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Advanced Higher Maths: Formulae
Green (G): Formulae you absolutely must memorise in order to pass Advanced Higher maths. Remember you get no formula sheet at all in the exam! Amber (A): You don’t have to memorise these formulae, as it is possible to derive them from scratch in the exam. But… it will save you a lot of time if you do choose to memorise them, and I advise that you do. Red (R): Don’t worry about memorising these. Just use this sheet to help jog your memory in classwork and homework. One or two of these formulae are on the syllabus, but are sufficiently obscure that I don’t think it essential to memorise them.
Essential Trigonometric Identities : (from Intermediate 2 and Higher)
Essential Formulae to know by heart for the exam (G)
Other useful ones that may be useful for homework/classwork etc. Links between ratios
cos^2 A + sin 2 A = 1 tan sin cos
A A
A
1 + tan 2 A =sec^2 A cot 2 A + 1 = cosec^2 A (A)
Compound Angle
sin( A ± B ) = sin A cos B ±cos A sin B cos( A ± B ) = cos A cos B msin A sin B tan( ) tan^ tan 1 tan tan A B A^ B A B ± = ± m
(R)
Double Angle (^) 2 2
sin(2 ) 2sin cos cos(2 ) cos sin
A A A
A A A
tan(2 ) 2 tan 1 tan
A A
A
(R)
Squared^2 (^2 ) 2
cos (1 cos 2 ) sin (1 cos 2 )
x x x x
Unit 1.1: Binomial Theorem
The coefficient of the r th^ term in the binomial expansion ( x + y ) n is n^ r^ r n x y r
Ê ˆ -
ÁË ˜¯
n r C n^ n r r n r
= Ê^ ˆ=
ÁË ˜¯ -
Unit 2.3: Complex Numbers
For the complex number, , ∑ the modulus is given by z = a^2^ + b^2 ∑ and the argument is given by tan b a
q= - p < q <p
De Moivre’s Theorem says that for any z = r (cos q + i sin q), then z n^ = r n (cos n q + i sin n q)
z = a + bi
Units 1.2 and 2.1: Differentiation
2
2
tan sec sec sec tan cosec cosec cot cot cosec ln ( ) '( ) ( )
f x f x x x x x x x x x x x f x f^ x f x
1 2 1 2 1 2
sin^1 1 cos^1 1 tan 1 1
f x f x x x x x x x
To differentiate an inverse function: 1 ( ) (^11) '( ( ))
d (^) f x dx f f x
Parametric Equations (where x = f t ( ), y = g t ( ) ): ∑ Gradient (direction of movement) =
dydt dxdt
dy dx
∑ Speed = ( ) ( )
dy^2 dx^2 dt + dt ∑
2 2
d y d dy dt dx dt dx dx = ÊÁ^ ˆ˜¥ Ë ¯
or
2 2 3
d y x y y x dx x
= & &&^ - & &&
(A)
Units 1.3 and 2.2: Integration
(G) Essential Integrals to Learn
2
1 2 1 2
( ) ( ) sec tan tan ln sec '( ) (^) ln ( ) ( ) (^1) tan 1 (^1) sin 1
f x f x dx x x C x x C f x (^) f x C f x x C x x C x
Ú
(A) Could use substitution if needed:
( )
( )
1 1 2 2 1 2 2
(^1) tan
(^1) sin
a xa
ax
f x f x dx C a x C a x
Ú
(R) To save you time in hard questions for homework/classwork, no need to memorise: ( ) ( ) cosec ln cosec cot cot ln sin sec ln sec tan
f x f x dx x x x C x x C x x x C
Ú
Volume of solid of revolution f(x) about x axis: ( )^2
b a
V = pÚ f x dx
Unit 3.1: Vectors, Lines and Planes (G)
Angle between two vectors: (Higher) a ∑ b = a b cos q
Equations of a line:
Parametric form Symmetric/Cartesian form
( )
x a tl y b tm t z c tn
x a d x^ a^ y^ b^ z^ c^ t l m n
Equations of a plane:
Normal n is
l m n
Ê ˆ
Á ˜
ÁÁ ˜˜
Ë ¯
Point on line = P (with position vector a )
Vector equation Symmetric/Cartesian Parametric x • n = a • n lx + my + nz = k x = a + m b +l c or ( x - a • n ) = 0 where k = a • n ( b and c are any two nonpa rallel vectors in plane)
Angle between two planes = Angle between their normals
Angle between line and plane = (Angle between n and d ) – 90°
Cross product: (^) 1 2 3
1 2 3
a a a b b b
i j k a b
Scalar triple product:
1 2 3 1 2 3 1 2 3
a a a b b b c c c
a g b ¥ c =
Unit 3.2: Matrices (G)
Determinant and Inverse
2×2 matrices
a b A c d
=^ Ê^ ˆ
Á ˜
Ë ¯
det A = ad - bc and 1 1 d b A ad bc c a
- = Ê^ - ˆ
- Á^ - ˜
Ë ¯
3×3 matrices
a b c A d e f g h i
Ê ˆ
= Á^ ˜
ÁÁ ˜˜
Ë ¯
det e f d f d e A a b c h i g i g h
( AB )-^1 = B -^1 A -^1
Transformation Matrices
Reflection in x axis
Ê ˆ
Á - ˜
Ë ¯
, Reflection in y axis
Ê - ˆ
Á ˜
Ë ¯
Enlargement by scale factor a
a a
Ê ˆ
Á ˜
Ë ¯
, Rotation by θ degrees cos sin sin cos
q q q q
Ê - ˆ
Á ˜
Ë ¯
Unit 3.4: Differential Equations (G)
For^ dy^ P x y ( ) Q x ( ) dx
- = , the Integrating Factor I ( x ) is
P x dx ( )
e Ú
and the solution is given by I x y ( ) = Ú I x Q x dx ( ) ( )
Second Order Differential Equations
Nature of roots Form of general solution Two distinct real m and n (^) y = Ae mx^ + Benx Real and equal m (^) y = Ae mx^ + Bxemx Complex conjugate m = p ± iq y = e px ( A cos q + B sin qx )
