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AS and A Level Maths Formulae Sheet
Shapes
Trigonometry
!
x
"
Area of Triangle
Area of Parallelogram
Area of Rectangle
Area of Trapezoid
Circumference & Area: Circle
Cuboid Surface area
Cuboid Volume
Cylinder Surface Area
Cylinder Volume
Cone Surface Area
Cone Volume
Sphere Surface Area
Sphere Volume
Prism Volume
Pyramid Volume
Multiplication
Division
Negative Powers
Fractions
9 =Area of cross section x height
1
9 = × >8?" 80"8 × ℎ
3
Indices
4 $ × 4 % = 4 $&%
(4 $ )% = 4 $%
(,4 $ 5 % )( = , ( 4 $( 5 %(
4$
4 $ ÷ 4 % = % = 4 $)%
4
1
4 )* = *
4
4 *
4*
C D = *
5
5
)*
4
5*
C D = *
5
4
Double Angle
!
=E
(8+ )* = F √8H
!
!
*
!
(8* )+ = √8*
Series
u! = a + (n − 1)d
where + =first term, d= common diff
n
0
S! = [2a + (n − 1)d] = (+ + 1)
2
2
Arithmetic sequence:
IJK term
Arithmetic sequence:
sum of n terms
where + =first term, d= common diff,
1=last term
u! = ar !"#
where + =first term, r= common ratio
Geometric sequence:
IJK term
Geometric sequence:
sum of n terms
$(#"&")
#"&
Binomial Theorem:
integer powers
Binomial Theorem:
Fractional & Negative
powers
Binomial Coefficient
Straight Line: Equation
(gradient means slope)
Parallel⟹ same slope
Perpendicular⟹ “flip
fraction and change the sign”
(slopes multiply to make −1)
Straight Line: Gradient
$(&""#)
,
&"#
S! =
=
r≠1
where + =first term, r= common ratio
$
S( = #"& , |5| < 1
where + =first term, r= common ratio
Geometric sequence:
Sum to infinity
Compound Interest
)
(8 + >)*
=8* + F*!H8*)! > + ⋯ + F*,H8*), >, +…+>*
(8 + >)*
%
= 8* C1 + # M N +
$
*(*)!) % "
M$N
"!
+ ⋯D
0
0!
8 9 = 0<) =
(0 − 5)! 5!
5
Geometry
• Slope intercept form: 5 = Q4 + ,
• General form: 84 + >5 + * = 0
• Point slope form: 5 − 5! = Q(4 − 4! )
∆= C. − 4+<
> 0 (2 real distinct roots)
= 0 (2real repeated/double roots)
< 0 (no real roots)
•
•
•
•
> "
>"
D +,−
28
48
>"
,−
48
6
8 = " 6 7*$ ,
![$$ 8 6 = 4 = 8789$ 4
where, 8, 4 > 0, 8 ≠ 1
, ![$$ > ⟺ ![$$ >5
log $ > = , ⟺ 85 = >, 8, >, > 0,8 ≠ 1
log $ > + log $ , ⟺ log $ >,
%
log $ > − log $ , ⟺ log $
•
log $ > ⟺
•
Solving a power of @: log both sides if 2
terms or use substitution if 3 terms
Solving an exponential : ln both sides
Solving a logarithm: raise e both sides or
write as log 0 as procced as usual for 1NO
8 C4 ±
•
•
:;<% %
:;<% $
or
(radians)
Identity of bcd S
Reciprocal
UV0
C
.
= ±`
#"AB: C
.
C
.
<NU
5
Derivatives
D$! 5±D$! 7
:+0(T ± b) = #∓D$! 5 D$! 7
5"7
5"7
9
.
<NUT − <NUb ≡ −2UV0 8 . 9 UV0 8
g
g
Vectors: 2D vectors 8h9 year 1 and 3D vectors ihk year 2
j
+
Vector Form
+l + Cm + <n ≡ iCk
<
"
(
"±(
"
-"
- !#% = '-# +
!# % ± ' ) + = ' # ± ) +
$
*
$±*
$
-$
"
(
!# % . ' ) + = "( + #) + $*
$
*
Integrals
Unit Vector
?
B
A
D
G1#/6#/8 #
8
1/H 6/0 8/I
8 . , . , . 9
Midpoint ef C@D and CC D
Scalar Product
(not in syllabus but useful to
know)
If no frequency:
=
∑ F6
∑F
*
#
− 4̅ "
− 4̅ " =
Independent Events
h = √=80)8#,"
Binomial Distribution
Binompd (=)
Binomcd (≤)
Normal Distribution
Normcd (given x, want prob)
Invnorm (given prob, want x)
Interquartile Range
Outliers
SUVAT
(5 formulae)
=
F
*
∑ F(6)G)#
∑F
2&&
4
#
Addition rule becomes:P(A∪B)=P(A)+P(B)−P(A)P(B)
To find whether independent: Find P(A), P(B) and
P(A ∩ B) and see whether the former 2 multiply to
make the latter or show that P(AIb) = Ä(T)
Q(R∩T)
U(>)
If independent: P(AIq) = k(2)
Ä(b|T)Ä(T)
Ä(b|T)Ä(T) + Ä(b|TQ )Ä(TQ)
4~q(#, t)
E(X)=Mean= #t, Var(X)= #t(1 − t)
P(u = 4) = F*6Ht 6 (1 − t)6
4~v(w, h " )
6)G
Standardised variable 7 =
V
IQR= x# − x!
Any values
> UQ + 1.5(IQR) or < LQ − 1.5(IQR)
Mechanics
= = Å + 8%
? = =%
!
− 8% "
"
K&M
N%
"
?=M
!
"
? = Å% + 8% "
="
sin"# D(@) ⇒
•
cos"# D(@) ⇒ −
=
Å"
+ 28?
I) (C)
S#"TI(C)U#
I) (C)
tan"# D(@)) ⇒
sec "# D(@) ⇒
•
<NU]< "# D(@) ⇒ −
•
cot "# D(@) ⇒ −
•
•
•
#
#/TI(C)U
I)(C)
#
I(C)STI(C)U "#
I) (C)
#
I(C)STI(C)U "#
I)(C)
#
#/TI(C)U
C*+(
∫ @ É@ = 4/# + <, 0 ≠ −1
#
#
∫ ,C É@ = V ln|@| + <
#
∫ sin ;@ É@ = − V cos ;@ + <
#
∫ cos ;@ É@ = V sin ;@ + <
#
∫ ] ,C É@ = , ] ,C + <
#
∫ +,C É@ = VW! 1 +,C + <
#
∫ sec . ;@ É@ = V tan ;@ + <
#
∫ sec ;@ tan ;@ É@ = , sec ;@ + <
#
∫ cosec ;@ cot ;@ É@ = − V cosec ;@ + <
#
∫ cosec . ;@ É@ = − , cot ;@ + <
#
∫ U]< ;@ É@ = , ln| sec ;@ + tan ;@| + <
#
∫ <NU]< ;@ É@ = − , ln|<NU]<;@ + cot ;@|+c
#
#
6C
É@ = 6 sin"# 8 1 9 + <
∫ G1#
#
"(6C)
#
#
"# 6C
É@ = 6 cos 8 1 9 + <
∫
"G1#"(6C)#
#
#
6C
∫ 1#/(6C)# É@ = 16 tan"# 8 1 9 + <
4
Integration by parts
*K+%L, 8F ON,IPO
I)(C)
S#"TI(C)U#
•
•
Newton Raphson
IQ(C)
I(C)
•
•
P(A ∩ B) = P(A)P(B)
Ä(T|b) =
•
•
Addition rule becomes: P(A∪B)=P(A)+P(B)
P(AIq) =
sin D(@) ⇒ D Q(@) cos D(@)
cos D(@) ⇒ −D Q (@) sin D(@)
] I(C) ⇒ D Q (@)] I(C)
+ I(C) ⇒ D Q(@) + I(C) ln +
tan D(@) ⇒ D Q (@) sec . D(@)
sec D(@) ⇒ D Q(@) sec D(@) tan D(@)
<NU]< D(@) ⇒ −D Q (@)<NU]< D(@) cot D(@)
cot D(@) ⇒ −D Q (@)<U< .D(@)
•
P(A’)=1−P(A) i.e. probabilities add to 1
P(A∪B)=P(A)+P(B)−P(A ∩ B)
P(A ∩ B) = 0
Conditional
“A given B”
Bayes Theorem
*(=)
*(J)
lnÑD(@)Ö ⇒
•
•
•
•
•
•
•
•
•
(∑ 6' )
*
*K+%L, 8F F$M8K,$%7L 8KN58+LO
*K+%L, 8F P8OOI%7L 8KN58+LO
∑(4I − 4̅ )" = ∑ 4I" −
k(2) =
•
•
∑(6)G)#
=
h[ h[ hp
=
×
h_ hp h_
h[
ht
hp
=p
+t
h_
h_
h_
hp
ht
p
h[ t h_ − p h_
[= ⟹
=
t
h_
t,
@ 4 ⟹ 0@ 4"#
4
4"#
ÑD(@)Ö ⇒ 0ÑD(@)Ö D Q (@)
•
Trapezium Rule
%)$
h=
=
H# 2
H- #
•
•
•
<
HR
H-
[ = pt ⟹
•
1
86 9
1 H
Angle Between 2 vectors
J6K.M 0 N
8 I
This is just a re-arrangement of
Z = cos"# q 1 H r
OJ6 KOPM 0 NP
above.
8
I
(not in syllabus but useful to
know)
H
1
Vector Equation of a line
5 = e6f + s 8I0 9
8
(not in syllabus but useful to
know)
Probability and Statistics
∑6
∑ F6
Mean
If no frequency: 4̅ =
, If frequency: 4̅ = ∑
∑ 6#
(
H2
[ = o(p), p = i(_) ⟹
•
" (
"
(
!# %.' ) +=0!# %0 1' ) +1 $234
$
*
$
*
"
(
where, Z is the angle between !#% and ' ) +
$
*
h"
-
h[
i(_ + ℎ) − i(_)
= i ( (_) = lim
)→+
h_
ℎ
•
#
1
Unit vector of e6f =
-
(
•
(last formula not in syllabus but useful to know)
Magnitude of a vector
-
Quotient rule
UV0T + UV0b ≡ 2UV0 8 . 9 <NU 8 . 9
T+b
T−b
UV0T − UV0b ≡ 2<NU e
f UV0 e
f
2
2
5/7
5"7
<NUT + <NUb ≡ 2<NU 8 . 9 <NU 8 . 9
+
oiCko = √+. + C. + < .
<
"%&
Distance=∫- # |Ç(:)|É:, Displacement=∫- # Ç(:)É:
Product Rule
AB: C/#
.
5/7
#%&
Acceleration=
= ±`
5/7
!"
“every time we differentiate a [ we write !# "
curve & _ c_de: ∫#%' [ h_ curve & [ axis: ∫"%' _ h[
(take + answer if neg)
#%&
Between 2 curves: ∫#%' (top curve-bottom curve)h_
Differentiation 1st
Principles
Chain Rule
<NU(T ± b) = <NUT<NUb ∓ UV0TUV0b
Properties
(addition/subtraction,
multiplication and scalar
product)
[ − [$ = ^(_ − _$ )
Differentiate to get ^ (tangent means ∥, Normal means ⊥)
(
sin(T ± b) = sin T cos b ± cos T sin b
Factor Formula: sum to product
Note: For product t to sum re=&>
=)>
arrange and let
and
"
"
equal your given angles and
solve for A and B simultaneously
!!"
To find where concave down/concave: solve !# ! < 0
Velocity: ∫- # +(:)É: or H-
@
1 − <NU@ 1 − <NU@
UV0 @
:+0 = ±a
=
=
2
1 + <NU @
UV0@
1 + <NU @
Compound Angle
!!"
To find where concave up/convex: solve !# ! > 0
Remember to split up if separate areas
Kinematics:
sin 2@ = 2 sin @ cos @
<NU2@ = <NU .@ − UV0.@
AB: .C/#
= 2 cos. @ − 1 ⟹ cos. @ =
.
1 − cos 2@
= 1 − 2 sin. Z ⟹ sin. @ =
2
. D$! C
tan 2@ = #"-14# C
Half Angle
!"
To find where decreasing: solve !# < 0
Implicit
Area between
Z.
2
!
(6 #
!"
To find where Increasing: solve !# > 0
Increasing/Decreasing
(use number line to solve)
Convex/Concave
(use number line to solve)
Tangents and Normals
#=*
<
1 + :+0.@ = U]< .@
1 + <N: .@ = <NU]< .@
cos @ = sin(90 − @)
sin @ = cos (90 − @)
sin @
tan @ =
cos @
#
#
#
sec @ = AB: C , csc @ = :;! C, cot @ = D$! C
Pythagorean identity 2
Pythagorean identity 3
Cofunction
!
!
tan Z ≈ Z
UV0. @ + <NU . @ = 1
Pythagorean identity 1
Complementary Events
Combined Events (Addition Rule)
Mutually Exclusive Events
•
•
•
Exponential and Logarithmic
Functions
# .
5 Z
.
cos Z ≈ 1 −
Probability of event A
C
2+
!
5Z (radians)
sin Z ≈ Z
Small Angle Approximations
Standard Deviation
iHH
,8 ≠0
D(@) = @ . + C@ + < ⟹ @ = −
Completing The Square
US1 ± VS + W = X
Max/Min Value
Exponentials & Logarithm
Rules`
"$
=
Note: can also use the formula
(@ − +). + (? − C). = 5 .
centre (+, C), radius 5
:;! 9
8
R to D: ×
or
If frequency: h " =
@#/@. ?#/?.
8
,
9
2
2
4=
Area of a Sector
Variance
Quadratics
Quadratic Function:
Solutions to
US1 + VS + W = X
Quadratic Function:
Axis of Symmetry
Quadratic Function:
Discriminant
>
× 2Y5 (degrees)
?@*
>
× Y5 . (degrees)
?@*
Length of an arc
*
?." ?#
>=
@." @#
.
A(@." @#) + (?." ?# ).
)%±√%#)4$5
Degrees ⟷ radians
,-
FV = PV81 + #** ,9
FV=future value
PV=present value
:=no. of years
5=nominal annual interest rate
; =no. of compounding periods per year
Distance between 2 points
(S0 , T0 ), (S1 , T1 )
Coordinates of midpoint of
(S0 , T0 ), (S1 , T1 )
Circles
Finding an angle:T =
1
+CUV0W
2
<
D to R: × #=*
of parallel sides) x height
Area of Triangle
*
8+
Calculus (Differentiation and Integration) Continued
(W
Turning/Stationary Points
Solve = 0
(6
(Max/Min)
! "
! "
If !# > 0 min and !# < 0 max
Proving whether
!"
Max/Min
Or can do sign change test for !# using number line
#
( W
Points of Inflection
solve
=0
8
:;! 7
= 6
.
6#/8 #"1#
cos"# 8 .68 9
, = 2/0, 2 = /0 "
32 = 245 + 247 + 257
Where 4, 5, 8#* 7 are side lengths
9 = 457
where 4, 5, 8#* 7 are side lengths
32 = 2/0ℎ + 2/0 "
Note: Curved part: 2/0ℎ
9 = /0 " ℎ
32 = /0! + /0 "
Note: Curved part: /0!
where ! is slant length
1
9 = /0 " ℎ
3
32 = 4/0 "
Note: Hemisphere= 2/0 " + /0 " = 3/0 "
4
= = /0 #
3
"
Note: Hemisphere= /0 #
Rational Powers
:;! 5
1
.
inding a side: +. = C + < − 2C< cos T
Cosine Rule
#
6
Finding a side: 2345 = 2347 = 2349
Finding an angle:
base x height
!"#$%ℎ × ()*%ℎ
!
(sum
"
1
Sine Rule
base x height
up
hp
hp
h_ = pt − u t
h_
h_
h_
ℎ
[? + 2(?#/?./??/?X + ⋯ ) + ?4 ]
2 *
#
Simply put, . ℎ[1U: ? + 2(>VÉÉ1] ? Q U) + 1+U:?]
I(C )
For solving D(@) = 0: @4/# = @4 − IQ(C* )
*
Functions
Inverse
Composite
Odd and Even Functions
Transformations
+D(C@ + <) + É
“anything in a bracket affects
@ and does the opposite”
Replace D(@) with ?, swap @ & ?, solve for ?
DO(@) means plug O(@) into D(@)
Even: D(−@) = D(@) Odd: D(−@) = −D(@)
#
a=vertical stretch sf +, b=horizontal stretch sf 6
c=translation c units x direction, d=translation d
units in y direction
D(−@)=reflcn in ?, axis − D(@)=reflcn in @ axis
Linear: ä = ãå + j
Domain: x∈ ℝ
Range: y∈ ℝ
Quadratic: ä = ±g(hå + j)Y+ d
Domain: x∈ ℝ
Range: ? ≥ É if min, ? ≤ É if max
Exponential: ä = gëZ[/\ + í
Domain: x∈ ℝ (Hint: power of exp can
be anything, so no restriction)
Range: ? > É if + > 0, ? < É if + < 0
(Hint: exp can’t be zero)
Asymptote: ? = É
Logarithm: ä = gìî(hå + j)+d
8
Domain: @ > − 6 (Hint: 10 can’t take
a neg number so C@ + < > 0)
Range: y ∈ ℝ
8
Asymptote: @ = − 6
Root: ä = g√hå + j + í:
8
Domain: @ ≥ − 6 (Hint: underneath
root must be positive so b@ + < ≥0)
Range: ? ≥ É if + > 0 and ? ≤ É if
a<0
Modulus ä = g|hå + j| + í:
Domain: @ ∈ ℝ
Range: ? ≥ É : ? ≥ É if + > 0 and ? ≤
É if + <0
@, @ ≥ 0
Note: Definition of |@| = ï
−@, @ < 0
© MyMathsCloud
][/Z
Rational: ä = \[/^ + ë
H
Domain: x∈ ℝ, @ ≠ − 8 (hint:denom≠0)
1
Range: y∈ ℝ, ? ≠ 8 + ]
H
1
Asymptotes: @ = − 8 , ? = 8 + ]
Note: often + and or ] are zero
Trigonometry:ä = gñlî(hå + j)+d
ä = gjóñ(hå + j)+d
Domain: @ ∈ ℝ
Range: −+ + É ≤ ? ≤ + + É
Note: If asked to find values of a,b,c d
_$` a"_;! a
+ = amplitude=
.
.<
?@*
C = b0)3cH or b0)3cH
_$` a/_;! a
É = principal axis=
.
< =phase shift (plug in point to find)
Trigonometry:ä = gògî(hå + j)+d
<
Domain: @ ∈ ℝ, @ ≠ . + 0Y
Range:−∞ ≤ ? ≤ ∞
Inverse trig: ä = ñlî"då
Domain: −1 ≤ @ ≤ 1
<
<
Range: − . ≤ @ ≤ .
Inverse trig: ä = jóñ"då
Domain: −1 ≤ @ ≤ 1
Range: 0 ≤ @ ≤ Y
Inverse trig: ä = ògî"då
Domain: −∞ ≤ @ ≤ ∞
<
<
Range: − . < @ < .