Arc Length

s = rθ

Area of a Sector of a Circle

A = ½r²θ

Linear Speed of an Object Traveling in Circular Motion

v = s/t

Angular Speed of an Object Travelling in Circular Motion:

ω = θ/t

Sin(π/6)

Sin(π/6) = 1/2

Cos(π/6)

Cos(π/6) = √(3)/2

Tan(π/6)

Tan(π/6) = √(3)/3

Sin(π/3)

Sin(π/3) = √(3)/2

Cos(π/3)

Cos(π/3) = ½

Tan(π/3)

Tan(π/3) = √(3)

Sinθ

Domain:

Range:

Sinθ

Domain: All real numbers

Range: -1 ≤ sinθ ≤ 1

Cosθ

Domain:

Range:

Cosθ

Domain: All real numbers

Range: -1 ≤ cosθ ≤ 1

Tanθ:

Domain:

Range:

Tanθ:

Domain: All real numbers, except odd interger multiples of π/2

Range: All Real Numbers

Pythagorean Identity:

sin²θ + cos²θ = 1

tan²θ + 1 = ?

tan²θ + 1 = sec²θ

Cot²θ + 1 = ?

Cot²θ + 1 = Csc²θ

Equation for a Graph of the Sine/Cos Function:

 y = Asin(ωx - φ) + B

y = Asin[ω(x-(φ/ω))]

Amplitude = |A|

Period = T = 2π/ω

Phase Shift = φ/ω

B = Vertical Shift

Equation for the Graph of a Tangent Function:

y = Atan(ωx) + B

Vertical Stretch = |A|

Period = T = π/ω

Vertical Shift = B

Translate:

sin^-1(x) = y

sin(y) = x

-1 ≤ x ≤ 1 and -π/2 ≤ y ≤ π/2

Sum and Difference Formulas:

cos(α + β) = 

sin(α + β) =

tan(α + β) = 

cos(α + β) = cosα cosβ - sinα sinβ

sin(α + β) = sinα cosβ + cosα sinβ

tan(α + β) = (tanα + tanβ)/(1- tanα tanβ)

Sum and Difference Formulas:

cos(α - β) = 

sin(α - β) = 

tan(α - β) = 

cos(α - β) = cosα cosβ + sinα sinβ

sin(α - β) = sinα cosβ - cosα sinβ

tan(α - β) = (tanα - tanβ)/(1 + tanα tanβ)

Double Angle Formulas:

sin (2θ) = 

sin(2θ) = 2 sin(θ) cos(θ)

Double Angle Formulas:

Cos(2θ) =

Cos(2θ) = cos²θ - sin²θ

Cos(2θ) = 1 - 2sin²θ

Cos(2θ) = 2cos²θ - 1

Double Angle Formulas:

Tan(2θ) = 

Tan(2θ) = 2tan(θ)/(1-tan²θ)

Half Angle Formulas:

sin(α/2) = 

cos(α/2) =

tan(α/2) = 

sin(α/2) = ±√((1 - cosα)/2)

cos(α/2) = ±√((1 + cosα)/2)

tan(α/2) = ±√((1 - cosα)/(1 + cosα))

tan(α/2) = (1- cosα) / sinα = sinα / (1+ cosα)

± determined by the quadrant of α/2

Product - to - Sum Formulas:

sinα sinβ = 

cosα cosβ = 

sinα cosβ = 

sinα sinβ = ½[cos(α - β) - cos(α + β)]

cosα cosβ = ½[cos(α - β) + cos(α + β)]

sinα cosβ = ½[sin(α + β) + sin(α - β)]

Sum - to - Product Formulas:

sinα + sinβ =

sinα - sinβ =

cosα + cosβ =

cosα - cosβ =

sinα + sinβ = 2sin((α + β)/2) cos((α - β)/2)

sinα - sinβ = 2sin((α - β)/2) cos((α + β)/2)

cosα + cosβ = 2cos((α + β)/2) cos((α - β)/2)

cosα - cosβ = -2sin((α + β)/2) sin((α - β)/2)

Complementary Angle Theorem:

Cofunctions of Complementary Angles are Equal:

tan(x) = cot(90-x)

tan(40°) = cot(50°)

The Law of Sines:

SinA = SinB = SinC

a          b           c

The Law of Cosines:

c² = a² + b² - 2ab cosC

b² = a² + c² - 2ac cosB

a² = b² + c² - 2bc cosA

Convert from Polar to Rectangular Coordinates:

x = rcosθ

y = rsinθ

Convert from Rectangular to Polar Coordinates:

r² = x² + y²

tanθ = y/x          if x ≠ 0

Find the Magnitude of a Point in the Complex Plane:

General Form of a Complex Number: x + yι

|z| = √(x² + y²)

Convert a Complex Number Between Rectangular and Polar Form:

z = x + yi = (r cosθ) + (r sinθ)i = r(cosθ + i sinθ)

Find the Product of Complex Numbers in Polar Form:

z₁z₂ = r₁r₂[cos(θ₁ + θ₂) + i sin(θ₁ + θ₂)]

Find the Quotient of Complex Numbers in Polar Form:

z₁/z₂ = r₁/r₂[cos(θ₁ - θ₂) + i sin(θ₁ - θ₂)]

De Moivre's Theorem:

(Used to raise a complex number to a power)

zⁿ = rⁿ[cos(nθ) + i sin(nθ)]

where n ≥ 1 (any positive interger)

Find a Complex Root:

zⁿ = w

where z is the complex n^th root of w

z_k = (ⁿ√r)[cos((θ₀/n) + ((2kπ)/n)) + i sin((θ₀/n) + ((2kπ)/n))]

where k = 0, 1, 2, 3, ... n-1

Define the horizontal and vertical components of a vector v in terms of the unit vectors i and j:

i = the unit vector on the positive x-axis

j = the unit vector on the positive y-axis

v = ‹a,b› = ai + bj

Operations with Vectors:

v + w =

v - w = 

αv = 

||v|| = 

v + w = (a₁ + a₂)i + (b₁ + b₂)j

v - w = (a₁ - a₂)i + (b₁ - b₂)j

αv = (αa₁)i + (αb₁)j

||v|| = √(a²₁ + b²₁)

Find a Unit Vector u That Has the Same Direction as a Given Vector v:

u = v

       ||v||

Find the Dot Product of Two Vectors:

v · w = 

given v = ‹a₁,b₁› and w = ‹a₂,b₂›

v · w = a₁a_{2 +} b₁b₂

Find the Angle Between Two Vectors:

cosθ =    u · v

               ||u|| ||v||

two vectors are parallel if cosθ = ±1

two vectors are perpendicular if v · w = 0