Definition of \$\mu\$
This formula seems to be misinterpreted by you: 

$$x(t) = A_c[1+\mu x_m(t)]\cos(\omega_c t)$$
\$x_m(t)\$ is any message signal, not necessarily a sine wave. Therefore, 

$$x(t)=[A_c + A_m x_m(t)]cos(\omega_c t)$$
\$A_c\$ : carrier amplitude
\$A_m\$ : message amplitude

Then the carrier amplitude \$A_c\$ term is taken common to yield: 

$$x(t)=A_c[1+\mu x_m(t)]cos(\omega _ct)$$
Where \$\mu\$ = modulation index \$\dfrac{A_m}{A_c}\$ and \$0 \leq\mu \leq 1\$.

So according to this link you cannot give both the carrier and message amplitude and expect a signal with a modulation index within reality.