AP PreCalculus Formulas

Difference b/w Geometric and Arithmetic Sequences?

Geometric: Sequence that continues by multiplying by a number (1,3,9,27…)

Arithmetic: Sequence that continues by adding or subtracting by a number. (10,8,6,4,2,…)

Difference b/w Explicit (General term) and Recursive?

Explicit: formula used to find the value of a term based on it’s position

Recursive: formula used to find value of a term based on the previous term

The Arithmetic Sequence formulas are?

Explicit (General term): t_n = a + (n − 1)(d)

Recursive: t₁=a ; t_n= t_n-1 + d

The Geometric Sequence Formulas are?

Explicit (General term): t_n = a x r^n-1

Recursive: t₁ = a ; t_n = t_n-1 x r

What do “d” “r” & ‘“a” represent in sequence formulas?

a = initial term

d = common addition/difference (for Arithmetic)

r = common multiplication (for Geometric)

Difference b/w series and sequences?

Sequence: a group of terms following a specific order

Series: the sum of terms in a sequence (S₃₂ = the sum of the first 32 terms in a sequence)

What are the Arithmetic Series formulas?

Explicit (General term): S_n = n/2 [2a + (n-1)d]

Recursive: S_n = n/2 (a + t_n)

What are the Geometric Series formulas?

Explicit (General term): S_n = a(rⁿ - 1) / r - 1

Recursive: None.

All zeros (x-intercepts/roots) that are _____ numbers have a conjugate pair that is also a root.

Conjugate pairs.

Ex: (if 0.472 - 0.931i is a root of a function, 0.472 + 0.931i is also a root of the function)

How do you find the average rate of change (slope) for a poler function?

Use the slope formula (y₂ - y₁ / x₂ - x₁) but interchange y with radians (r) and x with degrees (θ), (r₂ - r₁ / θ₂ - θ₁).

What formula can be used to find the radius of a circle, when given the distance a terminal arm has moved in the circle, the point the arm intersects with the circle, and the length of arc isolated by the distance the terminal arm has moved?

θ x r = arc

What is the equation of a secant line?

y = m(x - x_#) + y_#

x: an x input

x_# and y_# = a point on the secant line Ex: (2,3)

An Odd function reflects over ___

the origin. (Max to Min)

An Even function reflects over ___

the y - Axis (Max to Max)

sin²θ = ?

1 - cos²θ

csc²θ = ?

1 + cot²θ

sec²θ = ?

tan²θ +1

sin (2θ) = ?

2(sinθ)(cosθ)

cos²θ - sin²θ = ?

cos (2θ)

2 cos²θ - 1 = ?

cos (2θ)

1 - 2 sin²θ = ?

cos (2θ)

sin (A + B) = ?

(sin (A) x cos (B)) + (sin (B) x cos (A))

cos (A ∓ B) = ?

(cos (A) x cos (B)) + (sin (A) x sin (B))