4 APPLIED MECHANICS 
Logarithms.—If a®=N, then # is the logarithm of N to the base a. 
In the common system of logarithms the base is 10. In the Napierian 
system of logarithms the base is e= 2°71828. . . . Napierian logarithms 
are also called natwral and also hyperbolic logarithms. 
log (A x Bx C) =log A+log B+ log C. 
log }}=log A —log B. log A” =n log A. log hws log A. 
aa = _ log b 
If a*=0, then x log a=log b, and oS loa ; 
The foregoing rules are true whatever be the system of logarithms 
used. 
If ~=log,m, and y elegy: then y= log,a = log : 
x 
logy @ 
log, 10 = 2°3026 nearly, and log,,¢ = 0°4343 nearly. 
If x=log,,m, and y=log.m, then y= log,10 = 
6. Trigonometrical Formule.— 
cosec A = pany ee sec A = fore 
sin A 1 cos A ti 
tan A= A eta. cotA = A tanA’ 
sin? A +cos? A= 1. 
sec? A=1+ tan? A. cosec? A = 1 + cot? A. 
sin (A +B) =sin A cos B + cos A sin B. 
sin (A — B) =sin A cos B -- cos A sin B. 
cos (A + B)=cos A cos B — sin A sin B. 
cos (A — B) =cos A cos B+ sin A sin B, 
_ tan A+tan B 
tan (A+B)=i~4tan A tan B’ 
tan A -—tan B 
tan (A-B)=Tyitan A tan B’ 
sin 2A =2 sin A cos A. 
cos 2A =cos? A — sin? A =2 cos? A—~1=1 —-2 sin? A, 
2 tan A 
1-tan? A’ 
2 tan A oA = La tan? A 
1+tan? A’ 08 ae TF ten AL 
sin 3A=3 sin A—4 sin? A. 
cos 3A = 4 cos? A — 3 cos A. 
3tanA—tan? A 
tan3A= 1-3 tan? A ‘ 
. 
tan 2A = 
sin 2A = 
