Quadratic formula
If a x 2 + b x + c = 0 , a x 2 + b x + c = 0 , x = − b ± b 2 − 4 a c 2 a x = − b ± b 2 − 4 a c 2 a
Triangle of base b b h h
Area = 1 2 b h = 1 2 b h
Circle of radius r r
Circumference = 2 π r = 2 π r
Area = π r 2 = π r 2
Sphere of radius r r
Surface area = 4 π r 2 = 4 π r 2
Volume = 4 3 π r 3 = 4 3 π r 3
Cylinder of radius r r h h
Area of curved surface = 2 π r h = 2 π r h
Volume = π r 2 h = π r 2 h
Table
E1
Geometry
Trigonometry
Trigonometric Identities
sin θ = 1 / csc θ sin θ = 1 / csc θ cos θ = 1 / sec θ cos θ = 1 / sec θ tan θ = 1 / cot θ tan θ = 1 / cot θ sin ( 90 0 − θ ) = cos θ sin ( 90 0 − θ ) = cos θ cos ( 90 0 − θ ) = sin θ cos ( 90 0 − θ ) = sin θ tan ( 90 0 − θ ) = cot θ tan ( 90 0 − θ ) = cot θ sin 2 θ + cos 2 θ = 1 sin 2 θ + cos 2 θ = 1 sec 2 θ − tan 2 θ = 1 sec 2 θ − tan 2 θ = 1 tan θ = sin θ / cos θ tan θ = sin θ / cos θ sin ( α ± β ) = sin α cos β ± cos α sin β sin ( α ± β ) = sin α cos β ± cos α sin β cos ( α ± β ) = cos α cos β ∓ sin α sin β cos ( α ± β ) = cos α cos β ∓ sin α sin β tan ( α ± β ) = tan α ± tan β 1 ∓ tan α tan β tan ( α ± β ) = tan α ± tan β 1 ∓ tan α tan β sin 2 θ = 2 sin θ cos θ sin 2 θ = 2 sin θ cos θ cos 2 θ = cos 2 θ − sin 2 θ = 2 cos 2 θ − 1 = 1 − 2 sin 2 θ cos 2 θ = cos 2 θ − sin 2 θ = 2 cos 2 θ − 1 = 1 − 2 sin 2 θ sin α + sin β = 2 sin 1 2 ( α + β ) cos 1 2 ( α − β ) sin α + sin β = 2 sin 1 2 ( α + β ) cos 1 2 ( α − β ) cos α + cos β = 2 cos 1 2 ( α + β ) cos 1 2 ( α − β ) cos α + cos β = 2 cos 1 2 ( α + β ) cos 1 2 ( α − β ) s = r θ s = r θ
Triangles
Law of sines: a sin α = b sin β = c sin γ a sin α = b sin β = c sin γ
Law of cosines: c 2 = a 2 + b 2 − 2 a b cos γ c 2 = a 2 + b 2 − 2 a b cos γ
Pythagorean theorem: a 2 + b 2 = c 2 a 2 + b 2 = c 2
Series expansions
Binomial theorem: ( a + b ) n = a n + n a n − 1 b + n ( n − 1 ) a n − 2 b 2 2 ! + n ( n − 1 ) ( n − 2 ) a n − 3 b 3 3 ! + ··· ( a + b ) n = a n + n a n − 1 b + n ( n − 1 ) a n − 2 b 2 2 ! + n ( n − 1 ) ( n − 2 ) a n − 3 b 3 3 ! + ···
( 1 ± x ) n = 1 ± n x 1 ! + n ( n − 1 ) x 2 2 ! ± ··· ( x 2 < 1 ) ( 1 ± x ) n = 1 ± n x 1 ! + n ( n − 1 ) x 2 2 ! ± ··· ( x 2 < 1 ) ( 1 ± x ) − n = 1 ∓ n x 1 ! + n ( n + 1 ) x 2 2 ! ∓ ··· ( x 2 < 1 ) ( 1 ± x ) − n = 1 ∓ n x 1 ! + n ( n + 1 ) x 2 2 ! ∓ ··· ( x 2 < 1 ) sin x = x − x 3 3 ! + x 5 5 ! − ··· sin x = x − x 3 3 ! + x 5 5 ! − ··· cos x = 1 − x 2 2 ! + x 4 4 ! − ··· cos x = 1 − x 2 2 ! + x 4 4 ! − ··· tan x = x + x 3 3 + 2 x 5 15 + ··· tan x = x + x 3 3 + 2 x 5 15 + ··· e x = 1 + x + x 2 2 ! + ··· e x = 1 + x + x 2 2 ! + ··· ln ( 1 + x ) = x − 1 2 x 2 + 1 3 x 3 − ··· ( | x | < 1 ) ln ( 1 + x ) = x − 1 2 x 2 + 1 3 x 3 − ··· ( | x | < 1 )
Derivatives
d d x [ a f ( x ) ] = a d d x f ( x ) d d x [ a f ( x ) ] = a d d x f ( x ) d d x [ f ( x ) + g ( x ) ] = d d x f ( x ) + d d x g ( x ) d d x [ f ( x ) + g ( x ) ] = d d x f ( x ) + d d x g ( x ) d d x [ f ( x ) g ( x ) ] = f ( x ) d d x g ( x ) + g ( x ) d d x f ( x ) d d x [ f ( x ) g ( x ) ] = f ( x ) d d x g ( x ) + g ( x ) d d x f ( x ) d d x f ( u ) = [ d d u f ( u ) ] d u d x d d x f ( u ) = [ d d u f ( u ) ] d u d x d d x x m = m x m − 1 d d x x m = m x m − 1 d d x sin x = cos x d d x sin x = cos x d d x cos x = − sin x d d x cos x = − sin x d d x tan x = sec 2 x d d x tan x = sec 2 x d d x cot x = − csc 2 x d d x cot x = − csc 2 x d d x sec x = tan x sec x d d x sec x = tan x sec x d d x csc x = − cot x csc x d d x csc x = − cot x csc x d d x e x = e x d d x e x = e x d d x ln x = 1 x d d x ln x = 1 x d d x sin −1 x = 1 1 − x 2 d d x sin −1 x = 1 1 − x 2 d d x cos −1 x = − 1 1 − x 2 d d x cos −1 x = − 1 1 − x 2 d d x tan −1 x = 1 1 + x 2 d d x tan −1 x = 1 1 + x 2 Integrals
∫ a f ( x ) d x = a ∫ f ( x ) d x ∫ a f ( x ) d x = a ∫ f ( x ) d x ∫ [ f ( x ) + g ( x ) ] d x = ∫ f ( x ) d x + ∫ g ( x ) d x ∫ [ f ( x ) + g ( x ) ] d x = ∫ f ( x ) d x + ∫ g ( x ) d x ∫ x m d x = x m + 1 m + 1 ( m ≠ − 1 ) = ln x ( m = −1 ) ∫ x m d x = x m + 1 m + 1 ( m ≠ − 1 ) = ln x ( m = −1 ) ∫ sin x d x = − cos x ∫ sin x d x = − cos x ∫ cos x d x = sin x ∫ cos x d x = sin x ∫ tan x d x = ln | sec x | ∫ tan x d x = ln | sec x | ∫ sin 2 a x d x = x 2 − sin 2 a x 4 a ∫ sin 2 a x d x = x 2 − sin 2 a x 4 a ∫ cos 2 a x d x = x 2 + sin 2 a x 4 a ∫ cos 2 a x d x = x 2 + sin 2 a x 4 a ∫ sin a x cos a x d x = − cos 2 a x 4 a ∫ sin a x cos a x d x = − cos 2 a x 4 a ∫ e a x d x = 1 a e a x ∫ e a x d x = 1 a e a x ∫ x e a x d x = e a x a 2 ( a x − 1 ) ∫ x e a x d x = e a x a 2 ( a x − 1 ) ∫ ln a x d x = x ln a x − x ∫ ln a x d x = x ln a x − x ∫ d x a 2 + x 2 = 1 a tan −1 x a ∫ d x a 2 + x 2 = 1 a tan −1 x a ∫ d x a 2 − x 2 = 1 2 a ln | x + a x − a | ∫ d x a 2 − x 2 = 1 2 a ln | x + a x − a | ∫ d x a 2 + x 2 = sinh −1 x a ∫ d x a 2 + x 2 = sinh −1 x a ∫ d x a 2 − x 2 = sin −1 x a ∫ d x a 2 − x 2 = sin −1 x a ∫ a 2 + x 2 d x = x 2 a 2 + x 2 + a 2 2 sinh −1 x a ∫ a 2 + x 2 d x = x 2 a 2 + x 2 + a 2 2 sinh −1 x a ∫ a 2 − x 2 d x = x 2 a 2 − x 2 + a 2 2 sin −1 x a ∫ a 2 − x 2 d x = x 2 a 2 − x 2 + a 2 2 sin −1 x a ∫ 1 ( x 2 + a 2 ) 3 / 2 d x = x a 2 x 2 + a 2 ∫ 1 ( x 2 + a 2 ) 3 / 2 d x = x a 2 x 2 + a 2 ∫ x ( x 2 + a 2 ) 3 / 2 d x = – 1 x 2 + a 2 ∫ x ( x 2 + a 2 ) 3 / 2 d x = – 1 x 2 + a 2
