# Why do we use Radian measurement for Time?

I had a problem where we had to write out the sinusoidal function for the rotation of a ferris wheel. My teacher said that because the rotation was a function of time (x-axis was time), we had to use radians and not degrees.
Could someone explain to me if/why you need to use radian measurement for time when calculating things like the period and frequency, or can you just as easily scale the graph for degrees. Thanks!

jambaugh
Gold Member
Radians are actually dimensionless units. They are chosen so that sector area and arclength have the simple forms: $s = r\theta$ and $A = \frac{1}{2}r^2 \theta$. If you used degrees these would be "messy".

I teach my students that "we always use radians" we just hide the fact. Treat the degree symbol as a number (pi/180). Thus e.g. $360^o = 360 \cdot \frac{\pi}{180} = 2\pi$. The only problem with this is that calculators do not explicitly show the degree symbol so there you have to be careful about modes.

As for using time units, you should never have bare time units inside a trig function. The time unit should be multiplied by an angular frequency unit $\omega$ (radians per second ) so that $\sin(\omega t)$ is evaluating sine of a pure radian value.

Now you actually could work with "degrees per second" angular frequency but that too makes for bad formulas, e.g. speed (v) of a point rotating on a circle is $v = r \omega$ provided $\omega$ is radians per time unit.

As far as graphing goes if it is a function of time then it should be scaled in time units. If you would post the format of the question I might better understand what your teacher is insisting upon.

The main thing is that radians are the natural units of the trigonometric functions. (for example there is a small angle approximation sin(x) ~ x for small values, but only when you're working in radian units!!!!) Get used to working in radians and converting between degrees and radians (and decimal cycles i.e. 360deg = 2 pi radians = 1 cycle).

rcgldr
Homework Helper
You don't need to scale the graph, since you're plotting sin() versus time, but as previously posted, you'd need to specify the rotation of the ferris wheel in terms of radians per second or degrees per second and convert to the type of angle that the sin function uses.

Assuming that a sin function only takes radians as an input, you'd need to convert degrees to radians anyway:

For degrees, if the ferris wheel rotates d (degrees / second), you use

sin(t seconds x d (degrees / second) x (π radians / (180 degrees)))

For radians, if the ferris wheel rotates at ω radians per second, you use

sin(t seconds x ω (radians / second))

In either case, the graph of sin(t ...) (y axis) versus time (x axis) will be the same.

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