Gold's Logatrigmic Formula

It is planned to be part of SY3,an upcoming A2 Symbolic Mathematics module.

Gold's Logatrigmic Formula or just Martin's Formula finds the missing link between logarithms and trigonometry for finding powers of sin represented as sin sin(x) and sin sin sin(x) and is mainly used to find more complex ones, than can be found on a calculator but works for the simpler functions too like sin^1.5.It is important to note that the use of the formula for calculating sines was later replaced by the Complex Sine Theorem.

The formula says that if we want Z= Trig Trig ....x...Trig Trig (y), then we can find this by computing Z is approximately d^((x-INT(x))+log(Trig Trig ....INT(x)...Trig Trig (y))/log(d)) with d= log(Trig Trig ....INT(x+1)...Trig Trig (y))/log(Trig Trig ....INT(x)...Trig Trig (y)).

Trig throughout the whole equation represents one of cos, tan or sin.INT represents the integer part of a number.&nbsp

One of the beauties of the equation is that it works the same for radians, gradians and degrees but be careful the value of Z will still be different each time because you calculating a different function each time but the formula doesn’t have to be rectified for angles measured in different ways.

The formula can be extended to encompass most similar functions. The formula can results in close approximations of roots. To do this, instead of finding x amounts of a function, the outputs of the functions of two close inputs are used. , with the approximations of the answer getting closer if the INT is removed and replaced, to 1dp, 2dp, making the approximation better and better but it will never be 100% correct. The integer approximation of the square root of 35.5, using the formula is roughly 0.00005% too small and the 1dp approximation of the square root 35.5, using the formula is roughly 0.000002% too small.

Instead of Logarithms,the formula uses Logatrigims-the TLog,calculator notation of (sine 1 radian)=1.42730424058 .The TLog 1=0 Tlog (2 radians)=-0.87937149060.Tlog(x)=((log x/(log (sinx/x))-this works for all measuremeants although the Tlog of (2 radians) isn't the same as the Tlog of (2 degrees).The examples represented are in respect ot the Sine Function, to use another function change sinx to f(x).