0195136047.pdf

MATHEMATICAL RELATIONS 839

4 cos^3 (x)=3 cos(x)+cos( 3 x)

8 sin^4 (x)= 3 −4 cos( 2 x)+cos( 4 x)

8 cos^4 (x)= 3 +4 cos( 2 x)+cos( 4 x)

Acos(x)−Bsin(x)=Rcos(x+θ)

where, R=

√

A^2 +B^2

θ=tan−^1

B

A

A=Rcosθ

B=Rsinθ

Ifx 1 , sinx∼=x;cosx∼= 1 −

x^2

2

D.2 EXPONENTIAL AND LOGARITHMIC FUNCTIONS

e±jx=cosx±j sinx

exey=e(x+y)

ex/ey=e(x−y)

logxy=logx+logy

log

x

y

=logx−logy

logxn=nlogx

logax=logbx/logba=(logbx)(logab)

Base of natural (Naperian or hyperbolic) logarithms:e∼= 2. 71828

lnx=logex=(loge 10 )(log 10 x)∼= 2 .3026 log 10 x

log 10 x=(log 10 e)(logex)= 0 .4343 loge x= 0 .4343 lnx

alneb=ab

Ifx 1 ,ex∼= 1 +x;ln( 1 +x)∼=x