Rewrite −5sin(x)−4cos(x) as Asin(x+ϕ)

Question

Rewrite −5sin(x)−4cos(x) as Asin(x+ϕ). 

A= sqrt41

ϕ=?



 Note: ϕ should be in the interval −π < ϕ < π

Answer 1

To write

-5 sin(Φ) - 4 cos(Φ) in the form of A sin(x + Φ), we start with

A sin(x + Φ) = A sin(x) cos(Φ) + A cos(x) sin(Φ)

Then

A cos (Φ) = -5

A sin(Φ) = -4

The ratio of the second to the first equation is

sin(Φ) / cos(Φ) = -4 / -5

or simply

tan(Φ) = 4/5

Φ = arc-tan(4/5) ≈ 0.214776*π

By squaring and adding the equations, we get

A² ( 1 ) = 25 + 16 = 41

or A = sq-rt(41)