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MATHEMATICS
& STATISTICS/
A-LEVEL CORE
FORMULA SHEET
Pythagorian Identities
➢ sin2 (x) + cos2 (x) = 1
➢ 1 + tan2 (x) = sec 2 (x)
➢ 1 + cot 2 (x) = cosec 2 (x)
Co − function Identities
π
2
π
cos ( 2
π
tan (
2
➢ sin ( − x) = cos(x)
➢
➢
Surds, for a, b > 0
√a
√b
=
π
π
➢ cot ( 2 − x) = tan(x)
Indices
➢ pa × pb = pa+b
➢ pa ÷ pb = pa−b
➢ (pa )b = pab
Even − Odd Identities
➢ sin(−x) = −sin(x)
➢ cos(−x) = cos(x)
➢ tan(−x) = −tan(x)
➢ cosec(−x) = −cosec(x)
➢ sec(−x) = sec(x)
➢ cot(−x) = −cot(x)
Logs and Exponents (where ln = log )
➢ eln (x) = ln(ex ) = x
➢ log(x) + log(y) = log(xy)
➢ log(x) − log(y) = log(x/y)
➢ alog(x) = log(x a )
4
π
2
➢ sec ( 2 − x) = cosec(x)
a
b
➢ log b x =
− x) = cot(x)
➢ cosec ( − x) = sec(x)
➢ a × √b = √a2 × b
➢ (√a)2 = a
➢
− x) = sin(x)
–
➢ sin(u ± v) = sin(u) cos(v) ± sin(v) cos(u)
➢ cos(u ± v) = cos(u)cos(v) ∓ sin(u) sin(v)
log a x
log a b
tan (u)±tan (v)
➢ tan(u ± v) = 1∓tan (u)tan (v)
➢ y = mx + c
change in y
in x
➢ where m = gradient of the line = change
➢ c = y axis intercept
➢
+
Double Angle Formulas
➢ sin(2u) = 2 sin(u) cos(u)
2
2
➢ cos(2u) = cos (u) − sin (u)
2 (u)
= 2cos
➢
−1
= 1 − 2sin2 (u)
➢
2tan (u)
➢ tan(2u) = 1−tan 2 (u)
Power − Reducing Identities
1
➢
Quadratic Formula
For ax² + bx + c = 0,
➢ x = −b±√b
➢
2 −4ac
2a
Geometric Series
Sine rule
➢ sin (A) = sin (B) = sin (C)
a
2
b
2
1−cos (2u)
2
1+cos (2u)
cos2 (u) =
2
1−cos (2u)
tan2 (u) =
1+cos (2u)
2
➢ sin (u) =
➢ a + ar + ar 2 + ar 3 + …+ ar n−1
k
= ∑n−1
k=0 ar = a
2
➢ a = b + c − 2bc cos(A)
1−r n
1−r
Arithmetic Series
➢ a = 5irst term, d = common difference,
➢ a + (a + d) + (a + 2d) + … + (a + (n − 1)d)
n
= ∑n−1
k=0 a + kd = (2a + (n − 1)d)
2
➢
➢
Notes
dy
du
dv
= ×
dx
dv
dx
dy
dv
du
u(x)v(x), dx = u dx + v dx
➢ y = u(v(x)),
➢ y=
u(x)
➢ y = v(x) ,
dy
dx
1
cot(x) = tan (x)
=
v
du
dv
−u
dx
dx
2
v
Integration by Parts
dv
du
➢ ∫ u dx dx = uv − ∫ v dx dx
Derivatives Table
(for a constant c and any real coef5icient m)
( )
c
x
cx
xm
sin(mx)
cos(mx)
emx
ln(mx)
ln(ax + b)
tan(mx)
sec(mx)
cot(mx)
cosec(mx)
′( )
0
1
c
mx m−1
m cos(mx)
−m sin(mx)
m emx
1
x
a
ax + b
m sec 2 (mx)
m sec(mx) tan(mx)
−m cosec 2 (mx)
−m cosec (mx) cot (mx)
Integrals Table
(for a constant c and any real coef5icient m)
x
m
( )
(ax + b)m
1
x
1
ax + b
ex
emx
cos(mx)
sin(mx)
sec 2 (mx)
➢
c
sin (x)
cos (x)
tan(x) = (x) cot(x) = (x)
cos
sin
1
1
cosec(x) = sin (x) sec(x) = cos (x)
Differentiation Rules
∫ ( )
m+1
x
+c
m+1
m+1
(ax + b)
+c
a(m + 1)
ln| x| + c
1
ln| ax + b| + c
a
ex + c
1 mx
e + c
m
1
sin(mx) + c
m
1
− m cos(mx) + c
1
tan(mx) + c
m
Average of data with n values
➢ mean = x̅ =
x 1 +x 2 +x 3 +…+x n
n
1
= n ∑nk=1 xk
➢ median = middle value when data is
arranged in order
➢ mode is the most commonly occurring value
20 Range = highest value − lowest value
21 Frequency = how often a given value occurs
Notes
Skewness is the shape of a set of data
➢ if mean < median < mode then the data skews
to the left, known as negative skew
➢ if mean > median > mode then the data skews
to the right, known as positive skew
➢ if mean = median = mode then the data values
are symmetrically distributed
www.kent.ac.uk/smsas
@unikentsmsas
[email protected]
www.tiny.cc/SMSASALevel
School of
Mathematics, Statistics
and Actuarial Science
MATHEMATICS
&Skewness
STATISTICS/
is the shape of a set of data
++
arar+ …+
ar
⋯
if
mean
< median < mode then the data skews
➢
A-LEVEL
CORE
to the left, known as negative skew
=a
➢ if mean > median >SHEET
FORMULA
mode then the data skews
21 Frequency = how often a given value occurs
s
3 n−1
1−r n
1−r
Circles − Arcs and Sectors
Volume of Revolution
Circles − Arcs and Sectors
➢ Arc length =
d
n
= (2a
2
)d
ies
)d)
+ (n − 1)d)
1 n
∑ (x
n k=1 k
23 Standard Deviation = σ =
360
➢ Sector Area =
➢ Sector Area =
to the right, known as positive skew
➢
if
mean = median = mode then the data values
Continued
s
are symmetrically distributed
m, d = common difference,
mmon
difference,
(a + 2d)
+ … + (a + (n − 1)d)
x2
➢ Arc length = 360 x 2
= ∫ π y 2 dx
n−1
segment
360
360
x
x
2
2
r
− x̅)2
1
n
24 Variance = σ2 = ∑nk=1(xk − x̅)2
Volume of Revolution
y = f(g(x)),
u = g(x)so
y
u = g(x)so
y
y = f(g(x)),
y = f(g(x)), u = g(x)so y
f(u) then
= f(u)=then
= f(u) then
Common
graphs:
Common
graphs:
Common graphs:
Common
Common
graphsgraphs
Common graphs
= ∫ π y 2 dx
Common graphs
Circles − Arcs and Sectors
➢ Arc length =
360
➢ Sector Area =
Circles − Arcs and Sectors
➢ Arc length =
360
➢ Sector Area =
x2
360
x
x2
360
x
2
2
=
www.kent.ac.uk/smsas
@unikentsmsas
2
[email protected]
www.tiny.cc/SMSASALevel
Formula sheet by SMSAS Student Ambassadors Rachael Whyman and Patrick Ryan
=
3
School of
Mathematics, Statistics
and Actuarial Science