Revolutions, Cycles, Radians, Degrees; Oh My

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I always battled with the choice of using degrees or radians for trigonometric functions. I also wondered why we had both.

Radians are usually used in math and science due to the fact they are "unitless." The next question is usually, "How can they be unitless?" I think about the definition of $\mathrm{\pi \; <!--\; greek\; small\; letter\; pi\; -->}$.

If you take a some pieces of string the length of a circle's radius and wrap them around the circumference, how many strings to you need to get clear around the circle? It turns out that you need $2·\; <!--\; middle\; dot\; -->\mathrm{\pi \; <!--\; greek\; small\; letter\; pi\; -->}$ (roughly 6.28) pieces of string to do the trick.

$2·\; <!--\; middle\; dot\; -->\mathrm{\pi \; <!--\; greek\; small\; letter\; pi\; -->}·\; <!--\; middle\; dot\; -->r\left(\right)=1·\; <!--\; middle\; dot\; -->\mathrm{Circumference}\left(\right)$

Now solve for $\mathrm{\pi \; <!--\; greek\; small\; letter\; pi\; -->}$ and make up a name for the angle measure associated with $\mathrm{\pi \; <!--\; greek\; small\; letter\; pi\; -->}$.

$2·\; <!--\; middle\; dot\; -->\mathrm{\pi \; <!--\; greek\; small\; letter\; pi\; -->}\left(\right)=\frac{\mathrm{Circumference}\left(\right)}{r\left(\right)}$

So, the made up unit of radian is equivalent to a $\frac{\mathrm{meter}}{\mathrm{meter}}$, or unitless.

Why is using a unitless measure angle so nice? If calculus is involved, not having units is extremely handy. I have no idea what the integral of degrees would be. However, the integral of nothing is very nice to deal with.

Ways we describe going clear around a circle.

• One Cycle
• One Revolution
• 360 degrees
• $2·\; <!--\; middle\; dot\; -->\mathrm{\pi \; <!--\; greek\; small\; letter\; pi\; -->}\left(\right)$

A discussion of a sinusoidal wave encouraged this web page. To graph a sinusoidal wave form, plot $y=\sin \left(\right)$.

PLOT?

If the frequency of the sine wave is given, then the following is true:

$F·\; <!--\; middle\; dot\; -->\left(\right)=F·\; <!--\; middle\; dot\; -->\left(\frac{\mathrm{cycles}}{\mathrm{second}}\right)=2·\; <!--\; middle\; dot\; -->\mathrm{\pi \; <!--\; greek\; small\; letter\; pi\; -->}·\; <!--\; middle\; dot\; -->F·\; <!--\; middle\; dot\; -->\left(\frac{\mathrm{radians}}{\mathrm{second}}\right)=\mathrm{\omega \; <!--\; greek\; small\; letter\; omega\; -->}·\; <!--\; middle\; dot\; -->\left(\frac{\mathrm{radians}}{\mathrm{second}}\right)$.
where $\mathrm{\omega \; <!--\; greek\; small\; letter\; omega\; -->}$ is referred to as the "circular frequency."

In the real world, we tend to use "frequency" in "Hertz" and in the world of mathematics, they use "circular frequency" in $\frac{\mathrm{radians}}{\mathrm{second}}$, because of the unitless nature of radians.

All this "units" stuff is important because your calculator can calculate trigonometric functions using either degrees or radians. You really do need to know which mode you need to set your calculator. Many teachers/professors love to put problems on exams to find out if you know which setting to use.