Radians¶

Level: 101 → 201 · for anyone who knows what an angle is, and has wondered why calculus stops using degrees

One line: An angle in radians is the length of the arc it cuts from the circle of radius 1, so a point walking round that circle at speed 1 is at angle t after time t; that is why d/dt sin t = cos t holds only in radians, and why e^{iπ} = −1 has a π in it and not 180.

The talk's half circle¶

One of the talk's first pictures is a circle of radius 1 with the upper half of it drawn in red and labelled π, next to e^{πi} = −1. The label is a length. Half of that circle is π long, and the picture says: walk π along the circle from the point 1, and you arrive at −1. Nothing about degrees appears, and this page is about why.

What the program prints¶

Verified output of radians.py — regenerated by tools/run_examples.py, never hand-typed.

1. HOW LONG IS THE CIRCLE? ARCHIMEDES' POLYGONS
   Regular polygons inside the circle of radius 1, sides doubled each
   time, starting from the hexagon, whose side is the radius:
         6 sides   perimeter = 6.000000   half of it = 3.000000
        12 sides   perimeter = 6.211657   half of it = 3.105829
        24 sides   perimeter = 6.265257   half of it = 3.132629
        48 sides   perimeter = 6.278700   half of it = 3.139350
        96 sides   perimeter = 6.282064   half of it = 3.141032
      6144 sides   perimeter = 6.283185   half of it = 3.141593
   The perimeters settle on 2 pi = 6.283185. The whole circle is
   2 pi long, half of it pi, a quarter pi/2.

2. A RADIAN IS THE ANGLE WHOSE ARC IS ONE RADIUS LONG
   Measure an angle by the length of arc it cuts from the unit circle:
     turn         degrees   radians   as a number
     a quarter    90        pi/2      1.570796
     a half       180       pi        3.141593
     a whole      360       2 pi      6.283185
     1 radian     57.2958   1         1.000000
   One radian is a little under 60 degrees: the arc as long as the
   radius. It is not a round number of degrees because 2 pi is not
   a round number.

3. IN RADIANS, THE ANGLE IS THE DISTANCE WALKED
   Walk from (1, 0) to the point at angle t in n straight steps along
   the circle, and add up the steps:
     n      t = 1        t = pi/2     t = pi
     1      0.958851     1.414214     2.000000
     2      0.989616     1.530734     2.828427
     4      0.997398     1.560723     3.061467
     10     0.999583     1.569182     3.128689
     100    0.999996     1.570780     3.141463
     1000   1.000000     1.570796     3.141591
     t      1.000000     1.570796     3.141593
   The walk settles on t itself. That is what radians are for: the
   point at angle t has walked a distance t from (1, 0).

4. THE VELOCITY OF A TURNING POINT
   Let the point be at angle t at time t: position (cos t, sin t). Its
   velocity, measured over the time from t - 0.000001 to t + 0.000001:
     t      position                 velocity                 |velocity|   position . velocity
     0      (1.000000, 0.000000)     (0.000000, 1.000000)     1.000000     0.000000
     1      (0.540302, 0.841471)     (-0.841471, 0.540302)    1.000000     0.000000
     pi/2   (0.000000, 1.000000)     (-1.000000, 0.000000)    1.000000     0.000000
     pi     (-1.000000, 0.000000)    (0.000000, -1.000000)    1.000000     0.000000
   The velocity is (-sin t, cos t): the position turned a quarter turn,
   at right angles to it, and of length 1. In radians,
     d/dt cos t = -sin t        d/dt sin t = cos t
   Measure the same turning in degrees, the point at angle d degrees at
   time d, and every velocity shrinks by the same factor:
     at d = 30:  |velocity| = 0.017453    pi/180 = 0.017453
   so in degrees d/dd sin = (pi/180) cos, with a constant that radians
   make 1.

5. sin(x) / x FOR A SMALL ANGLE
     x          in radians     x in degrees
     0.1        0.998334       0.017453
     0.01       0.999983       0.017453
     0.001      1.000000       0.017453
   In radians a small sine is its angle, and the ratio settles on 1;
   in degrees it settles on pi/180 = 0.017453. A small arc is almost
   straight, and its height is its length, only when the angle IS the
   length.

How long is the circle?¶

Section 1 measures it without using π, the way Archimedes did around 250 BC. A regular hexagon inside the circle of radius 1 has sides of length 1, so its perimeter is 6. Doubling the number of sides pushes the polygon out toward the circle: 12 sides give 6.2117, 24 give 6.2653, 96 give 6.2821, which is as far as Archimedes went, and 6144 give 6.283185. The perimeters settle on 2π = 6.283185…, so the whole circle is 2π long, half of it π, and a quarter of it π/2. (Each doubling uses the new side s′ = s / √(2 + √(4 − s²)), the form that avoids subtracting nearly equal numbers; the textbook form does subtract them, and loses its digits the way catastrophic cancellation describes.)

A radian is one radius of arc¶

Degrees cut the circle into 360 parts because the Babylonians counted in sixties. Radians measure an angle by the arc it cuts: the angle whose arc is as long as the radius is one radian, and an angle of t radians cuts an arc of length t from the unit circle. Section 2 lines the two units up. A quarter turn is 90° and π/2 radians, a half turn is 180° and π, a whole turn is 360° and 2π, and one radian is 57.2958°, a little under 60. It is not a round number of degrees because 2π is not a round number.

The angle is the distance walked¶

Section 3 checks the definition by walking. From (1, 0) to the point at angle t it takes n straight steps along the circle and adds up their lengths. With one step the walk is a single chord, too short; with a thousand it is 1.000000 for t = 1, 1.570796 for t = π/2 and 3.141591 for t = π. The walk settles on t itself: in radians, the point at angle t has walked a distance t. That sentence is the only thing radians add, and everything else on this page follows from it.

The velocity of a turning point¶

Now let a point turn: at time t it is at angle t, position (cos t, sin t). Section 4 measures its velocity the way the derivative is a velocity does, and gets (−sin t, cos t) at every t. That is the position turned a quarter turn, at right angles to it (the dot product is 0) and of length 1. Read one coordinate at a time, it is the two derivatives every calculus book lists:

d/dt cos t  =  −sin t          d/dt sin t  =  cos t          t in radians

They hold because the point walks a distance t in time t, so its speed is 1. Measure the same turning in degrees, with the point at angle d degrees at time d, and it walks only π/180 of a unit per degree: section 4 measures a speed of 0.017453, which is π/180, and in degrees every derivative of sin and cos carries that factor. Radians are the unit that makes it 1.

Section 5 shows the same thing in one line: for a small angle x, sin x / x is almost exactly 1 in radians, because a small arc is almost straight and its height is almost its length; in degrees the ratio is π/180 = 0.017453 however small x gets.

This is the step Euler's identity needs. The motion e^{it} has velocity i · e^{it}, the position turned a quarter turn, of length 1: it is this page's turning point. After a time t it has walked t round the circle, so it is at angle t radians, (cos t, sin t); and after a time π it has walked half the circle, to −1. In degrees the identity would read e^{i·180°} = −1 with a conversion factor hidden inside the e, which is why nobody writes it that way.

Run it yourself¶

From the root of your clone of this repository:

python3 09_Calculus/radians/examples/radians.py

Po polsku, w skrócie¶

Kąt w radianach to długość łuku, który ten kąt wycina z okręgu o promieniu 1. Program mierzy okrąg wielokątami Archimedesa, bez użycia π: sześciokąt ma obwód 6, a każde podwojenie liczby boków przybliża obwód do 2π = 6,283185. Pół okręgu ma długość π, ćwierć π/2, a jeden radian to kąt, którego łuk jest równy promieniowi, około 57,3°.

Stąd jedyne, co radiany wnoszą: punkt pod kątem t przeszedł po okręgu drogę t. Jeśli punkt w chwili t jest pod kątem t, to porusza się z prędkością 1, a jego prędkość (−sin t, cos t) to położenie obrócone o 90°. Dlatego pochodna sin t to cos t tylko w radianach; w stopniach każda pochodna ma dodatkowy czynnik π/180. I dlatego w tożsamości Eulera jest π, a nie 180: ruch e^{it} to właśnie taki obracający się punkt, a po czasie π przeszedł pół okręgu i jest w −1.

Auf Deutsch: Stichwörter¶

Im Bogenmaß ist der Winkel die auf dem Einheitskreis zurückgelegte Strecke, und nur dann gilt (sin t)′ = cos t.

Stichwörter: Bogenmaß (Radiant), Gradmaß, Einheitskreis, Bogenlänge, Umfang 2π, Ableitung des Sinus, Umrechnung.