The arc measure of an angle

The radian is - like the degree - a unit for measuring the size of angles. It is defined on the unit circle and its magnitude corresponds to the central angle of an arc whose length is equal to the radius of that arc.

Definition

To define arc measure, we will need a unit circle. On this circle we plot two points A and B so that the arc they form has length one, i.e. the length of the arc is the same as the radius of this circle. If we mark the centre of the circle with the point S, then the angle ASB has a magnitude of just one radian.

The line AS represents the radius of the circle, it is coloured red. The other red line, the arc AB, must be the same length. The important thing is that we are not interested in the distance between the points A and B directly, i.e. we are not interested in the line AB, but really in the length of this arc.

What is the value of radians in degrees? Approximately $57^\circ 17^{\prime} 45^{\prime\prime}$. We cannot quantify the exact value. The question then is what such a unit is good for.

Practical calculation with radians

The first question to answer is how many radians are equivalent to a whole circle, i.e. how many radians are equivalent to 360 degrees. We know that the arc length of a radian is just the radius of the circle. How many such arcs are we able to fit on the whole circle? The whole circle has a length (i.e., circumference) of just

$$o=2\pi r.$$

Our arc has length r, so to find how many times the arc fits on the circle, we need to divide obvod / polomer, which gives us:

$$\frac{2\pi r}{r}=2\pi.$$

Thus, the answer is that we are able to fit such arcs on the whole circle 2π. That's quite a nice number. So we can write that

$$360^\circ=2\pi \mbox{ rad}\quad\mbox{ a }\quad180^\circ=\pi\mbox{ rad}.$$

We also often omit the unit rad itself and just write an angle of size, for example 2π.

The following figure shows three angles of size one radian, these are the alpha, beta and gamma angles. You can see that the sum of the sizes of their angles is almost 180 degrees. But since we would need π radians to get to 180 degrees, and π has an approximate value of 3,1415…, we are still missing a bit.

Converting from degrees to radians and vice versa

Most instruments and programs that calculate goniometric functions work with radians by default. This is a bit of a problem if you have an angle specified in degrees. You have two options: if you can, switch the instrument to count in degrees or convert degrees to radians.

For example, the Google internet search engine has a built-in calculator, so you can calculate the value of the sine function there. But by default it just works with radians, so when we try to calculate sin(30), we don't get the correct result 0,5, because Google doesn't understand it as thirty degrees, but thirty radians. However, if we add the word "degrees" (which is English for "degree") after the number, it already calculates correctly: sin(30 degrees). Most calculators then have just two modes, one working in degrees and the other in radians. Usually this is under a button that says rad and deg. Rad is short for radian, deg is just degree.

If there's no rest, you have to convert degrees to radians or vice versa. Let's start with the first one: converting from degrees to radians. We already know that the equation holds $180^\circ=\pi\mbox{ rad}$. We'd like to know what part of a radian corresponds to one degree, so we divide the equation by 180:

$$1^\circ=\frac{\pi}{180}\mbox{ rad}$$

We already know how many radians one degree is. If we need to calculate thirty degrees, for example, we multiply that fraction by thirty:

$$30^\circ=30\cdot\frac{\pi}{180}\mbox{ rad}=\frac{\pi}{6}\mbox{ rad}.$$

Similarly for other degrees.

Now the opposite case, converting from radians to degrees. Again, we start from the equation $180^\circ=\pi\mbox{ rad}$. If we have 2π radians, we get twice as many degrees, or 360. Conversely, if we have only half π radians, we get half the number of degrees - 90. So first we divide our value in radians by π, to find what multiple of 180 degrees the angle represents. Then we multiply the value by 180 and we have the angle in degrees. So if we have an angle of magnitude 3π/2 radians, after dividing we get:

$$\frac{3\pi}{2}:\pi=\frac{3\pi}{2}\cdot\frac{1}{\pi}=\frac32$$

Multiply this result by 180 and we have the angle in degrees: 3/2 · 180 = 270. If we have two radians, then we get:

$$2\mbox{ rad} = \frac{2}{\pi}\cdot180=\left(\frac{360}{\pi}\right)^\circ = 114{,}59\ldots^\circ$$

The basic relationship between the value in radians and the value in degrees is shown by this formula (rad is the value in radians, deg is the value in degrees):

$$deg=rad\cdot\frac{180^\circ}{\pi}.$$

History

The concept of the radian was probably first developed by Roger Cotes in 1714. At that time his unit was not yet called "radian", but the other definitions and properties were identical. The term "radian" first appeared on paper on 5 June 1873 in an article by Jameson Thomson. The unit could still be called "radial" or just "rad". (Source: wiki)

Table of basic conversion relationships

Some angles are used quite often, so the following table gives the values in degrees and their equivalent in radians:

$$ \LARGE \begin{matrix} \mbox{deg: }&0&30&45&60&90&180&270&360\\ \mbox{rad: }&0&\frac{\pi}{6}&\frac{\pi}{4}&\frac{\pi}{3}&\frac{\pi}{2}&\pi&\frac{3\pi}{2}&2\pi \end{matrix} $$