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AS and A Level Maths Formulae Sheet
Shapes
Trigonometry
!
x base x height
"
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
base x height
!"#$%ℎ × ()*%ℎ
!
(sum of parallel sides) x height
"
, = 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 #
#
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
!
*
!
*
(8+ )* = F √8H
8+ = E
!
!
* +
*
(8 ) = √8
Rational Powers
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
$(#"&")
$(&""#)
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
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
)
(8 + >)*
=8* + F*!H8*)! > + ⋯ + F*,H8*), >, +…+>*
(8 + >)*
%
$
*(*)!) % "
M$N + ⋯ D
"!
0
0!
8 9 = 0<) =
(0 − 5)! 5!
5
Geometry
• Slope intercept form: 5 = Q4 + ,
• General form: 84 + >5 + * = 0
• Point slope form: 5 − 5! = Q(4 − 4! )
8
Quadratic Function:
Solutions to
US1 + VS + W = X
Quadratic Function:
Axis of Symmetry
Quadratic Function:
Discriminant
4=
C
2+
∆= C. − 4+<
> 0 (2 real distinct roots)
= 0 (2real repeated/double roots)
< 0 (no real roots)
•
•
•
•
> "
>"
D +,−
28
48
>"
,−
48
6
8 = " 6 7*$ ,
![$$ 8 6 = 4 = 8789$ 4
where, 8, 4 > 0, 8 ≠ 1
, ![$$ > ⟺ ![$$ >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 ±
•
•
:;<% %
:;<% $
R to D: × <
>
× 2Y5 (degrees)
?@*
>
× Y5 . (degrees)
?@*
Length of an arc
Area of a Sector
or
5Z (radians)
# .
5 Z
.
or
sin Z ≈ Z
Small Angle Approximations
cos Z ≈ 1 −
Pythagorean identity 2
Pythagorean identity 3
Cofunction
Identity of bcd S
Reciprocal
(radians)
Half Angle
C
UV0 . = ±`
#"AB: C
.
C
5
Kinematics:
Derivatives
D$! 5±D$! 7
5"7
5"7
<NUT − <NUb ≡ −2UV0 8 . 9 UV0 8 . 9
g
g
Vectors: 2D vectors 8h9 year 1 and 3D vectors ihk year 2
j
+
Vector Form
+l + Cm + <n ≡ iCk
<
"
(
"±(
"
-"
- !#% = '-# +
!# % ± ' ) + = ' # ± ) +
$
*
$±*
$
-$
"
(
!# % . ' ) + = "( + #) + $*
$
*
Integrals
1
Unit vector of e6f =
?
B
A
D
Midpoint ef C@D and CC D
Scalar Product
(not in syllabus but useful to
know)
#
G1#/6#/8 #
8
1/H 6/0 8/I
8 . , . , . 9
Independent Events
Conditional
“A given B”
Bayes Theorem
Binomial Distribution
Binompd (=)
Binomcd (≤)
Normal Distribution
Normcd (given x, want prob)
Invnorm (given prob, want x)
•
cos"# D(@) ⇒ −
2&&
4
#
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
Interquartile Range
Outliers
IQR= x# − x!
Any values
> UQ + 1.5(IQR) or < LQ − 1.5(IQR)
Mechanics
SUVAT
(5 formulae)
= = Å + 8%
!
"
"
"
= = Å + 28?
? = Å% + 8% "
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
S#"TI(C)U#
•
•
Trapezium Rule
%)$
h=
I)(C)
sin"# D(@) ⇒
•
∑F
IQ(C)
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(@)
•
•
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)
K&M
N%
"
lnÑD(@)Ö ⇒ I(C)
•
P(A ∩ B) = P(A)P(B)
?=M
•
•
Addition rule becomes: P(A∪B)=P(A)+P(B)
!
? = =% − 8% "
"
@ 4 ⟹ 0@ 4"#
4
4"#
ÑD(@)Ö ⇒ 0ÑD(@)Ö D Q (@)
•
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
Ä(T|b) =
•
•
•
<
(∑ 6' )
*
*(=)
*K+%L, 8F F$M8K,$%7L 8KN58+LO
k(2) =
=
*(J)
*K+%L, 8F P8OOI%7L 8KN58+LO
P(AIq) =
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,
[ = pt ⟹
•
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 =
∑(4I − 4̅ )" = ∑ 4I" −
(
H2
[ = o(p), p = i(_) ⟹
•
" (
"
(
!# %.' ) +=0!# %0 1' ) +1 $234
$
*
$
*
"
(
where, Z is the angle between !#% and ' ) +
$
*
h = √=80)8#,"
-
h[
i(_ + ℎ) − i(_)
= i ( (_) = lim
)→+
h_
ℎ
•
1
86 9
∑ 6#
∑(6)G)#
If no frequency: h " =
− 4̅ " =
*
*
∑ F6 #
∑ F(6)G)#
If frequency: h " = ∑ − 4̅ " =
∑F
F
-
(
•
(last formula not in syllabus but useful to know)
Unit Vector
-
•
•
•
•
•
•
•
•
:+0(T ± b) = #∓D$! 5 D$! 7
Magnitude of a vector
"%&
Distance=∫- # |Ç(:)|É:, Displacement=∫- # Ç(:)É:
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. + < .
<
#%&
Product Rule
<NU(T ± b) = <NUT<NUb ∓ UV0TUV0b
Properties
(addition/subtraction,
multiplication and scalar
product)
!"
“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
AB: C/#
.
5/7
[ − [$ = ^(_ − _$ )
Differentiate to get ^ (tangent means ∥, Normal means ⊥)
HR
H# 2
Acceleration= H- = H- #
<NU . = ±`
5/7
!!"
To find where concave down/concave: solve !# ! < 0
(
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 up/convex: solve !# ! > 0
Velocity: ∫- # +(:)É: or H-
@
1 − <NU@ 1 − <NU@
UV0 @
:+0 = ±a
=
=
2
1 + <NU @
UV0@
1 + <NU @
Compound Angle
!"
To find where decreasing: solve !# < 0
Remember to split up if separate areas
1 + :+0.@ = U]< .@
1 + <N: .@ = <NU]< .@
cos @ = sin(90 − @)
sin @ = cos (90 − @)
sin @
tan @ =
cos @
#
#
#
sec @ = AB: C , csc @ = :;! C, cot @ = D$! C
sin 2@ = 2 sin @ cos @
<NU2@ = <NU .@ − UV0.@
AB: .C/#
= 2 cos. @ − 1 ⟹ cos. @ =
.
1 − cos 2@
= 1 − 2 sin. Z ⟹ sin. @ =
2
. D$! C
tan 2@ = #"-14# C
Double Angle
!
(6 #
!"
To find where Increasing: solve !# > 0
Increasing/Decreasing
(use number line to solve)
Convex/Concave
(use number line to solve)
Tangents and Normals
Implicit
Area between
Z.
2
tan Z ≈ Z
UV0. @ + <NU . @ = 1
Pythagorean identity 1
Complementary Events
Combined Events (Addition Rule)
Mutually Exclusive Events
•
•
•
Exponential and Logarithmic
Functions
#=*
D to R: × #=*
Probability of event A
,8 ≠0
D(@) = @ . + C@ + < ⟹ @ = −
Completing The Square
US1 ± VS + W = X
Max/Min Value
Exponentials & Logarithm
Rules`
"$
<
Degrees ⟷ radians
Note: can also use the formula
Quadratics
)%±√%#)4$5
1
+CUV0W
2
Area of Triangle
!
!
6#/8 #"1#
Finding an angle:T = cos"# 8 .68 9
Standard Deviation
iHH
(@ − +). + (? − C). = 5 .
centre (+, C), radius 5
!
inding a side: + = C + < − 2C< cos T
Cosine Rule
Variance
@#/@. ?#/?.
,
9
2
2
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
:;! 5
:;! 7
:;! 9
Finding an angle: 1 = 6 = 8
.
.
.
*
?." ?#
>=
@." @#
.
A(@." @#) + (?." ?# ).
Distance between 2 points
(S0 , T0 ), (S1 , T1 )
Coordinates of midpoint of
(S0 , T0 ), (S1 , T1 )
Circles
6
Finding a side: 2345 = 2347 = 2349
,-
FV = PV81 + #** ,9
FV=future value
PV=present value
:=no. of years
5=nominal annual interest rate
; =no. of compounding periods per year
= 8* C1 + # M N +
1
Sine Rule
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
Replace D(@) with ?, swap @ & ?, solve for ?
DO(@) means plug O(@) into D(@)
Even: D(−@) = D(@) Odd: D(−@) = −D(@)
Transformations
+D(C@ + <) + É
“anything in a bracket affects
@ and does the opposite”
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: − . < @ < .