[Math] Why are logarithms of trigonometric functions useful

Question

I have noticed that in many trigonometric tables the logarithm of the trigonometric values are given.

Why this is given and not the actual values of the trigonometric functions? For example, instead of listing the value of $\sin(43^\circ)$, it is the value of $\log(\sin(43^\circ))$ that is listed.

The only reason I cam come up with is that in the use of the law of $\sin$:
$$
\frac{\sin(A)}{a} = \frac{\sin(B)}{b}
$$
taking logarithms on both sides one gets
$$
\log(\sin(A)) – \log(a) = \log(\sin(B)) – \log(b)
$$
making it easier to use that specific formula.

But this wouldn't work with the law of cosine.

Are there other reasons to prefer logarithms of trigonometric functions over just the actual values of the trigonometric functions?

Answer 1

Short answer:

The main reason is the simplification of reducing multiplication and division to addition and subtraction.

Historical aspects:

One application which is heavily based upon trigonometric formulas is Spherical Geometry. This realm e.g. important for astronomy and geodesy used logarithmic tables of trigonometric functions right from the beginning since logarithms have been published in $1614$ by John Napier.

If we take a look e.g. at the Cosine Rules \begin{align*} \cos a &= \cos b \cos c + \sin b\sin c \cos A\\ \cos b &= \cos c \cos a + \sin c\sin a \cos B\\ \cos c &= \cos a \cos b + \sin a\sin b \cos C\\ \end{align*} or one of Napier's analogies \begin{align*} \tan \frac{1}{2}(A+B)=\frac{\cos \frac{1}{2}(a-b)}{\cos \frac{1}{2}(a+b)}\cot \frac{1}{2}C \end{align*}

we can get a glimpse of the amount of time which could be saved by calculating these multiplications and divisions with the help of logarithms.

In fact, the first presentation of logarithms in John Napiers famous book Mirifici Logarithmorum Canonis Descriptio / The Description of the Wonderful Canon of Logarithm already contains calculation of logarithms of sines and tangent values.

Napier wrote in the preface of his Mirifici Logarithmorum:

Since nothing is more tedious, fellow mathematicians, in the practice of the mathematical arts, than the great delays suffered in the tedium of lengthy multiplications and divisions, the finding of ratios, and in the extraction of square and cube roots– and in which not only is there the time delay to be considered, but also the annoyance of the many slippery errors that can arise ...

(ending the preface with)

ON LOGARITHMS. By which all the sines, tangents, and secants, are set out for you from great labour and prolixity; And which this little table of Logarithms, gentle reader, Gives to you all at once, without great labour.