Tangents and 18/pi

Main Question or Discussion Point

In degrees, look at what the result is when you take tan(89), tan(89.9), tan(89.99) etc.

The value as the number of 9's gets larger converges to 18/pi, with a shifted decimal point.

Why?

Simon Bridge
Homework Helper
Observe:
Code:
octave:10> t
t =

89.900
89.990
89.999
90.000
90.000
90.000
90.000
90.000
90.000
90.000
90.000
90.000
90.000
90.000
90.000

octave:11> tan(t*pi/180)
ans =

5.7296e+02
5.7296e+03
5.7296e+04
5.7296e+05
5.7296e+06
5.7296e+07
5.7296e+08
5.7296e+09
5.7295e+10
5.7296e+11
5.7278e+12
5.7535e+13
5.4419e+14
3.5301e+15
1.6332e+16
18/pi is 5.72957795130823

To 4dp that 5.6296 ... but notice that the calculation departs from the pattern at the e+10 stage?
(That is when theta is 89.999999999 ... 9 "9"'s in a row.)

Lets try it without rounding up:
Code:
octave:12> format long

octave:14> tan(t*pi/180)
ans =

5.72957213354303e+02
5.72957789312165e+03
5.72957795072129e+04
5.72957795104345e+05
5.72957794122192e+06
5.72957798144568e+07
5.72957787343207e+08
5.72957898008453e+09
5.72953173262481e+10
5.72956950454798e+11
5.72776101541460e+12
5.75350505624601e+13
5.44191874731457e+14
3.53013952228678e+15
1.63317787283838e+16
What you are seeing is that the tangent function behave a bit like an exponential function for values close to 89.999999deg ... just like it behaves like y=x for values close to 0. What you are not seeing is any convergence to a pattern - far from it, continuing the calculation shows divergence as the argument approaches 90deg.

lurflurf
Homework Helper
We have the identity

tan(90-x)=1/tan(x)

it is better to work in radians so that

tan(pi/2-x)=1/tan(x)

when x is small we have (~ mean approximately)
tan(x)~x
so
tan(pi/2-x)=1/tan(x)~1/x

back in degrees

tan(90-x)=1/tan(x)~180/(pi x)

as you observed