Math Equations

Math is mainly how comfortable you are with the whole thing. However, you do need to know some formulas.

The formulas below are hopefully all that you will need to know. (These formulas were organized for preparation for the Junior Semester 2 final.)

Trigonometry

Basic Trig
sin θ = y/r
cos θ = x/r
tan θ = y/x

Trig Identities
sin² θ + cos² θ = 1
1 + tan² θ = sec² θ
1 + cot² θ = csc² θ

Triangle Area
K = ½ ab sin C = ½ bc sin A = ½ ac sin B
K = √(s(s – a)(s – b)(s – c)), when s = ½ (a + b + c)

Trig Laws

The Law of Sines
(sin A) / a = (sin B) / b = (sin C) / c
The Law of Cosines
c² = a² + b² – 2 ab cos C

Trig Formulas

Sine
sin (α + β) = sin α cos β + cos α sin β
sin (α – β) = sin α cos β – cos α sin β
Cosine
cos (α + β) = cos α cos β – sin α sin β
cos (α – β) = cos α cos β + sin α sin β
Tangent
tan (α + β) = tan α + tan β / 1 – tan α tan β
tan (α – β) = tan α – tan β / 1 + tan α tan β

Polar
Component form- (magnitiude, direction)

Polar Conversion

Rectangular to Polar
x = r cos θ
y = r sin θ
x² + y² = r²
Polar to Rectangular
r = ± √(x² + y²)
sin θ = y/r
cos θ = x/r
tan θ = y/x

Dot Product
v₁ • v₂ = x₁x₂ + y₁y₂

Properties of the Dot Product
1. u • v = v • u
2. u • u = |u|²
3. k(u • v) = (ku) • v
4. u • (v + w) = u • v + u • w

Angle Between Two Vectors
cos θ = u • v / |u||v|

Series

Sum of a Finite Arithmetic Series (t₁ + t₂ + t₃ + ... + t_n)
S_n = n(t₁ + t_n) / 2
Sum of a Finite Geometric Series (t₁ + t₁r + t₁r² + ... + t₁rⁿ)
S_n = t₁(1 – rⁿ) / (1 – r)
Sum of an Infinite Geometric Series (t₁ + t₁r + t₁r² + ...)
S = t₁ / (1 – r), If |r| < 1

Sigma

Sum of Integers

  n
Σ k   = n(n + 1) / 2
k=1

Sum of Squares

  n
Σ k²   = n(n + 1)(2n + 1) / 6
k=1

Sum of Cubes

  n
Σ k³   = [n(n + 1) / 2]²
k=1