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.