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MATHEMATICS SPECIALIST
ATAR COURSE
FORMULA SHEET
2016
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2016/1834[v2]
Mathematics Specialist Formula Sheet 2016
Ref: 16-054
MATHEMATICS SPECIALIST
2
Formula SHEET
Index
Vectors
3
Trigonometry
4
5–6
Functions
Complex numbers
6
Exponentials and logarithms
7
Mathematical reasoning
7
Measurement
8
Chance and data
8
See next page
Formula SHEET
3
MATHEMATICS SPECIALIST
Vectors
Magnitude:
(a1, a2, a3) = √ a12 + a22 + a32
Dot product:
a·b = a b cosθ = a1b1 + a2b2 + a3b3
Triangle inequality:
a+b≤ a+b
Vector equation of a line in space: one point and the slope:
two points A and B:
Cartesian equations of a line in space:
x – a1
y – a2
z – a3
=
=
b1
b2
b3
Parametric form of vector equation of
a line in space:
r = a + λb
r = a + λ(b − a)
x = a1 + λb1......(1)
y = a2 + λb2......(2)
z = a3 + λb3......(3)
Vector equation of a plane in space:
r•n = c
Cartesian equation of a plane:
ax + by + cz = d
Vector cross product:
a × b = a b sinθ n where
a = the magnitude of vector a
or
r = a + λb + µc
b = the magnitude of vector b
θ is the angle between a and b and
n is the unit vector perpendicular to vectors a and b
ax
bx
Given a = ay and b = by then
az
bz
ay bz – az by
ax
bx
a × b = ay × by = az bx – ax bz
ax by – ay bx
az
bz
See next page
MATHEMATICS SPECIALIST
4
Trigonometry
In any triangle ABC:
Formula SHEET
a
b
c
sin A = sin B = sin C
a2 = b 2 + c 2 − 2bc cosA
b2 + c2 – a2
cosA =
2bc
1
A = 2 ab sin C
In a circle of radius r, for an arc subtending angle θ (radians) at the centre:
Length of arc
= rθ
Area of segment =
Identities:
1 r 2 (θ − sinθ)
2
Area of sector
1
= 2 r2 θ
cos 2θ = cos 2 θ – sin 2θ
cos2 θ + sin2 θ = 1
= 2cos 2 θ – 1
1 + tan2 θ = sec2 θ
= 1 – 2sin2 θ
–
+
cos (θ +
– φ) = cosθ cosφ sinθ sinφ
sin (θ +
– φ) = sinθ cos φ +
– cosθ sinφ
tanθ +– tanφ
tan (θ +– φ) = 1 +– tanθ tan φ
sin2θ = 2sinθ cosθ
2tanθ
tan 2θ = 1 – tan 2 θ
lim
x→0
sin x
=1
x
lim
x→0
1 − cos x
=0
x
d2 x
Simple Harmonic Motion: If dt 2 = −k 2x then x = Asin(kt +α) or x = Acos(kt +β) and
v 2 = k 2 (A2 − x2), where A is the amplitude of the motion, α and β are phase angles, v is the velocity
and x is the displacement.
See next page
Formula SHEET
Functions
Differentiation:
5
If f(x) = y then f '(x) =
MATHEMATICS SPECIALIST
dy
dx
If f(x) = x n then f '(x) = nx n–1
1
If f(x) = e x then f '(x) = e x
If f(x) = ln x then f '(x) = x
If f(x) = sin x then f '(x) = cos x
If f(x) = cos x then f '(x) = −sin x
1
If f(x) = tan x then f '(x) = sec2 x = cos2 x
Product rule:
If y = f (x) g(x)
or
then y' = f '(x) g(x) + f(x) g'(x)
Quotient rule:
Incremental formula:
δy ~
dv
dy du
=
v
+
u
dx
dx dx
u
f '(x) g(x) – f(x) g'(x)
(g(x))2
dy
δx
dx
or
or
then y' = f '(g(x)) g'(x)
If y = v
dy
dx =
then
or
If y = f (g(x))
Chain rule:
then
f(x)
If y = g(x)
then y' =
If y = uv
dv
du
v–u
dx
dx
2
v
f (x + h) − f (x) ~ f '(x)h
If y = f (u) and u = g(x)
then
dy
dy
du
=
×
dx
dx
du
Integration:
Powers:
x n+1
∫ x ndx = n +1 + c, n = −1
1
Logarithms: ∫ x dx = ln x + c
Exponentials:
∫ e xdx = e x + c
Trigonometric:
∫ sin x dx = −cos x + c
∫ cos x dx = sin x + c
∫ sec2 x dx = tan x + c
Fundamental Theorem of Calculus:
d x
dx ∫ a f(t) dt = f (x)
and
∫ a f'(x) d x = f (b) − f (a)
b
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MATHEMATICS SPECIALIST
Functions
Quadratic function:
6
Formula SHEET
−b ± √ b2 −4ac
for x
If y = ax + bx + c and y = 0, then x =
2a
2
Absolute value function:
C
x, for x ≥ 0
x = –x, for x < 0
Complex numbers
For z = a + ib, where i 2 = –1
Argument: b
arg z = θ, where tan θ = a and −π < θ ≤ π
Modulus: mod z = z = a + ib = √a2 + b2 = r
Product: z1 z 2 = z1 z 2
arg (z1 z2) = arg z1 + arg z2
Quotient:
z 1 z1
z2 = z2
z
arg z1 = arg z1 − arg z2
2
Polar form:
For z = r cisθ, where r = |z| and θ = arg z:
cis(θ + φ) = cis θ cis φ
1
cis(−θ) = cis θ
z1z2 = r1r2 cis (θ + φ)
cis θ = cos θ + i sin θ
cis(0) = 1
z1 r1
z2 = r2 cis (θ – φ)
For complex conjugates:
z = a + bi
z = a − bi
z = r cis θ
z = r cis (–θ)
1 z
z−1= = 2
z | z|
z1 z2 = z 1 z2
z z = |z|2
z1 + z2 = z 1 + z 2
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Formula SHEET
7
Exponentials and logarithms
For a, b > 0 and m, n real:
MATHEMATICS SPECIALIST
am
m–n
an = a
1
a –n = an
(ab)m = am bm
am an = a m+n
a0 = 1
n
(am ) = a mn
For m an integer and n a positive integer:
1
an = √a m = ( √a )
m
n
an = √a
n
n
m
For a, b, y, m and n positive real and k real:
y = ax
loga y = x
logamn = logam + logan
a = a1
loga a = 1
log bm
logam = log a
b
loga(mk) = k loga m
1= a0
loga 1 = 0
(change of base)
dP
If dt = kP, then P = P0 e kt
Mathematical reasoning
De Moivre's theorem:
(cis θ)n = (cos θ + isin θ)n
(cis θ)n = cos nθ + isin nθ
z n = z n cis (nθ)
1
q
1
q
z = z cos
θ + 2πk
θ + 2πk
+ isin
for k an integer
q
q
See next page
MATHEMATICS SPECIALIST
8
Formula SHEET
Measurement
Circle:
C = 2πr = πD, where C is the circumference, r is the radius and
D is the diameter
A = πr 2, where A is the area
1
Triangle: A = 2 bh, where b is the base and h is the perpendicular height
Parallelogram:
A = bh
Trapezium:
1
A = 2 (a + b)h, where a and b are the lengths of the parallel sides
Prism:
V = Ah, where V is the volume and A is the area of the base
Pyramid: V = 1 Ah
3
Cylinder: S = 2 πrh + 2 πr 2, where S is the total surface area
V = πr 2h
S = πrs + πr 2, where s is the slant height
1
V = 3 πr 2h
Cone:
Sphere: S = 4πr 2
V =
4 3
3 πr
Chance and Data
A confidence interval for the mean of a population is:
σ
σ
X – z √n ≤ μ ≤ X + z
√n
where μ is the population mean,
σ is the population standard deviation,
X is the sample mean,
n is the sample size,
z is the cut off value on the standard normal distribution corresponding to the confidence level.
Sample size: n =
( z ×d σ ) where d is the required value of the difference from the mean.
2
Note: Any additional formulas identified by the examination panel as necessary will be
included in the body of the particular question.