Why are the power series for trig functions in radians? As an example, the function $\cos(\theta)$ represents the ratio of, relative to the angle $\theta$, the adjacent-side and hypotenuse of a right triangle.  Strictly speaking, $\theta$ is measured in any units we want - it's the interpretation/definition of the $\cos(\theta)$ function relative to our chosen "units" for $\theta$ that matters. That's why it's just a matter of economics when choosing between radians, degrees, or anything else.
Of course, when finding arcs on a circle, it's convenient to define the angle $\theta$ in terms of the ratio between the radius of a circle and its circumference, $1/2\pi$ (e.g. when you've covered an angle $\pi$ in these units along the circumference of a circle, you've traveled $\pi/(2\pi)=1/2$ of the circumference). That way, calculating arc-lengths becomes simple multiplication by the angle $\theta$ (this obviously isn't true for degrees!). 
However, I don't know how to interepret the power (Taylor/Maclaurin) series for trig functions like $\cos(\theta)$ and $\sin(\theta)$. 
$$\cos(\theta)=1-\frac{\theta^2}{2}+\cdots,\,\,\,\,\sin(\theta)=\theta-\frac{\theta^3}{6}+\cdots$$
Why must we use radians in the above series representation? Why don't we use for the "units" of $\theta$, for example, the fraction of the entire circle that it covers (e.g. $1/4$ instead of $\pi/2$, $1$ instead of $2\pi$, etc.)? That would seem more natural and "unitless".
 A: Think of radians as having a special identity multiplication property.
$radians\cdot radians=radians$.  The 'unit' is unaffected unlike other units.
In fact, I learned in physics the other day that $radians\cdot meters=meters$, which means that 'radians' aren't even really units.
This allows the Taylor series for trig functions to make sense in terms of units.
Since degrees have the same property, it is simply how we derive the Taylor series of our trigonometric functions that determines that we use radians over degrees.
It has to do with derivatives, which makes a well defined Taylor series for our trig functions if we use radians.
If we use degrees, then the Taylor series for the trig functions are
$$\cos(\theta)=1-\frac{\left(\frac{90}{\pi}\right)^2\theta^2}{3!}+\frac{\left(\frac{90}{\pi}\right)^4\theta^4}{5!}-\dots$$
Of course, this makes sense and all, but the reason it is like this is because $\frac d{d\theta}\cos(\theta)=-\frac{90}{\pi}\sin(\theta)$, if $\theta$ is given in degrees.  This results from simple chain rule, which you will learn about in Calculus I.
A: To go off what Simply Beautiful Art said,
This works because radians multiplied by a meters is still meters. This works for any translational distance. Why? Take the instance where we are moving $\theta_{\text{radians}}$ radians around a circle of radius $r$ meters. Let us take look at the case where $\theta_{\text{radians}} = \pi$. This is the same as going around half of the circle's circumference. Since 
$$\text{circumference} = 2\pi r$$
we know
$$\frac{\text{circumference}}{2} = \pi r$$
Clearly, $\frac{\text{circumference}}{2}$ is in meters, even though it is the product of radians and meters. Thus, radians isn't a unit. Radians are the ratio of circle arc length to circle radius.
This is not true for degrees, however. If we take the product of degrees and meters, we get $180^{\circ} r$. Note how we write a $^\circ$, where we don't write a unit for radians. This is because degrees is a unit, not a ratio between distances. You might wonder why this is important. If you modify degrees so you multiply the raw 
number of degrees by some scalar, you can still get meters when multiplying by meters. You are right! However, we can't treat it the same as radians as this is like saying 180 dollars is equal to 180 pennies. They are different units. One is a ratio, and one is not. Thus, we actually need to cancel the degrees out by multiplying by $\frac{1}{1} = \frac{\text{ratio}}{\text{degrees}} = \frac{\text{radians}}{\text{degrees}})$.
Given an angles in degrees, we know the proportion we go around a circle is $\theta_\text{degrees} \cdot \frac{1}{360^{\circ}}$. Thus,
$$\text{circumference} = \left(\theta_\text{degrees} \cdot\frac{1}{360^{\circ}}\right)2\pi r = \frac{\theta_\text{degrees}}{180^{\circ}} \pi r$$
Thus, we need to multiply $180^{\circ} r$   by $\frac{\pi r}{180^{\circ}}$ to get a value in meters. 
This is the same reason the power series is in radians. If we wanted it in degrees, we would need to factor in $\frac{\pi r}{180^{\circ}}$ when applying The Chain Rule. 
