Examples of sinusoidal in the following topics:

 Determine factors responsible for the sinusoidal behavior of uniform circular motion
Identify types of solutions that exist for the equations of motion of simple harmonic oscillators
The solutions to the equations of motion of simple harmonic oscillators are always sinusoidal, i.e., sines and cosines.
 Solving the differential equation above always produces solutions that are sinusoidal in nature.
 For example, x(t), v(t), a(t), K(t), and U(t) all have sinusoidal solutions for simple harmonic motion.
 Only the top graph is sinusoidal.
 Uniform circular motion is therefore also sinusoidal, as you can see from .
 sinusoidal (adjective) In the form of a wave, especially one whose amplitude varies in proportion to the sine of some variable (such as time).

 Phasors greatly reduce the complexity of expressing sinusoidally varying signals.
 In the sinusoidal steady state, every voltage and current in a system is sinusoidal with angular frequency ω.
 This can be particularly useful because the frequency factor (which includes the timedependence of the sinusoid) is often common to all the components of a linear combination of sinusoids.
 In the sinusoidal steady state, every voltage and current (or force and velocity) in a system is sinusoidal with angular frequency ω.
 However, the amplitudes and phases of these sinusoidal voltages and currents are all different.
 sinusoidal steady state (noun) Indicates every voltage and current in a system is sinusoidal with the same angular frequency ω.

 For instance, when we studied the forced harmonic oscillator, we first solved the problem by assuming the forcing function was a sinusoid (or complex exponential).
 We then argued that since the equations were linear this was enough to let us build the solution for an arbitrary forcing function if only we could represent this forcing function as a sum of sinusoids.
 Later, when we derived the continuum limit of the coupled spring/mass system we saw that separation of variables led us to a solution, but only if we could somehow represent general initial conditions as a sum of sinusoids.

 One of the most common waveforms in physics is the sinusoid.
 Consider one of the most common waveforms, the sinusoid.
 We looked closely into the sinusoidal wave.
 Since a wave with an arbitrary shape can be represented by a sum of many sinusoidal waves (this is called Fourier analysis), we can generate a great variety of solutions of the wave equation by translating and summing sine waves that we just looked closely into.
 This graph shows the general form of a sinusoid wave.

 The extrema (peaks and troughs) of a sinusoid of frequency $f_s$ will lie exactly $1/2f_s$ apart.

 The RMS current and voltage (for sinusoidal systems) are the peak current and voltage over the square root of two.
 It is especially useful when the function alternates between positive and negative values, e.g., sinusoids.The RMS value of a set of values (or a continuoustime function such as a sinusoid) is the square root of the arithmetic mean of the squares of the original values (or the square of the function).
 Consider the case of sinusoidally varying voltage :
(a) DC voltage and current are constant in time, once the current is established.
 The voltage and current are sinusoidal and are in phase for a simple resistance circuit.
 $P_{ave}=I_{rms}V_{rms}=\frac{I_{0}}{\sqrt{2}}\frac{V_{0}}{\sqrt{2}}=\frac{1}{2}V_{0}I_{0}$
The RMS values are also useful if the voltage varies by some waveform other than sinusoids, such as with a square, triangular or sawtooth waves .

 $
(Eq. 1.1.29)
These give the following solutions for the motion (using $x(0) = x_0$ )
$\displaystyle{ x(t) = x_0 e^{\frac{\gamma}{2} t} e^{\pm i t \omega _0 \sqrt{1  \left( \frac{\gamma}{2\omega_0} \right)^2 }} . }$
(Eq. 1.1.30)
This looks like the equation of a damped sinusoid.
 But the second term may or may not be a sinusoid, depending on whether the square root is positive.
 First if $\frac{\gamma}{2\omega_0} 1$ , corresponding to small damping, then the argument of the square root is positive and indeed we have a damped sinusoid.


 Later we will see that it is no loss to treat sinusoidal forces; the linearity of the equations will let us build up the result for arbitrary forces by adding a bunch of sinusoids together.

 Consider the continuous sinusoidal signal:
$x(t) = A \cos(2 \pi f t + \phi)$
Suppose we sample this signal at a sampling period of $T_s$ .
 $
Now consider a different sinusoid of the same amplitude and phase, but sampled at a frequency of $f + \ell f_s$ , where $\ell$ is an integer and $f_s = 1/T_s$ .
 Let the samples of this second sinusoid be denoted by $y[n]$ .
 These two sinusoids have exactly the same samples, so the frequency of one appears to be the same.