Cheat Sheet 2
Math 114
Circle with center (h, k) and radius r: (x − h)2 + (y − k)2 = r2 .
Circumference: 2πr, Area: πr2 .
2
2
Ellipse: xa2 + yb2 = 1. Area: πab.
Area of a parallelogram: base * height.
Area of a triangle: 1/2∗ base * height.
Area of a trapezoid: 12 (b1 + b2 )h.
Area Stretch Theorem: Suppose R is a region in the coordinate plane, c, d positive numbers. Let R0
be the region obtained from R by stretching horizontally by a factor of c and stretching vertically by a
factor of d. Then
the area of R0 equals cd times the area of R.
Definition of e: e is the number such that the area under the graph of f (x) = x1 between x = 1 and
x = e is 1.
Definition of ln c for c > 0: it’s the area under the graph of f (x) = x1 between x = 1 and x = c.
Approximations: if t is close to 0, then ln(1 + t) ≈ t, and et ≈ 1 + t.
If |x| is much greater than |r|, then (1 + xr )x ≈ er .
Compound Interest: The accumulation of an initial amount P invested at annual interest rate r for t years
is: P (1 + nr )nt .
If interest is compounded continuously at annual interest rate r for t years, then an initial amount P
grows to P ert .
The same is true for any quantity that grows at rate r per time unit, for t time units with initial amount P .
Trigonometry:
Length of a circular arc on the unit circle corresponding to an angle of θ degrees:
θπ
.
180
You need to know the sines and cosines of 30, 45, 60, 90, 180, 270 degrees.
Radians: 2π radians = 360 degrees.
Conversions: θ radians = 180θ
, θ degrees =
π
θπ
180
.
A slice with angle θ radians inside a circle of radius r has area 12 θr2 .
A point on the unit circle, representing the endpoint of the radius of angle θ has coordinates (cos θ, sinθ).
More Trig functions: tan θ =
sin θ
,
cos θ
Trigonometry in a right triangle:
adjacentside
cos θ = hypothenuse
,
sec θ =
1
,
cos θ
sin θ =
csc θ =
oppositeside
,
hypothenuse
1
,
sin θ
cot θ =
cos θ
.
sin θ
tan θ =
oppositeside
.
adjacentside
Trigonometric Identities:
cos2 θ + sin2 θ = 1, tan2 θ + 1 = sec2 θ, cos(−θ) = cos θ, sin(−θ) = − sin θ, tan(−θ) = − tan θ.
cos( π2 − θ) = sin θ, sin( π2 − θ) = cos θ, tan( π2 − θ) = tan1 θ ,
cos(θ + π) = − cos θ, sin(θ + π) = − sin θ, tan(θ + π) = tan θ,
cos(θ + 2π) = cos θ, sin(θ + 2π) = sin θ, tan(θ + 2π) = tan θ,
Arccosine: For t ∈ [−1, 1], the arccosine of t, cos−1 t, is the unique angle in [0, π], whose cosine equals t.
, π ], whose sine equals t.
Arcsine: For t ∈ [−1, 1], the arcsine of t, sin−1 t, is the unique angle in [ −π
2 2
Arctangent: For t ∈ R, the arctangent of t, tan−1 t, is the unique angle in ( −π
, π ), whose tangent equals t.
2 2
Inverse Trigonometric identities:
cos−1 (−t) = π − cos−1 t, sin−1 (−t) = − sin−1 t, tan−1 (−t) = − tan−1 t.
cos−1 t + sin−1 t = π2
π
− tan−1 t
if t > 0
−1 1
2
tan ( t ) =
− π2 − tan−1 t if t < 0
The area of a triangle with sides of length a and b and with angle θ between the two sides is 12 ab sin θ.
The area of a parallelogram with adjacent sides of length a and b and with angle θ hetween those two sides
is ab sin θ.
Law of Sines:
In a triangle with sides whose lengths are a, b, and c, with corresponding angles A, B, C:
sin A
a
sin B
b
=
=
Law of Cosines:
In a triangle with sides whose lengths are a, b, and c, and with an angle C opposite to side b:
c2 = a2 + b2 − 2ab cos C.
Double angle formulas:
cos(2θ) = 1 − 2 sin2 θ = 2 cos2 θ − 1 = cos2 θ − sin2 θ,
Half AngleqFormulas:
q
θ
θ
1−cos θ
cos 2θ = ± 1+cos
,
sin
=
±
, tan 2θ =
2
2
2
sin θ
1+cos θ
sin(2θ) = 2 sin θ cos θ,
=
tan(2θ) =
1−cos θ
.
sin θ
Addition/Subtraction Formulas
cos(u + v) = cos u cos v − sin u sin v, cos(u − v) = cos u cos v + sin u sin v,
sin(u + v) = sin u cos v + cos u sin v, sin(u − v) = sin u cos v − cos u sin v,
tan(u + v) =
tan u+tan v
1−tan u tan v
tan(u − v) =
tan u−tan v
1+tan u tan v
Given the trigonometric function f (x) = a cos(bx + c) + d,
is the period, −c/b is the phase shift, d is the vertical shift.
|a| is the amplitude, 2π
b
Polarpcoordinates: x = r cos θ, y = r sin θ.
r = x2 + y 2 , tan θ = xy .
Dot Product: u · v = |u||v| cos θ, where |u| =
p
u21 + u22 .
2 tan θ
,
1−tan2 θ
sin C
.
c