Why is radian so common in maths?

Solution 1:

The reasons are mostly the same as the fact that we usually use base $e$ exponentiation and logarithm. Radians are simply the natural units for measuring angles.

• The length of a circle segment is $x\cdot r$, where $x$ is the measure and $r$ is the radius, instead of $x\cdot r\cdot \pi/180$.
• The power series for sine is simply $\sin(x)=\sum_{i=0}^\infty(-1)^i{x}^{2i+1}/(2i+1)!$, not $\sin(x)=\sum_{i=0}^\infty(-1)^i(x\cdot \pi/180)^{2i+1}/(2i+1)!$.
• The differential equation $\sin$ (and $\cos$) satisfies is $f+f''=0$, not $f+f''\pi^2/(180)^2=0$.
• $\sin'=\cos$, not $\cos\cdot 180/\pi$.

You could add more and more to the list, but I think the point is clear.

Solution 2:

As I teach my trigonometry students: "Degrees are useless."

You want to know the length of a circular arc? It's $r \theta$ where $r$ is the radius of the circle and $\theta$ is the angle it subtends in radians. If you use degrees, you get ridiculous answers.

You want to know the area of a sector? It's $\frac{1}{2} r^2 \theta$, with $r$ and $\theta$ as above. Again, if you use degrees, you get ridiculous results.

To really understand this, move on to calculus and study arc length. The arc length of the graph of the circle gives radian results. Or, look at the power series expansion of the circular trigonometric functions: if you use radians, everything works with small coefficients; if you use degrees, extra powers of $\frac{\pi}{180}$ scatter around.

What are degrees any good for? Dividing circles into even numbers of parts. That's it. If you want to actually calculate something, degrees are useless.

Solution 3:

Radians naturally arise when you look at some circles (note that they are a dimensionless unit). On the contrary, full circle being $360^\circ$ is due to some dude dividing the circle to as many pieces as there are days in the year (for some historical reason this resulted in $360$).

Why people think in radians then? My personal guess is that the reason is simply that mathematicians prefer to work with things that are somehow intrinsic to the object in consideration.

Solution 4:

radians are the natural unit of measure for angles. it's no anthropocentric convention. aliens on the planet Zog that do calculus and solve physics problems will also be understanding the naturalness of describing angles in radians.

as mentioned previous, the angle, expressed in radians, is the amount of circular arc swept by the angle divided by the radial arm with both lengths expressed in the same units. so radians are dimensionless. they're just a number. no units.

if you have a wheel of radius $r$ on a slip-free surface, the distance the wheel moves on the surface $x$ is equal to the angle turned, $\theta$ (in radians) times the radius of the circle of the outer rim of the wheel, $r$.

$$ x = r \cdot \theta $$

if the wheel (of radius $r$) is spinning at a rate of $\omega = \frac{\text{d} \theta}{\text{d} t}$, the speed that the wheel moves relative to the surface is $$v = \frac{\text{d} x}{\text{d} t} = r \cdot \omega \ .$$

if the angle was measured in any other manner, there would be constants of proportionality necessary for those equations to be true but those constants of proportionality are equal to 1 (and go away) if the angle is measured in radians.

radians are as natural to angles as $e \ \approx \ $ 2.718281828... is natural as a base for logarithms and exponentials in calculus. Euler's equation

$$ e^{i \theta} = \cos(\theta) + i \sin(\theta) $$

would need more nasty constants of proportionality if the base was not $e$ or the angle $\theta$ was not in radians.

so it's the opposite of human convention that in calculus the measure of angles are expressed in terms of radians.

Solution 5:

Degrees are a mistake of history (not speaking of minutes and seconds). Division in four quadrants of ninety degrees is quite arbitrary and inconvenient, but for one thing: it allows an easy representation of the remarkable angles, $30°$ and $45°$. In this respect, it is a little better than the $4\times100$ subdivisions in grades.

As explained by many others, radians are a natural unit as they avoid a constant factor when taking derivatives and make the formulas for the arc length or sector area the simplest.

IMO, the opportunity to use an interesting alternative is gone forever: the revolution. Counting angles in revolutions is pretty convenient as you can handle them modulo $1$, i.e. just using the decomposition in integer and fractional part.