Wondering if there is a way to translate some old-style trigonometric calculations that used tables and notation like log. cos. or arith. comp. of log. cos. to something that can be done on a computer, and without tables?
To put it simpler, does anybody know an algorithm to calculate:
log.cos. θarith. comp. of log. cos. θ
as in this screenshot
taken from the book Plane Trigonometry and Mensuration, for the use of the Royal Military College
Base 10 logarithms in the old days were expressed as $-10$ to avoid negative numbers, and it made calculations from tables easier. $\log \sin 49°23'$ is represented as $9.880289$, but it really meant $9.880289-10$ (the actual value is $-0.1191711$).
The "arith. comp." (arithmetic complement) is also known as the cologarithm ("colog"), and is the reciprocal of the log function $\left (\log \dfrac {1} {x} = -\log x\right)$.
If you wanted to do it on a computer, you could define functions as follows...
$\text {function (OldLogSin(x))} := \log(\sin(x))+10$
$\text {function (OldLogCos(x))} := \log(\cos(x))+10$
$\text {function (Colog(x))} := -\log(x)$
If your computer uses natural (base $e$) logarithms, use $\dfrac {\ln x}{\ln 10}$ to get a base $10$ logarithm; if your computer uses radians, multiply $x$ by the factor $\dfrac {180}{\pi}$ to get degrees.
$\text {function (OldLogSin(x))} := \dfrac {\left[\ln\left(\sin \dfrac {180 \cdot x}{\pi}\right)\right]}{\ln 10}+10$
$\text {function (OldLogCos(x))} := \dfrac {\left[\ln\left(\cos \dfrac {180 \cdot x}{\pi}\right)\right]}{\ln 10}+10$