Download MA 108 FORMULAS (FINAL EXAM) For these formulas, assume θ to

MA 108 FORMULAS (FINAL EXAM)
For these formulas, assume θ to be measured in radians unless specified otherwise.
• Given a circle of radius r with a central angle θ, θ subtends an arc of length s where s = rθ .
1
• The area A of the sector formed by a central angle θ in a circle of radius r is A = r2 θ .
2
• If an object moves at constant speed around a circle of radius r, where s is the distance traveled
by the object, θ is the angle swept out by the object, and t is the elapsed time, the linear speed
v and the angular speed W of the object satisfy these equations.:
W =
• Radian-degree unit conversions:
θ
t
v=
s
t
v = rW .
180 ◦
π
= 1 radian ,
radians = 1 degree .
π
180
2π
• The graphs of y = A sin(ωx − φ) + B or y = A cos(ωx − φ) + B, where ω > 0, have period
ω
φ
and phase shift .
ω
π
• The period of a graph of the form A cot(ωx) + B or A tan(ωx) + B is
ω
Sum and Difference formulas:
cos(α + β) = cos α cos β − sin α sin β
cos(α − β) = cos α cos β + sin α sin β
sin(α + β) = sin α cos β + cos α sin β
sin(α − β) = sin α cos β − cos 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
Double Angle Formulas:
sin(2θ) = 2 sin θ cos θ
tan(2θ) =
2 tan θ
1 − tan2 θ
cos(2θ) = cos2 θ − sin2 θ = 1 − 2 sin2 θ = 2 cos2 θ − 1
Half Angle Formulas:
r
α
1 − cos α
=±
sin
2
2
cos
α
2
r
=±
More Half Angle Formulas:
α 1 − cos α
sin2
=
2
2
α 1 − cos α
tan2
=
=
2
1 + cos α
•
1 + cos α
2
cos2
α
2
tan
=
1 − cos α
=
sin α
180 ◦
π
= 1 radian ,
radians = 1 degree
π
180
1
α
2
1 + cos α
2
sin α
1 + cos α
r
=±
1 − cos α
1 + cos α
2
MA 108 FORMULAS (FINAL EXAM)
• If θ is a central angle (in radians) of a circle of radius r, then the area A of the sector of the circle
1
swept out by θ is given by the formula A = r2 θ
2
• The Law of Sines: For a triangle with sides a, b, c and opposite angles labeled A, B, C respectively,
sin B
sin C
sin A
=
=
a
b
c
• The Law of Cosines : For any triangle with sides a, b, c opposite angles A, B, C, the following
formulas hold:
a2 = b2 + c2 − 2bc cos A
b2 = a2 + c2 − 2ac cos B
c2 = a2 + b2 − 2ab cos C
• The area K of the SAS triangle with sides a, b and included angle C is given by the formula
1
K = ab sin C
2
• Heron’s formula: The area K of a triangle with sides a, b, and c is given by the formula
p
K = s(s − a)(s − b)(s − c)
1
where s = (a + b + c).
2
• An object moves with simple harmonic motion if it moves on a coordinate axis where the
distance d from the rest position at time t is given by either d = a cos ωt or d = a sin ωt, where
2π
ω
a, ω are constants with ω > 0. The motion has amplitude |a|, period
and frequency
.
ω
2π
• Let a be the displacement at t = 0 of an object of mass m with damping factor b. Assume that
2π
is the period of the simple harmonic motion model for the object. Then the displacement d
ω
of the oscillating object from its equilibrium position at time t is given by
" r
! #
2
b
−bt/2m
ω2 −
d(t) = ae
cos
t
4m2
Four cases for Parabolas with vertex at the origin, axis of symmetry a coordinate axis:
Vertex Focus Directrix Equation
Direction
Axis of Symmetry
(0, 0)
(a, 0)
x = −a
y 2 = 4ax
Opens Right
x-axis
(0, 0) (−a, 0)
x=a
y 2 = −4ax
Opens Left
x-axis
(0, 0)
(0, a)
y = −a
x2 = 4ay
Opens Upward
y-axis
(0, 0) (0, −a)
y=a
x2 = −4ay Opens Downward
y-axis
Four analogous cases for parabolas with axis of symmetry parallel to a coordinate axis, vertex
at (h, k)
Vertex
(h, k)
(h, k)
(h, k)
(h, k)
Focus
(h + a, k)
(h − a, k)
(h, k + a)
(h, k − a)
Directrix
x=h−a
x=h+a
y =k−a
y =k+a
Equation
(y − k)2 = 4a(x − h)
(y − k)2 = −4a(x − h)
(x − h)2 = 4a(y − k)
(x − h)2 = −4a(y − k)
Direction
Opens Right
Opens Left
Opens Upward
Opens Downward
Axis of Symmetry
y=k
y=k
x=h
x=h