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Polar Coordinates
r2 = x2 + y 2
x = r cos θ
y = r sin θ
tan θ =
y
x
The point represented by polar coordinates (r, θ) is also represented by (r, θ +2nπ) or (−r, θ +
(2n + 1)π), if n is an integer.
Polar curves A polar curve is the graph of a polar equation r = f (θ).
Areas of polar regions
∫b
A=
1
[f (θ)]2 dθ
2
a
If a region R is bounded by two polar curves of equations r = f (θ) and r = g(θ), with
a ≤ θ ≤ b, where f (θ) ≥ g(θ) ≥ 0 and 0 < b − a < 2π, then its area is given by
1
A=
2
∫b
[ 2
]
f (θ) − g 2 (θ) dθ
a
Arc Length of a polar curve
∫b
√
(
r2
L=
+
dr
dθ
)2
dθ
a
Vectors If P = (x1 , y1 , z1 ) and Q = (x2 , y2 , z2 ), then the vector P⃗Q is given, in coordinates,
by P⃗Q =< (x2 − x1 ), (y2 − y1 ), (z2 − z1 ) >.
Length (magnitude) If ⃗v =< v1 , v2 , v3 > then |⃗v | =
√
v12 + v22 + v32 . The vector ⃗u =
1
⃗v
|⃗v |
a unit vector, pointing in the direction of ⃗v .
Dot Product If ⃗u =< u1 , u2 , u3 > and ⃗v =< v1 , v2 , v3 > then ⃗u · ⃗v = u1 v1 + u2 v2 + u3 v3 .
(
)
⃗u · ⃗v
⃗u · ⃗v
cos(θ) =
proj⃗v (⃗u) =
⃗v
|⃗u||⃗v |
|⃗v |2
Cross Product If ⃗u =< u1 , u2 , u3 > and ⃗v =< v1 , v2 , v3 > then
⃗ ⃗ ⃗ i j k
⃗u × ⃗v = u1 u2 u3 =< u2 v3 − v2 u3 , u3 v1 − v3 u1 , u1 v2 − v1 u2 >
v1 v2 v3 1
is
Moreover, |⃗u × ⃗v | = |⃗u||⃗v | sin(θ).
The volume of the box determined by ⃗u, ⃗v and w
⃗ is given by:
(⃗u × ⃗v ) · w
⃗ = (⃗v × w)
⃗ · ⃗u = (w
⃗ × ⃗u) · ⃗v
Lines and Planes
The line L going through P (x0 , y0 , z0 ) parallel to ⃗v =< v1 , v2 , v3 > is given by:
< x, y, z >=< x0 , y0 , z0 > +t < v1 , v2 , v3 >
with t ∈ R. The distance from the point S to the line through P parallel to ⃗v is given by:
d=
|P⃗S×⃗v |
.
|⃗v |
The plane going through P (x0 , y0 , z0 ) and normal to ⃗n =< A, B, C > is given by:
A(x − x0 ) + B(y − y0 ) + C(z − z0 ) = 0 or Ax + By + Cz = D.
The distance from the point S to the plane through P normal to ⃗n is given by: d = P⃗S ·
⃗
n .
|⃗
n| Vector Functions ⃗r(t) =< f (t), g(t), h(t) >. Taking limits, differentiation and integration
is done component by component.
Notation: ⃗r(t) is the position, ⃗v (t) = ⃗r′ (t) =
and ⃗a(t) = ⃗r′′ (t) =
d2
⃗r(t)
dt2
d
⃗r(t)
dt
is the acceleration.
2
is the velocity, |⃗v (t)| = |⃗r′ (t)| is the speed
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TRIGONOMETRIC IDENTITIES
• Reciprocal identities
1
1
cos u =
sin u =
csc u
sec u
1
1
cot u =
tan u =
cot u
tan u
1
1
sec u =
csc u =
sin u
cos u
• Pythagorean Identities
sin2 u + cos2 u = 1
1 + tan2 u = sec2 u
1 + cot2 u = csc2 u
• Quotient Identities
cos u
sin u
cot u =
tan u =
cos u
sin u
• Co-Function Identities
π
π
sin( − u) = cos u cos( − u) = sin u
2
2
tan(
csc(
π
π
− u) = cot u cot( − u) = tan u
2
2
π
− u) = sec u
2
sec(
π
− u) = csc u
2
• Parity Identities (Even & Odd)
sin(−u) = − sin u cos(−u) = cos u
tan(−u) = − tan u cot(−u) = − cot u
csc(−u) = csc u
sec(−u) = sec u
• Sum & Difference Formulas
sin(u ± v) = sin u cos v ± cos u sin v
cos(u ± v) = cos u cos v ∓ sin u sin v
tan u ± tan v
tan(u ± v) =
1 ∓ tan u tan v
• Double Angle Formulas
sin(2u) = 2 sin u cos u
cos(2u) = cos2 u − sin2 u
= 2 cos2 u − 1
= 1 − 2 sin2 u
2 tan u
tan(2u) =
1 − tan2 u
• Power-Reducing/Half Angle Formulas
1 − cos(2u)
2
1 + cos(2u)
cos2 u =
2
1 − cos(2u)
tan2 u =
1 + cos(2u)
sin2 u =
• Sum-to-Product Formulas
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u−v
u+v
cos
sin u + sin v = 2 sin
2
2
sin u − sin v = 2 cos
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cos u + cos v = 2 cos
u+v
2
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cos u − cos v = −2 sin
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u+v
2
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sin
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u+v
2
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cos
�
u−v
2
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sin
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u−v
2
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u−v
2
• Product-to-Sum Formulas
sin u sin v =
1
[cos(u − v) − cos(u + v)]
2
cos u cos v =
1
[cos(u − v) + cos(u + v)]
2
sin u cos v =
1
[sin(u + v) + sin(u − v)]
2
cos u sin v =
1
[sin(u + v) − sin(u − v)]
2
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