In right triangles: SOH CAH TOA
In any triangle sin A sin B sin C
Law of Sines: =
=
a
b
c
Law of Cosines: c 2 = a 2 + b 2 − 2ab cos C Area of a Triangle: 12 ab sin C K
K
Given: u = ( x1 , y1 , z1 ) and v = ( x2 , y2 , z2 ) Vector Equation of a Line: ( x, y, z ) = ( x0 , y0 , z0 ) + t (a, b, c) Parametric Equations of a Line: x = x0 + at ; y = y0 + bt ; z = z0 + ct Velocity: (a, b, c) Speed: velocity Angle between two K K
u•v
vectors: cos θ = K K u v
Vector Cross Product: i
j k
K K
u × v = x1 y1 z1 x2 y2 z2
Polar x2 + y2 = r 2
x = r cos θ ( x, y ) ( r ,θ ) y = r sin θ
Complex rcisθ ( x + yi ) n
DeMoivre’s Theroem: If z = rcisθ , then z = r n cis (nθ ) Vectors JJK
uv = ( x2 − x1 , y2 − y1 , z2 − z1 ) JJK
uv = ( x2 − x1 ) 2 + ( y2 − y1 ) 2 + ( z2 − z1 ) 2 K K
Vector Dot Product: u • v = x1 x2 + y1 y2 + z1 z2 Properties: K K K K
1. u • v = v • u
K K
K K
2. If u • v = 0, then u ⊥ v
K K
K K
3. k (u • v) = ku • v
K K JK K K K JK
4. u • (v + w) = u • v + u • w
Cartesian Equation of a Plane: ax + by + cz = d , d = ax0 + by0 + cz0 where (a, b, c) is the Real Rectangular perpendicular direction vector to the plane and ( x0 , y0 , z0 ) is a point on the plane. Vector Properties: K K
K
K
1. u × v is ⊥ to the plane containing u and v
K K
K K
K K JK
K K
K JK
2. u × v = −(v × u )
4. u × (v + w) = (u × v) + (u × w)
K K
u×v
K K
K K
v
&v
3.
sin
=
5.
If
u
×
=
0,
then
u
θ
K
K
u v
Polar Exponential Growth/Appreciation: Exponential Decay/Depreciation: Continuous Interest: Compound Interest: A = P (1 + r )
A = P (1 − r )
A = Pe
A = P 1 + nr
t
t
n(a1 + an )
an = a1 + (n − 1)d S n =
2
(
)
y = a ( x − h) 2 + k
Equation of a parabola with vertex ( h, k ) and axis of symmetry y = k : x = a( y − k )2 + h
Length of the latus rectum =
1
a
Properties of Logarithms log b (m ⋅ n) = log b m + log b n
log b ( mn ) = log b m − log b n
log b mn = n log b m
log b m =
log m
(change of base)
log b
b g
nt
Geometric Sequences and Series Formulas: Arithmetic Sequences and Series Formulas: The Distance Formula 2
2
d = ( x2 − x1 ) + ( y2 − y1 ) The Midpoint Formula x1 + x2 y1 + y2
, 2
2
Equation of a circle with center ( h, k ) and radius r : 2
2
2
( x − h) + ( y − k ) = r
Equation of an ellipse having foci ( ± c, 0 ) 2a : and sum of focal radii 2
2
y
2
ax 2 + b2 = 1 , where c = a 2 − b 2 . Equation of an ellipse having foci ( 0, ± c ) and focal radii 2a : x22 + y22 = 1 , where c 2 = a 2 − b 2 . b
a
Equation of a hyperbola having foci ( ± c, 0 ) and difference of focal radii 2a : x2 y 2
2
= a 2 + b 2 . 2 − 2 = 1 , where c
b
a
Equation of a hyperbola having foci ( 0, ± c ) and difference of focal radii 2a : y 2 x2
2
2
2
a 2 − b2 = 1 , where c = a + b . Equation of a parabola with vertex ( h, k ) and axis of symmetry x = h : rt
an = a1r
n −1
(1 − r n )
a
S n = a1
SG = 1 , r < 1 (1 − r )
1− r
Derivative Formula General Rules d
d
[ f ( x) + g ( x) ] = f '( x) + g '( x)
[ f ( x) − g ( x)] = f '( x) − g '( x)
dx
dx
d [ f ( x) ⋅ g ( x) ] = f '( x) ⋅ g ( x) + f ( x) ⋅ g '( x) product rule
dx
d
f ( g ( x)) ] = f '( g ( x)) ⋅ g '( x) chain rule
dx [
Power Rules d n
d ⎡ f ( x) ⎤ g ( x) ⋅ f '( x) − g '( x) ⋅ f ( x)
x = nx n −1
quotient rule ⎢
⎥=
2
[ g ( x)]
dx ⎣ g ( x) ⎦
d
[cf ( x)] = cf '( x)
dx
dx
( )
d
(c) = 0
dx
The Derivative of a Function f '( x) = lim
h →0
f ( x + h) − f ( x )
h
d
( cx ) = c
dx