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# Sine of time

If we introduce time as the argument to the sine function, we get a sine wave. Animated. Created by Willy McAllister.

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• Just to confirm that I understand, are the frequency and the angular frequency the same parameter (just in different units)? For example, we could measure distance in miles or kilometers. Angular frequency just means that the frequency has different units?
• You are correct. Frequency has units of cycles per second (also known as hertz), while angular frequency has units of radians per second. You use whichever unit is best suited to the problem at hand.
• We know that the frequency is the reciprocal of the period (f = 1/T and T = 1/f). In your calculation you obtain the frequency from the reciprocal of the period (1/0.02). So it seems like that 50 derives naturally from the period. Why can't it be used directly as omega value ?

You said that frequency and angular frequency can be used interchangeably. However it seems like that the frequency in Hertz need to be always converted to rad/s in order to use it with sinusoids. So from a calculator point of view you end up using rad/s always. Assuming that a Hertz value is inserted, then the calculator need to convert it first to rad/s before performing the actual calculation. Did I get it right ?
• what happens to the sine wave of the increasing frequency?
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• The sine function has a period of 2π. That means the sin function completes one cycle when its entire argument goes from 0 to 2π.

ω represents the frequency of a sine wave when we write it this way: sin(ωt).

If ω=1 the sin completes one cycle in 2π seconds.
If ω=2π the sin completes one cycle sooner, every 1 second.

For f>1: If ω=2πf the sin completes one cycle even faster, every 1/f seconds. Or, equivalently, sin completes f cycles in 1 second. So the higher f is, the more cycles happen in 1 second. More and more cycles get crammed into that 1 second interval. That's what higher frequency is.

Vocabulary:
ω is called the radian frequency with units of radians per second. 2π radians is all the way around a circle (360°). 1 radian is about 57° of the way around a circle.

f is the frequency with units of cycles per second. Cycles per second has the honorary name Hertz. ω = 2πf (good to memorize)
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• What is the difference between sin and cos
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• Sine and Cosine have exactly the same shape, waving up and down between +1 and -1. The only difference is that sin(x) starts at 0 when x=0 and cos(x) starts at 1. They are "90 degrees out of phase."
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• At about you state that "omega is 1/time", and later you state that "omega is radians/sec". You really made this simple topic very confusing. Could you explain?
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• The argument (the number inside) sin or cos is in radians, (a measure of the angle distance around a circle). Going around full circle is 2pi radians (of angle).

Radians are a unit of measure of circular angles. You might think that radians have some kind of dimension. However, oddly, radians are dimensionless. Recall the definition of radians: the ratio of the distance around the circumference of a circle divided by the circle's radius. That's the ratio of two distances, like meters/meters. The resulting dimension of a radian is: none.

If the argument of sin or cos is [\omega t] it has to be the case that the overall argument has dimension: none. That means \omega has dimension 1/t (1/time, 1/seconds, 1/minutes, whatever).

Since we know the argument to sin or cos is a radian value, it might be more precise to include "radian" in the dimension of \omega, like this: radian/sec. This reveals why \omega is called the radian frequency.

I perhaps made a small verbal slip by saying \omega is 1/t. It's quite common to forget the dimensionless "radians" when you are showing how \omega cancels out the seconds in t.
(1 vote)