Higher Engineering Mathematics

Ess-For-H8152.tex 19/7/2006 18: 2 Page 707

ESSENTIAL FORMULAE 707

Triangle formulae:

With reference to Fig. FA2:

Sine rule

a

sinA

=

b

sinB

=

c

sinC

Cosine rule a^2 =b^2 +c^2 − 2 bccosA

Figure FA2

Area of any triangle

(i)^12 ×base×perpendicular height

(ii)^12 absinCor^12 acsinBor^12 bcsinA

(iii)

√

[s(s−a)(s−b)(s−c)] wheres=

a+b+c

2

Compound angle formulae

sin (A±B)=sinAcosB±cosAsinB

cos(A±B)=cosAcosB∓sinAsinB

tan (A±B)=

tanA±tanB

1 ∓tanAtanB

IfRsin(ωt+α)=asinωt+bcosωt,

then a=Rcosα, b=Rsinα,

R=

√

(a^2 +b^2 ) andα=tan−^1

b

a

Double angles

sin 2A=2 sinAcosA

cos 2A=cos^2 A−sin^2 A=2 cos^2 A− 1

= 1 −2 sin^2 A

tan 2A=

2 tanA

1 −tan^2 A

Products of sines and cosines into sums or differences

sinAcosB=^12 [sin(A+B)+sin (A−B)]

cosAsinB=^12 [sin(A+B)−sin (A−B)]

cosAcosB=^12 [cos(A+B)+cos (A−B)]

sinAsinB=−^12 [cos(A+B)−cos (A−B)]

Sums or differences of sines and cosines into products

sinx+siny=2 sin

(

x+y

2

)

cos

(

x−y

2

)

sinx−siny=2 cos

(

x+y

2

)

sin

(

x−y

2

)

cosx+cosy=2 cos

(

x+y

2

)

cos

(

x−y

2

)

cosx−cosy=−2 sin

(

x+y

2

)

sin

(

x−y

2

)

For ageneral sinusoidal function

y=Asin(ωt±α), then

A=amplitude

ω=angular velocity= 2 πfrad/s

2 π

ω

=periodic timeTseconds

ω

2 π

=frequency,fhertz

α=angle of lead or lag (compared with

y=Asinωt)

Cartesian and polar co-ordinates

If co-ordinate (x,y)=(r,θ) thenr=

√

x^2 +y^2 and

θ=tan−^1

y

x

If co-ordinate (r,θ)=(x,y) thenx=rcosθ and

y=rsinθ.

The circle

With reference to Fig. FA3.

Area=πr^2 Circumference= 2 πr

πradians= 180 ◦