)sin(x) - (5/√41)cos(x)]

 

Let 4/√41 = cosθ

 

Then sinθ = √(1 - cos²θ) = √(1 - 16/41) = √[(41 - 16)/41] = 5/√41

 

Then

 

4sin(x) - 5cos(x) = √41 [cosθ sin(x) - sinθ cos(x)]

 

Use the identity 

 

cosθ sin(x) - sinθ cos(x) = sin(x - θ)

 

Then

 

4sin(x) - 5cos(x) = √41 sin(x - θ) = Asin(x + φ)

 

where

 

A = √41 and φ = -θ = -cos^-1(4/√41) ≅ - 0.8961 radians (be sure to work with radians here)

 

4sin(x) - 5cos(x) = √41 sin(x - 0.8961)