|
Below we inspect the approximation
sin(α) ≅ α
for
α (in radians) << 1
The table gives α
in degrees and in radians,
sin(α)
and the relative error, given by
( α - sin(α) ) / α
|
angle (degrees)
|
angle (radians)
|
sin(angle)
|
(angle-sin(angle))/angle
|
|
0
|
0
|
0
|
0
|
|
1
|
0.017453
|
0.017452
|
0.00005
|
|
2
|
0.034906
|
0.034899
|
0.0002
|
|
3
|
0.05236
|
0.05234
|
0.0005
|
|
4
|
0.06981
|
0.06976
|
0.0008
|
|
5
|
0.08727
|
0.08716
|
0.001
|
|
6
|
0.1047
|
0.1045
|
0.002
|
|
7
|
0.1222
|
0.1219
|
0.003
|
|
8
|
0.1396
|
0.1392
|
0.003
|
|
9
|
0.1571
|
0.1564
|
0.004
|
|
10
|
0.1745
|
0.1736
|
0.005
|
|
11
|
0.1920
|
0.1908
|
0.006
|
|
12
|
0.209
|
0.208
|
0.007
|
|
13
|
0.227
|
0.225
|
0.009
|
|
14
|
0.244
|
0.242
|
0.01
|
|
15
|
0.262
|
0.259
|
0.01
|
|
16
|
0.279
|
0.276
|
0.01
|
|
17
|
0.297
|
0.292
|
0.01
|
|
18
|
0.314
|
0.309
|
0.02
|
|
19
|
0.332
|
0.326
|
0.02
|
|
20
|
0.349
|
0.342
|
0.02
|
|
25
|
0.436
|
0.423
|
0.03
|
|
30
|
0.524
|
0.500
|
0.05
|
|
35
|
0.611
|
0.574
|
0.06
|
|
40
|
0.698
|
0.643
|
0.08
|
From the table we conclude that
sin(α)≅α
is a good approximation (with errors of at most 0.5%) for
α<100
and a reasonably good approximation (with errors of at most 2%) for
α<200.
This is sufficient for most practical applications of the pendulum.
|
|