Higher Engineering Mathematics, Sixth Edition

Essential formulae 661

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

A

B C

c

a

b

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:

sin2A=2sinAcosA

cos2A=cos^2 A−sin^2 A=2cos^2 A− 1

= 1 −2sin^2 A

tan2A=

2tanA

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=2sin

(

x+y

2

)

cos

(

x−y

2

)

sinx−siny=2cos

(

x+y

2

)

sin

(

x−y

2

)

cosx+cosy=2cos

(

x+y

2

)

cos

(

x−y

2

)

cosx−cosy=−2sin

(

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)then r=

√

x^2 +y^2 and

θ=tan−^1

y

x

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

y=rsinθ.