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Duncanrig Secondary School: Mathematics Department
Higher Mathematics Essentials
None of the above appear on the Higher Mathematics Formulae List. Everything on this sheet has to be learned off by heart.
distance formula
2 2 1
2 2 1
2 d (x 2 x 1 ) (y y) (z z)
mid-point formula M
x 1 x 2 y 1 y 2 z 1 z 2
gradient formula
2 1
2 1
x x
y y mAB
angle between line and +ve direction of x-axis mtan
perpendicular lines m 1 m 2 1 parallel linesm 1 m 2
equation of a line given point (a, b) and gradient m y – b = m(x – a)
degrees into radians multiply by 180
and simplify
exact values x 0 30 45 60 90
sin x 0 2
1
cos x 1 2
0
tan x 0 3
1 3 undefined
credit trig sin cos 1
2 2 x x x x
x tan cos
sin
sine rule C
c
B
b
A
a
sin sin sin
area of triangle A absinC 2
cosine rule a b c 2 bccosA
2 2 2 or bc
b c a A 2
cos
2 2 2
index laws
m n mn a a a
mn n
m
a a
a (^)
m n mn (a ) a
0 a n
n
a
a
n n^ m
m
a a
surds ab= a b e.g. 18 = 9 2 =^32
b
a
b
a x b
b
b
a b e.g. 6
x 6
Duncanrig Secondary School: Mathematics Department
Higher Mathematics Essentials
None of the above appear on the Higher Mathematics Formulae List. Everything on this sheet has to be learned off by heart.
rate of change same as derivative/gradient e.g.
dt
dx v , dt
dv a
i.e. differentiate displacement to get velocity and differentiate velocity to get acceleration
limit of a RR for u au b n 1 n a
b L
if – 1 < a < 1
quadratic formula if 0
2 ax bxc doesn’t factorise, use a
b b ac x 2
2
discriminant (i) 4 0 quadratichas 2 real(&unequal)roots
2 b ac
(ii) 4 0 quadratichas 2 equalroots
2 b ac
(iii) 4 0 quadratichasnorealroots
2 b ac
(iv) 4 0 quadratichasrealroots
2 b ac
(v) 4 ( ) quadratichasrationalroots
2 2 b acn n
circle circles touch if: r 1 r 2 C 1 C 2
circles don’t intersect if: r 1 r 2 C 1 C 2
circles intersect if: r 1 r 2 C 1 C 2
position vector P (a, b, c) has position vector OP p
c
b
a
vector magnitude p=
2 2 2 a b c
angle in 3D
ab
a.b cos perpendicular vectors if (^) a. (^) b = 0
chain rule for differentiation
n
y f( x) ( ) ( )
1 n f x f x dx
dy (^) n
integrate power of linear fn c an
ax b ax b dx
n n
1
log graphs y logax passes through (1,0), (a,1), (a
2 ,2) etc.
exp graphs
x y a passes through (0,1), (1,a), (2,a
2 ) etc.
log laws (i) log (^) a x logaylogaxy
(ii) y
x log (^) a x logayloga
(iii)
n n log (^) a xlogax
