Editing Trigonometric tables (section)

== A quick, but inaccurate, approximation ==

A quick, but inaccurate, algorithm for calculating a table of ''N'' approximations ''s''<sub>''n''</sub> for [[sine|sin]](2[[Pi|&pi;]]''n''/''N'') and ''c''<sub>''n''</sub> for [[cosine|cos]](2π''n''/''N'') is:

:''s''<sub>0</sub> = 0
:''c''<sub>0</sub> = 1
:''s''<sub>''n''+1</sub> = ''s''<sub>''n''</sub> + ''d'' &times; ''c''<sub>''n''</sub>
:''c''<sub>''n''+1</sub> = ''c''<sub>''n''</sub> &minus; ''d'' &times; ''s''<sub>''n''</sub>
for ''n'' = 0,...,''N''&nbsp;&minus;&nbsp;1, where ''d'' = 2π/''N''.

This is simply the [[Numerical ordinary differential equations#Euler method|Euler method]] for integrating the [[differential equation]]:

:<math>ds/dt = c</math>
:<math>dc/dt = -s</math>

with initial conditions ''s''(0) = 0 and ''c''(0) = 1, whose analytical solution is ''s'' = sin(''t'') and ''c'' = cos(''t'').

Unfortunately, this is not a useful algorithm for generating sine tables because it has a significant error, proportional to 1/''N''.

For example, for ''N'' = 256 the maximum error in the sine values is ~0.061 (''s''<sub>202</sub> = &minus;1.0368 instead of &minus;0.9757). For ''N'' = 1024, the maximum error in the sine values is ~0.015 (''s''<sub>803</sub> = &minus;0.99321 instead of &minus;0.97832), about 4 times smaller. If the sine and cosine values obtained were to be plotted, this algorithm would draw a [[logarithmic spiral]] rather than a circle.