Show that \(x(t)=a \cos \omega t-b \sin \omega t\) represents simple harmonic motion, as in Equation \(13.8,\) with \(A=\sqrt{a^{2}+b^{2}}\) and \(\phi=\tan ^{-1}(b / a)\)

Understanding trigonometric identities is essential when dealing with simple harmonic motion (SHM) equations. Trigonometric identities are equations involving trigonometric functions that are true for any value inserted into them.

For instance, the identity \(\cos(\theta - \phi) = \cos \theta \cos \phi + \sin \theta \sin \phi\) is invaluable when attempting to express an SHM equation in a more familiar form. For the given problem, this identity allows the equation \(x(t) = a \cos \omega t - b \sin \omega t\) to be rewritten in a form that closely resembles the general solution for SHM. By matching coefficients, we can use this identity to express the constants in terms of amplitude \(A\) and phase constant \(\phi\), making the physical parameters governing the motion more apparent.

Furthermore, manipulating these identities enables us to deduce crucial characteristics of the motion, such as amplitude and phase, from an equation that might not immediately appear in the standard SHM form. This underscores the importance of trigonometric identities not only in solving the problem but in understanding the deeper relationships between different forms of the same physical motion.

The amplitude of oscillation represents the maximum displacement from the equilibrium position in simple harmonic motion. It is a key characteristic that defines the extent of the motion. In our problem, once we have used trigonometric identities to express \(x(t)\) in terms of \(A\) and \(\phi\), we can determine the amplitude \(A\) by analyzing the coefficients \(a\) and \(b\).

In the expression \(x(t) = a \cos \omega t - b \sin \omega t\), \(a\) and \(b\) represent components along the cosine and sine axes, respectively, and the amplitude \(A\) can be thought of as the hypotenuse of a right triangle with legs \(a\) and \(b\). By applying the Pythagorean theorem, represented by the identity \(\sin^2\phi + \cos^2\phi = 1\), we find that \(A\) is equal to \(\sqrt{a^2 + b^2}\), giving us a clear geometric interpretation of the amplitude in terms of the original coefficients.

The phase constant \(\phi\) in simple harmonic motion is the initial angle that the system's state vector makes with the positive x-axis at time \(t = 0\). The phase constant determines where in its cycle the oscillation begins.

In the context of our exercise, we can decipher the phase constant by comparing the given SHM equation to the standard cosine form after applying trigonometric identities. Once we equate the coefficients \(a\) and \(b\) with \(A\cos\phi\) and \(A\sin\phi\), respectively, the phase constant \(\phi\) emerges naturally. By dividing \(b\) by \(a\), we obtain the tangent of the phase constant, \(\tan \phi\). The inverse tangent function, \(\tan^{-1}\), then gives us the value of the phase constant: \(\phi = \tan^{-1}(b / a)\).

This phase constant is a critical parameter because it affects how the motion aligns with the oscillation's driving force or with respect to time. Determining \(\phi\) allows us to completely describe the system's initial conditions and predict its future behavior.