AP Precalc Everything You Need to Know

sinθ

1/cscθ

cosθ

1/secθ

secθ reciprocal identity

1/cosθ

cscθ reciprocal identity

1/sinθ

cotθ reciprocal identity

1/tanθ

tanθ quotient identity

sinθ/cosθ

cotθ quotient identity

cosθ/sinθ

sin²θ + cos²θ Pythagorean identity

1

1 + tan²θ Pythagorean identity

sec²θ

1 + cot²θ Pythagorean identity

csc²θ

sin(α + β) sum identity

sinαcosβ + cosαsinβ

sin(α - β) difference identity

sinαcosβ - cosαsinβ

cos(α + β) sum identity

cosαcosβ - sinαsinβ

cos(α - β) difference identity

cosαcosβ + sinαsinβ

sin 2θ double-angle identity

2 sinθcosθ

cos 2θ double-angle identity

cos²θ - sin²θ

1/sin x

csc x

1/cos x

sec x

1/tan x

cot x

1/csc x

sin x

1/sec x

cos x

1/cot x

tan x

sin²x+cos²x

1

1-cos²x

sin²x

1-sin²x

cos²x

1+tan²x

sec²x

1+cot²x

csc²x

sec²x-1

tan²x

csc²x-1

cot²x

sin x / cos x

tan x

cos x/ sin x

cot x

Relative Extrema (local)

where it switches between decreasing and increasing, or end point if restricted domain

Absolute extrema (global)

the greatest from all the absolute max or the least of all local minima

Even and odd mulitplicity

no multiplicity= cross through

odd multiplicity= flattens but crosses

even = bounce (doesn’t cross)

Imaginary numbers come in _____________!!!!

PAIRS

if 2 - 3i is one then 2+3i is the conjugate

Even and odd functions

Even: symmetric over y-axis —> f(-x) = f(x)

Odd: symmetric about the origin —> g(-x) = -g(x)

Limit notation from graphs and equations

lim f(x) = L

x→ c

x approaches vertical asymptotes

f(x) approaches horizontal asymptotes

end behavior → horizontal asymptote

ax^n/bx^d

same degree = l.c/l.c

big bottom = y=0

big top = slant asymptote divide

Vertical asymptotes

denominator = 0

stronger than holes

Table best modeled by:

Transformations

Vertical translation: g(x) = f(x) + k

Vertical translation: g(x) = af(x)

Horizontal dilation: g(x) = f(bx) 1/b multiply

Horizontal translation: g(x) = f(x + h)

Arithmetic sequence

a_n = a₀ + dn

a_n = a_k +d(n - k)

Geometric sequence

g_n = g_kr(n - k)

Finding inverses from equations

switch x and y

an inverse function is a function when…

log rules

Sinusoidal graphs

period: 2b/ pi

amplitude: max - min/2

midline: max + min/2

Sine

Midline

High

Midline

Low

Midline

-sin

M

L

M

H

M

cosine

H

M

L

M

H

-cos

L

M

H

M

L

polar coordinates