Rewrite -6sin(x) - 2cos(x) as A sin ( x + ϕ )

Question

A=

ϕ=

Answer 1

-6sinx -2cosx

= about -6.32sin(x+16.32) where x is measured in degrees

let x = 73.68, then

-6.32sin(73.68 + 16.32)

= -632sin(90)

= -6.32

if x is measured in pi radians

then

-6sinx -3cosx = -6.32sin(x + .091 pi)

= -6(sin73.68)-2cos73.68

= -5.76 - .56

= - 6.32

or = -6(sin.41pi) -2cos(.41pi) = -6.32

or if you measure x in terms of radians (not pi radians)

then

-6.32sin(x+ 0.286 ) = -6sinx -2cosx

the angle phi = 0.091pi, 0.286 radians or 16.32 degrees

coefficient A = -6.32

Answer 2

y = - [6sinx + 2cosx]

Multiply and divide by √(6²+2²) = 2√10 to get -2√10 [ (3/√10)sinx + (1/√10)cosx] =

-2√10 [ sinxcosφ + cosxsinφ ] where φ = cos^-1(3/√10) = 0.32

So, y ≈ -2√10 sin(x + 0.32)

Answer 3

A = (36 + 4)^1/2 = 2·10^1/2 = 2√10; φ = π + tan^-1(-2/(-6)) = π + tan^-1(1/3)

-6sinx - 2cosx = 2√10(sinxcos(π + tan^-1(1/3)) + cosxsin(π + tan^-1(1/3))) = 2√10sin(x + π + tan^-1(1/3))

A = 2√10; φ = π + tan^-1(1/3) = 3.4633