Math note 4
Suppose you want to find the displacement at x = 21 m and t = 0 s for a
wave which is described by
y(x,t) = 2.0 sin(5.4 x -2.1 t) .
Here y and x are in meters and t in seconds. The displacement is
y(21,0) = (2.0 m) sin(113.4 rad) = 0.60 m.
Notice what happen when you round off the number 113.4 to 113, which might
make sense since you know the position of x only to two significant digits.
y(21,0) = (2.0 m) sin(113. rad) = - 0.19 m.
The two answers differ significantly from each other.
The reason is that since the value of the sine function repeats itself when
its argument is changed by an integer multiple of 2p,
the fractional part of its argument might be very important. So, do not
round off the argument of a trigonometric function but if you have to
do so, the correct way is, first, subtract a multiple number of 2p until
the argument becomes the closest possible value to zero, then, round it off.
For example, in 113.4 there are 18.05 2ps.
Subtracting 18(2p) from 113.4 leads to 113.4
-18(2p) = 0.3027.
Now, we can round 0.3037 to two significant digits, 0.3037
y(21,0)=(2.0 m) sin(113.4 rad) ≈
sin(0.30 rad) = 0.59 m,
which is very close to the correct answer 0.60 m.
Rounding off the argument of a trigonometric function