sin(−x) = − sin x; cos(−x) = cos x; sin( π2 ± x) = cos x; cos( π2 ± x) = ∓ sin x;

FORMULARIO TRIGONOMETRIA

sin ² x + cos ² x = 1; tan x = _{cos x} ^{sin x} ; coth x = ^{cos x} _{sin x}

sin(−x) = − sin x; cos(−x) = cos x; sin( ^π ₂ ± x) = cos x; cos( ^π ₂ ± x) = ∓ sin x;

sin(π ± x) = ∓ sin x; cos(π ± x) = − cos x; sin(x + 2π) = sin x; cos(x + 2π) = cos x;

sin(x ± y) = sin x cos y ± cos x sin y; cos(x ± y) = cos x cos y ∓ sin x sin y sin(2x) = 2 sin x cos x; cos(2x) = cos ² x − sin ² x = 2 cos ² x − 1 = 1 − 2 sin ² x cos ² x = ^1+cos(2x) ₂ ; sin ² x = ^1−cos(2x) ₂

sin u + sin v = 2 sin ^u+v ₂ cos ^u−v ₂ ; sin u − sin v = 2 cos ^u+v ₂ sin ^u−v ₂ ; cos u + cos v = 2 cos ^u+v ₂ cos ^u−v ₂ ; cos u − cos v = −2 sin ^u+v ₂ sin ^u−v ₂ ;

sin x cos y = ¹ ₂ [sin(x + y) + sin(x − y)]; cos x cos y = ¹ ₂ [cos(x + y) + cos(x − y)];

sin x sin y = − ¹ ₂ [cos(x + y) − cos(x − y)]

Posto t = tan(x/2), si ha: sin x = _1+t ^2t

; cos x = ^1−t _1+t

; tan x = _1−t ^2t

; sin 0 = 0 cos 0 = 1 sin ^π ₆ = ¹ ₂ ; cos ^π ₆ =

√ 3

2 ; sin ^π ₄ =

√ 2

2 ; cos ^π ₄ =

√ 2 2 ; sin ^π ₃ =

√ 3

2 ; sin ^π ₃ = ¹ ₂ ; sin ^π ₂ = 1; cos ^π ₂ = 0;

DISUGUAGLIANZE

|sin x| ≤ |x| per ogni x ∈ R; 0 ≤ 1 − cos x ≤ ^x ₂

per ogni x ∈ R;

log(1 + x) ≤ x per ogni x > −1; |xy| ≤ ^x

^+y ₂

; ^(x+y) ₂

≤ x ² + y ² ; x ⁴ + y ⁴ ≤ (x ² + y ² ) ² SVILUPPI DI MACLAURIN

e ^x = 1 + x + ^x _2!

+ ^x _3!

+ · · · + ^x _n!

+ o(x ⁿ )

log(1 + x) = x − ^x ₂

+ ^x ₃

+ · · · + (−1) ^{n+1 x} _n

+ o(x ⁿ ) sin x = x − ^x _3!

+ ^x _5!

+ · · · + (−1) ^{n x} _(2n+1)!

+ o(x ²ⁿ⁺² ) cos x = 1 − ^x _2!

+ ^x _4!

+ · · · + (−1) ^{n x} _2n!

+ o(x ²ⁿ⁺¹ ) tan x = x + ¹ ₃ x ³ + ₁₅ ² x ⁵ + o(x ⁶ )

arctan x = x − ^x ₃

+ ^x ₅

+ · · · + (−1) ^{n x} _2n+1

+ o(x ²ⁿ⁺² ) (1 + x) ^α = 1 + αx + ^α(α−1) _2! x ² + α(α−1)(α−2))

3! x ³ + · · · + α(α−1)···(α−n+1)

n! x ⁿ + o(x ⁿ )

1