Formula sheet for electrical engineering PDF

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Formula sheet for electrical engineering...

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2016

HIGHER SCHOOL CERTIFICATE EXAMINATION

REFERENCE SHEET

– Mathematics – – Mathematics Extension 1 – – Mathematics Extension 2 –

Mathematics Factorisation

Distance between two points

( )( ) = ( a + b )(a 2 − ab + b 2 ) = ( a − b ) (a 2 + ab + b 2 )

a2 − b2 = a + b a − b a 3 + b3 a 3 − b3

( x2 − x1 )2 + ( y2 − y1) 2

d=

Perpendicular distance of a point from a line

ax1 + by1 + c

d=

a2 + b 2

Angle sum of a polygon

S = ( n − 2) × 180°

Slope (gradient) of a line

y2 − y1 x 2 − x1

m= Equation of a circle

( x − h)2 + ( y − k) 2 = r 2

Point-gradient form of the equation of a line

y − y1 = m( x − x1 ) Trigonometric ratios and identities

sin θ =

opposite side hypotenuse

cos θ =

adjacent side hypotenuse

tanθ =

opposite side adjacent side

1 sin θ 1 secθ = cosθ sin θ tan θ = cosθ cosθ cot θ = sin θ cosec θ =

sin2 θ + cos 2 θ = 1 Exact ratios

45°

Tn = a + ( n − 1)d Sum to n terms of an arithmetic series

Sn =

n [ 2a + (n − 1) d ] or 2

2

n (a + l) 2

nth term of a geometric series

Tn = ar n−1

3

Sn =

(

a rn − 1 r −1

)

or

Sn =

(

a 1−rn 1−r

1

45°

60° 1

1

Limiting sum of a geometric series

S= Sine rule

a 1−r

a b c = = sin A sin B sinC

Compound interest

Cosine rule

r ⎞n An = P ⎛ 1 + ⎝ 100 ⎠

c2 = a 2 + b 2 − 2ab cosC Area of a triangle

Area =

Sn =

Sum to n terms of a geometric series

30° 2

nth term of an arithmetic series

1 ab sin C 2 –2–

)

Mathematics (continued) Differentiation from first principles

Integrals

ƒ ′( x ) = lim

⌠ (ax + b )n+1 n dx = +C ⎮ ( ax + b ) a ( n + 1) ⌡

Derivatives

⌠ ax+b 1 dx = eax+b + C ⎮e a ⌡

ƒ (x + h) − ƒ (x ) h→0 h

If y = x n , then

dy = nx n−1 dx

If y = uv , then

dy dv du =u +v dx dx dx

If y =

dy u , then = v dx

v

du dv −u dx dx 2 v

If y = F (u ) , then

dy du = F ′( u ) dx dx

If y = e ƒ ( x ), then

dy = ƒ ′ (x ) e ƒ ( x ) dx

⌠ ƒ ′( x ) ⎮ ƒ ( x ) dx = ln ƒ ( x ) + C ⌡ ⌠ 1 ⎮ sin( ax + b) dx = − a cos (ax + b ) + C ⌡ ⌠ 1 ⎮ cos (ax + b ) dx = a sin ( ax + b ) + C ⌡ ⌠ 1 2 ⎮ sec (ax + b )dx = a tan ( ax + b ) + C ⌡ Trapezoidal rule (one application)

dy ƒ ′( x ) If y = log e ƒ (x ) = ln ƒ (x ) , then = dx ƒ (x ) If y = sin ƒ ( x ) , then

dy = ƒ ′( x) cos ƒ ( x ) dx

If y = cos ƒ ( x ) , then

dy = − ƒ ′ ( x ) sin ƒ ( x ) dx

If y = tan ƒ ( x ) , then

dy = ƒ ′( x ) sec2 ƒ ( x ) dx

b

⌠ b−a ⎮ ƒ ( x ) dx ≈ 2 ⎡⎣ ƒ (a ) + ƒ ( b )⎤⎦ ⌡a Simpson’s rule (one application) b

⌠ b−a⎡ ⎤ ⎛ a + b⎞ ⎮ ƒ ( x ) dx ≈ 6 ⎢ ƒ (a ) + 4 ƒ ⎝ 2 ⎠ + ƒ ( b ) ⎥ ⎦ ⎣ ⌡a

Logarithms – change of base Solution of a quadratic equation

x=

loga x =

logb x logb a

−b ± b 2 − 4ac 2a Angle measure

Sum and product of roots of a quadratic equation

α+β=−

b a

αβ =

c a

180° = p radians Length of an arc

l = rq Equation of a parabola

( x − h )2 = ± 4a( y− k)

Area of a sector

1 Area = r 2q 2 –3–

Mathematics Extension 1 Angle sum identities

Acceleration

sin (θ + f ) = sinθ cos f + cosθ sin f

d 2x dv d ⎛ 1 2⎞ dv = v = v 2 = dt dt dx dx ⎝ 2 ⎠

cos (θ + f) = cosθ cos f − sin θ sin f tanθ + tan f 1− tan θ tan f

tan (θ + f ) =

Simple harmonic motion

(

x = b + a cos nt + α

t formulae

(

x = −n 2 x − b !!

θ If t = tan , then 2 s inθ =

2t 1+ t 2

)

)

Further integrals

⌠ x 1 dx = sin −1 + C ⎮ 2 a ⌡ a − x2

1− t 2 1+ t 2 2t t anθ = 1− t 2 cos θ =

⌠ 1 x 1 dx = tan −1 + C ⎮ 2 2 a a ⌡a +x

General solution of trigonometric equations

sin θ = a ,

θ = n π + ( −1) n sin−1 a

cos θ = a,

θ = 2n π ± cos−1 a

Sum and product of roots of a cubic equation

tan θ = a,

θ = n π + tan−1 a

α + β+ γ = −

b a

αβ + αγ + βγ = Division of an interval in a given ratio

αβγ = −

⎛ m x 2 + nx1 my2 + ny1 ⎞ ⎜⎝ m + n , m + n ⎟⎠

c a

d a

Estimation of roots of a polynomial equation

Newton’s method

Parametric representation of a parabola

For x 2 = 4ay, x = 2at,

(

x 2 = x1 −

y = at2

ƒ ( x1 ) ƒ ′( x1 )

)

At 2at , at 2 , tangent: y = tx − at2

Binomial theorem

3

normal: x + ty = at + 2at

(

At ( x1 , y1 ) ,

a+b

)

n

n

=

⎛ n⎞ ∑ ⎜⎝ k ⎟⎠ a k bn−k = k =0

tangent: xx1 = 2a( y+ y1 ) 2a normal: y − y 1 = − ( x − x1) x1

(

)

(

Chord of contact from x0 , y0 : x x0 = 2a y + y0

) –4–

© 2015 Board of Studies, Teaching and Educational Standards NSW

n

⎛ n⎞

∑ ⎜⎝ k ⎟⎠ an−k b k k =0...