# Why do people prefer cosine to sine when speaking of harmonic oscillation?

In almost all of the physics textbooks I have ever read, the author will write the oscillating function as

$$x(t)=\cos\left(\omega t+\phi\right)$$

My question is that, is there any practical or historical reason why we should prefer $\cos$ to $\sin$ here? One possible explanation I can think of is that, to trigger a harmonic oscillation movement, we usually push the mass (to the maximum displacement) from the balance point at the initial moment, for which the cosine function will be neater to use than sine ($\phi=0$). But is it really the case?

• My textbook actually uses sine :) – Wojowu Jul 5 '15 at 10:51
• @Wojowu Before I went to college, when I was self learning some conceptions about harmonic movement from some online courses I saw the professor use sine too. But by now almost all my textbooks use cosine to represent the harmonic oscillation and the wave function. – Vim Jul 5 '15 at 10:55
• Or is it for the compatibility with Euler's $\exp(ix)$, because we prefer Re to Im? Though I am not familiar with complex representation for the harmonic oscillation or the wave function. – Vim Jul 5 '15 at 10:57
• I would agree with you, the "most basic" set up in harmonic motion is for a mass released from rest with amplitude A. – Paul Jul 5 '15 at 11:49
• I think it comes from the general perception that the cosine is actually the primary trigonometric function, while the sine is somehow secondary. For example, cosine is an even function, which is a simpler and more basic kind of symmetry than being an odd function. Cosine is the real part of $e^{ix}$, which simpler than being the imaginary part. Cosine is related to the dot product, which is a more basic operation than the cross product. The cosine is the primary trigonometric function, and the sine is its companion. – Jim Belk Jul 5 '15 at 18:44

I agree with you that one good reason for using $\cos$ is that it corresponds well with the initial conditions for harmonic motion.

Another good reason is experimental observation. If I come across a physical system that is oscillating, I will want to measure its amplitude and period. Examples include a swinging pendulum, a planet observed rotating around it star, a pulsing wave etc.

Observing the phenomena from a distance, it is possible to note the extreme positions of the motion and determine the amplitude as half the distance between those positions.

To find the period, I want to start my stopwatch at a particular point in the motion and stop it again when that point is repeated, I can choose to do so when the motion is at one extreme or to do so at the moment when the object passes through a particular point. If I try to press the button as the object passes through its central point, I am doing so when it is at its maximum velocity and this can be difficult to observe accurately. It is therefore easier to start timing at one of the extreme values, which corresponds to a $\cos$ function.

• I don't get it. The object will be "right on" its extreme point—that is, moving imperceptibly—for a "long" time, and I have to push my stopwatch button right in the middle of that vaguely defined interval when the thing is "sitting" on its extreme point. Seems to me that would be less accurate than trying to time the instant it passes a particular point where it's moving fast. – bof Feb 3 '16 at 5:24
• @bof makes sense to me. But also note that although the $v_{max}$ point seems to be more conspicuous than extreme points where the motion is too vague to observe, it however doesn't imply that timing at the $v_{max}$ point will be easier: since the object passes this point the most quickly, it requires very swift response to stop the timer right on this point. – Vim Feb 3 '16 at 11:14
• If you have a consistent delay between observing the zero crossing and pressing the button, it will still cancel out when taking the difference between the two times. :) – user856 Feb 3 '16 at 20:05

Direct current (I = const, U = const, etc.) can be assumed as "alternating current with ω = 0 and ϕ = 0" only when using cosine, but not sine. This reason went from the electrical engineering.

Another reason is to use the initial position as a parameter.

Compare $z=z_0\lambda^t$ and $h=h_0\cos \omega t$

You can't do the same with $\sin$ - at best you can say perhaps $d=d_{max}\sin \omega t$

By the way, this argument fails once you introduce a phase shift!