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The amplitude in a trigonometric equation related to periodic motion is a crucial concept. It represents the maximum displacement from the equilibrium position, indicating how far the waveform reaches from its mean value. In the context of our exercise, the amplitude is calculated using the Pythagorean theorem principle applied to the coefficients of the sine and cosine functions.

To calculate amplitude, use the formula:

• \( A = \sqrt{A_1^2 + A_2^2} \),

where \(A_1\) and \(A_2\) are the coefficients of \(\sin(\omega t)\) and \(\cos(\omega t)\) respectively. For our given equation, \(x = \sqrt{2} \sin 2 \pi t + \sqrt{6} \cos 2 \pi t\),

• \(A_1 = \sqrt{2}\),
• \(A_2 = \sqrt{6}\).

Substituting these into the formula gives:

• \(A = \sqrt{(\sqrt{2})^2 + (\sqrt{6})^2} = \sqrt{2 + 6} = \sqrt{8} = 2\sqrt{2}\).

This value \(2\sqrt{2}\) indicates that our waveform reaches this maximum value above and below its central axis.

The phase angle, denoted as \(\theta\), is essential in shifting a trigonometric function horizontally along the time axis. It provides insight into the waveform's starting position at \(t = 0\).

To find the phase angle, we use the arctangent function:

• \(\theta = \tan^{-1}\left(\frac{A_2}{A_1}\right)\),

with \(A_1\) and \(A_2\) being the same coefficients as we used for the amplitude. From our example equation, the coefficients are again \(\sqrt{2}\) and \(\sqrt{6}\).

Thus, calculating gives:

• \(\theta = \tan^{-1}\left(\frac{\sqrt{6}}{\sqrt{2}}\right) = \tan^{-1}(\sqrt{3})\).

Using a calculator, this results in \(\theta = \frac{\pi}{3}\). This phase shift of \(\frac{\pi}{3}\) radians implies that the waveform is displaced to the left or right depending on the sign, starting earlier or later in its cycle than it would if \(\theta\) were zero.

Understanding the format and transformation of trigonometric equations is key when dealing with periodic motion problems. Our initial equation,

• \(x = \sqrt{2} \sin 2 \pi t + \sqrt{6} \cos 2 \pi t\),

is a combination of both sine and cosine functions, which are fundamental in modeling vibrations or oscillations.

Such an equation can always be represented in the form:

• \(x = A \sin(\omega t + \theta)\),

where \(A\) is the amplitude, \(\omega\) is the angular frequency, and \(\theta\) is the phase angle. The transformation involves expressing the sum of sine and cosine terms as a single sine function with a phase shift.

In our solution, this led us to:

• \(x = 2\sqrt{2} \sin(2\pi t + \frac{\pi}{3})\).

This form is simpler and highlights the key characteristics of the waveform: its amplitude, frequency (which is linked to \(\omega\)), and phase shift. Indeed, verifying the transformation involves graphing both the original and reformulated equations to ensure they overlap, confirming their mathematical equivalence.