Advanced Higher Mathematics
New Advanced Higher Mathematics: Formulae
Green (G): Formulae you must memorise in order to pass Advanced Higher maths as
they are not on the formula sheet.
Amber (A): These formulae are given on the formula sheet. But it will still be useful
for you to memorise them.
Red (R): Don’t worry about memorising these, but they might be useful to save time
in classwork and homework.
Trigonometric Identities: (from National 5 and Higher)
Essential Formulae to know
off by heart for the exam (G)
Other useful ones that may
be useful for
homework/classwork etc.
1  tan 2 A  sec 2 A
cot 2 A  1  cosec 2 A
cos 2 A  sin 2 A  1
sin A
tan A 
cos A
2
1
cos x  2 (1  cos 2 x )
Links
between
ratios
Squared
Compound
Angle
Double
Angle
sin 2 x  12 (1  cos 2 x )
sin( A  B )  sin A cos B  cos A sin B
cos( A  B )  cos A cos B  sin A sin B
sin(2 A)  2sin A cos A
tan( A  B ) 
tan(2 A) 
cos(2 A)  cos A  sin A
2
tan A  tan B
1  tan A tan B
2
2 tan A
1  tan 2 A
Exact Values(you should know all these, though there is no non-calculator paper, unlike Higher)
0
sin
0
cos
1
tan
0




6
1
2
4
1
3
3
2
2
3
2
1
1
3
2
2
1

3
2
2
1
0
1
0
0
1
0
1
undef.
0
undef.
0
1
2
3
Negative facts:
sin( )   sin( )
cos( )   cos( )
tan( )   tan( )
Complex Numbers
For the complex number, z  a  bi ,

the modulus is given by z  a 2  b 2

and the argument is given by tan  

The conjugate is z  a  bi
b
a
    
De Moivre’s Theorem says that
for any z  r (cos   i sin  ) , then z n  r n (cos n  i sin n ) ( n   )
Newbattle Community High School
© D Watkins 2015
Advanced Higher Mathematics
Differentiation
v
dv
du
Product Rule: u  v
dx
dx
f ( x)
Quotient Rule:
f '( x)
f ( x)
sec x
cosec x
cot x
1
-1
sin x

cos-1 x
tan -1 x
tan x
1  x2
1
1  x2
1
1  x2
sec2 x
ex
f '( x)
sec x tan x
cosec x cot x
cosec 2 x
f '( x)
f ( x)
ln f ( x)
To differentiate an inverse
dx 1
function:

dy dy
dx
1
x
ex
ln x x  0
du
dv
u
dx
dx
v2
Parametric Equations (where x  f (t ), y  g (t ) ):

dy

Gradient (direction of movement) =
dx

Speed =

d 2 y x 
y  y 
x

2
3
dx
x
dy
dt
dx
dt
   
dy 2
dt
dx 2
dt
Integration
On Formula Sheet
f ( x)
 f ( x )dx
1
tan ax  C
a
x
sin 1    C
a
sec 2 ax
1
a x
1
2
a  x2
2
2
1
x
tan 1    C
a
a
1 ax
e C
a
e ax
To save you time in hard questions for
homework/classwork, no need to memorise:
f ( x)
 f ( x)dx
tanx
ln sec x  C
cosecx
 ln cosec x  cot x  C
cot x
ln sin x  C
sec x
ln sec x  tan x  C
Integration by Parts
dv
 u dx
dx  uv   v
du
dx
dx
Volume of solid of revolution f(x) between a and b:
b
About x axis: V    f ( x)2 dx
a
Newbattle Community High School
b
About y axis: V    f ( y )2 dy
a
© D Watkins 2015
Advanced Higher Mathematics
Sequences and Series
Arithmetic Series
un  a  (n  1)d
th
n term
Sum of
n terms
Geometric Series
un  ar n 1
a (1  r n )
r 1
1 r
a
S 
r 1
1 r
S n  12 n  2a  ( n  1) d 
Sn 
Sum to
infinity
Important Identities
n
1  n
k 1
n
r 
r 1
n(n  1)
2
n
r
2

r 1
n
r
r 1
3

n (n  1)
4
2
n( n  1)(2n  1)
6
2
Maclaurin Series
f ( x)  f (0)  f (0) x 
f (0) 2 f (0) 3
f ( n ) (0) n
x 
x  ... 
x  ...
2!
3!
n!
and in particular:
Very useful to memorise:
x 2 x3
xn
ex  1  x 
  ... 
 ...
2! 3!
n!
x3 x5 x 7
sin x  x  

 ...
3! 5! 7!
x2 x4 x6
cos x  1 


 ...
2! 4! 6!
Less essential to memorise:
x3 x5 x7
 
 ...
3
5
7
x 2 x3 x 4
ln(1  x)  x 
 
 ...
2
3
4
tan 1 x  x 
Functions
Odd function: f ( x)   f ( x)
Even function: f ( x)  f ( x)
(180° rotational symmetry)
(line symmetry about the y-axis)
Newbattle Community High School
© D Watkins 2015
Advanced Higher Mathematics
Binomial Theorem
 n
The coefficient of the rth term in the binomial expansion ( x  y ) n is   xnr y r
r
n
n!
n
Cr    
 r  r !( n  r )!
Vectors, Lines and Planes
Angle between two vectors: (Higher)
a  b  a b cos 
Equations of a 3d line: through ( x1 , y1 , z1 ) and with direction vector d  ai  bj  ck
Parametric form
Symmetric form
x  x1  at
x  x1 y  y1 z  z1
y  y1  bt (x  a  td)


( t )
a
b
c
z  z1  ct
Equations of a plane:
l
Normal n is  m 
n
 
Vector equation
x •n  a •n
Point on line = P (with position vector a)
Symmetric/Cartesian
lx  my  nz  k
where k  a • n
Parametric (A)
x  a  s b  tc
(b and c are any two nonparallel vectors in plane)
Angle between two lines = Acute angle between their direction vectors
Angle between two planes = Acute angle between their normals
Angle between line and plane = 90° – (Acute angle between n and d)
Cross (vector) product:
i

a  b  a b sin  n  a1
b1
Scalar triple product:
Newbattle Community High School
j
a2
b2
k
a
a3  i 2
b2
b3
a3
a
j 1
b3
b1
a3
a
k 1
b3
b1
a1
a2
a3
a(b  c)  b1
c1
b2
c2
b3
c3
a2
b2
© D Watkins 2015
Advanced Higher Mathematics
Matrices
a
A
c
a

A  d
g

2×2 matrices
3×3 matrices
Determinant and Inverse
1  d b 
det A  ad  bc and A1 


ad  bc  c a 
b

d
b
e
h
( AB ) 1  B 1 A1
c

f
i 
det A  a
e
h
 cos
Anti-CW Rotation by θ degrees 
 sin 
 a 0
Dilatation by scale factor a 
,
 0 a
f
d e
c
i
g h
det AB  det A det B (A)
( AB )T  BT AT
Transformation Matrices
f
d
b
i
g
 sin  
 1
 , Reflection in y-axis 
cos 
0
1
Reflection in x-axis 
0
0

1
0

1
Differential Equations
For
dy
 P ( x ) y  Q ( x ) , the Integrating Factor I(x) is
dx
e
P ( x ) dx

and the solution is given by I ( x) y  I ( x)Q( x)dx
Second Order Differential Equations
COMPLEMENTARY FUNCTION (Homogeneous Equations)
Nature of roots
Two distinct real m and n
Form of general solution
y  Ae mx  Benx
Real and equal m
y  ( A  Bx )e mx
Complex conjugate m  p  iq
y  e px ( A cos qx  B sin qx )
PARTICULAR INTEGRAL (Inhomogeneous Equations)
Right-hand side contains…
sin ax or cos ax
eax
For Particular Integral, try…
y  P cos ax  Q sin ax
Linear expression y  ax  b
y  Pe ax
y  Px  Q
Quadratic expression y  ax 2  bx  c
y  Px 2  Qx  R
Newbattle Community High School
© D Watkins 2015