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R 2 γ 1 x 2 +y 2 = 1
θ
γ P cos θ P sin θ
P cos 2 θ + sin 2 θ = 1 P ∈ γ (x, y) = (cos θ, sin θ)
(x, y) (0, 0) (1, 0) θ/2
cos : R → [−1, 1] sin : R → [−1, 1]
2π cos(θ + 2π) = cos(θ) sin(θ + 2π) = sin(θ)
tan : R\{π/2+kπ : k ∈ Z} → R tan(x) = sin(x)
cos(x)
cot : R\{kπ : k ∈ Z} → R cot(x) = cos(x)
sin(x)
tan cot
π tan(x + π) = tan x cot(x + π) = cot x
sec : R \ {kπ : k ∈ Z} → R csc : R \ {π/2 + kπ : k ∈ Z} → R
sec(x) = 1
cos(x)
csc(x) = 1
sin(x) sec2 x + csc 2 x = csc 2 x · sec 2 x
x = kπ/2 k ∈ Z
y = arcsin x x = sin y x ∈ [−1, 1] y ∈ [−π/2, π/2]
y = arccos x x = cos y x ∈ [−1, 1] y ∈ [0, π]
y = arctan x x = tan y x ∈ R y ∈] − π/2, π/2[
y = arc sec x x = sec y y = arccos(1/x) |x| ≥ 1 y ∈]0, π[\{π/2}
y = arc csc x x = csc y y = arcsin(1/x) |x| ≥ 1 y ∈] − π/2, π/2[\{0}
y = arc cot x x = cot y y = arctan(1/x) x ∈ R y ∈]0, π[
d
sin x = cos x,
dx
d
cos x = − sin x,
dx
d
dx tan x = sec2 x = 1 + tan 2 x,
d
dx cot x = − csc2 x,
d
sec x = tan x sec x,
dx
d
csc x = − csc x cot x,
dx
d
arcsin x =
dx
1
√ 1 − x 2
d
−1
arccos x = √
dx 1 − x2 d
1
arctan x =
dx 1 + x2 d
−1
arc cot x =
dx 1 + x2 d
arc sec x =
dx
d
arc csc x =
dx
1
|x| √ x 2 − 1
−1
|x| √ x 2 − 1