# Why are logarithms of trigonometric functions useful?

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?

• Historically log of values were important because it enabled rapid(er) manual calculation of products. Not just log of trig values, but log of many standard functions. Honestly I'm glad those days are behind us. (Cue heart-warming, tedious stories about 'how back in the day we used a slide rule to calculate the trajectory of Apollo rockets ...' in 3, 2, 1 ...) – Simon S Apr 10 '15 at 0:18
• if you look deeper , the curve of periodic and some polynomials and logarithmic fts are similar at certain ranges , analyst researchers use to switch between these three functions to have approximates of specific values without limited developements – Abdou Abdou Apr 10 '15 at 0:22
• Let's say I want to find $\sin(a+b)$, for a certain $a$ and $b$. The formula $\sin a\cos b+\cos a\sin b$ is hard to do by hand, since it involves two multiplications. However, the equivalent formula $\exp(\ln\sin a+\ln\cos b)+\exp(\ln\cos a+\ln\sin b)$ involves only additions, as well as simply looking up $\ln\sin$, $\ln\cos$, and $\exp$. (Since $\exp$ is the inverse of $\ln$, you just need a $\ln$ table and to do a reverse lookup.) That's why $\ln\sin$ is important. (Note that the log doesn't necessarily have to be base $e$; I think a lot of log tables are base 10.) – Akiva Weinberger Apr 23 '15 at 14:02
• That's probably a bad example, 'cause you can just add $a$ and $b$ and take the sine. But I hope you get my point. – Akiva Weinberger Apr 23 '15 at 14:06

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.

• @JohnDoe: Again, thanks a lot for granting the bounty and accepting the answer! :-) – Markus Scheuer Apr 26 '15 at 20:29