Amplitude of a Product of Trigonometric Functions We know that $|a|$ is called the amplitude of $a \sin(bx\pm \delta)$ and $a \cos(bx\pm \delta)$. 
But what is the amplitude of a product of trigonometric functions like: $a \cos(bx\pm \delta_1) \cos(cy\pm \delta_2)$ or any other combinations of the trigonometric functions $\sin mx, \cos ny, \tan pz$ ?
 A: As already indicated in the comments, the basic definition of amplitude used to describe simple sinusoids breaks down for more general trigonometric functions. Travis offered the supremum as one possible generalization.
Here's a completely different way of generalizing the notion. Consider the product of cosines you mentioned,  $f(x)=\color{red}{a \cos(bx\pm \delta_1)}\color{blue}{ \cos(cy\pm \delta_2)}$. The original definition of amplitude says the amplitude is the absolute value of the term multiplying the sine or cosine function. We can just simply keep this definition for the most part, in which case we may view the red factor of $f(x)$ the amplitude: 
$$f(x)=\color{red}{A(x)}\color{blue}{\cos{(cy\pm\delta_2)}},\\
\color{red}{A(x):=a \cos(bx\pm \delta_1)}.$$
The cosine highlighted in blue is then called the carrier wave, and red one is called the modulation wave. Encoding signals in radio waves via amplitude modulation is the basis of AM radio.
As you might expect, different generalizations can vary from context to context.
