Amplitude and Phase Shift (C)

However, these formulae are extremely useful. For example, in the analysis of simple harmonic motion one encounters expressions such as $3\cos(2t) + 4\sin(2t)$. These are shifted cosine waves. To see why, observe that the ordered pair

$$\left(\frac{3}{3^2+4^2},\frac{4}{3^2+4^2}\right) = \left(\frac{3}{5},\frac{4}{5}\right)$$

is a point on the circle x² + y² = 1. Therefore there is an angle $\phi\in[0,2\pi)$ so that

$$(\cos(\phi),\sin(\phi)) = \left(\frac{3}{5},\frac{4}{5}\right).$$

Therefore

$$\begin{array} {rcl} 3\cos(2t) + 4\sin(2t) & = & 5\left(\fra... ...os(2t) + \sin(\phi)\sin(2t))\\ & = & 5\cos(2t-\phi).\end{array}$$

The coefficient 5 is the amplitude of the wave, and $\phi$ is the phase shift.

Exercises:

Find the amplitude and the sine and cosine of the phase shift for each of the following. Sketch a graph of the wave as a function of t:

1.

$4\cos(4t) + 3\sin(4t)$.

2.

$7\cos(5t) + 24\sin(5t)$.

3.

$5\cos(t/3) - 12\sin(t/3)$.