How to Calculate Amplitude Modulation Index Without Prefix

[Padmal]

Homework Statement

Let's say we have a carrier signal x_c(t) = A_cCos(ω_ct)
And there is a modulating signal or we can call it the information signal x_m(t) = A_m Cos(ω_mt) and the modulation index is given as 0.8

How to write the modulated signal? It's really confusing as many people give different definitions and can't figure out which is correct. At least this one might make me understand the basic principle as it contains an amplitude not equal to 1 in the modulating signal.

Homework Equations

x(t) = A_c[1 + μx(t)]cos(ω_ct)

The Attempt at a Solution

x(t) = A_c[1 + (μ×A_m/A_c)cos(ω_mt)]cos(ω_ct)[/B]

 

[haruspex]

This is regarding amplitude modulation, yes? Perhaps you are confused because the definition of the index depends on the type of modulation. If you are seeing different definitions of the index for AM, can you post links or quote them?
I would trust https://en.m.wikipedia.org/wiki/Amplitude_modulation#Modulation_index

 

[NascentOxygen]

x(t) = Ac[1 + (μ×Am/Ac)cos(ωmt)]cos(ωct)

μ is defined as $\frac{A_m}{A_c}$ so you won't use both together in your expression