Propiedades trigonométricas e identidades trigonométricas.

Dominios de las funciones trigonométricas.

arcsin (x) = y ⇔ x = sin (y)
y = sin (x) ; -π/2 ≤ x ≤ π/2
y = arcsin (x) ; -1 ≤ x ≤ 1

y = arccos (x) ⇔ x = cos (y)
y = arccos (x) ; -1 ≤ x ≤ 1
x = cos (y) ; 0 ≤ y ≤ π
y = cos (x) ; 0 ≤ x ≤ π
y = arccos (x) ; -1 ≤ x ≤ 1

y = arctan (x) ⇔ x = tan (y)
x = arctan (y) ; -∞ < y < ∞
x = tan (y) ; -π/2 < y < π/2
y = tan (x) ; -π/2 < x < π/2
y = arctan (x) ; -∞ < x < ∞

y = arccot (x) ; -∞ < x < ∞
y = arcsec (x) ; |x| ≥ 1
y = arccsc (x) ; |x| ≥ 1

Propiedades de las funciones trigonométricas inversas.

Si -1 ≤ x ≤ 1 ^ -π/2 ≤ y ≤ π/2 ∴ sin [arcsin (x)] = x ; arcsin [sin (y)] = y

Si -∞ ≤ x ≤ ∞ ^ -π/2 ≤ y ≤ π/2 ∴ tan [arctan (x)] = x ; arctan [tan (y)] = y

Si |x| ≥ 1 ^ 0 ≤ y ≤ π/2 ∴  sec [arcsec (x)] = x ; arcsec [sec (y)] = y

Si -1 ≤ x ≤ 1 ^ 0 ≤ y ≤ π ∴ cos [arccos (x)] = x ; arccos [cos (y)] = y

Si -∞ ≤ x ≤ ∞ ^ 0 ≤ y ≤ π ∴ cot [arccot (x)] = x ; arccot [cot (y)] = y

Si |x| ≥ 1 ^ -π/2 ≤ y ≤ 0 ∴  csc [arccsc (x)] = x ; arccsc [csc (y)] = y

Identidades recíprocas.

sinθ = 1/cscθ
cosθ = 1/secθ
tanθ = 1/cotθ
cotθ = 1/tanθ
secθ = 1/cosθ
cscθ = 1/sinθ

Identidades tangente y cotangente.

tanθ = sinθ/cosθ
cotθ = cosθ/tanθ

Identidades pitagóricas.

sin²θ + cos²θ = 1
1 + tan²θ = sec²θ
1 + cot²θ = csc²θ

Fórmulas para negativos.

sin(-t) = -sint
cos(-t) = cost
tan(-t) = -tant
cot(-t) = -cott
sec(-t) = sect
csc(-t) = -csct

El coseno y la secante son funciones pares, las otras son impares.

Fórmulas de suma y resta.

sin(u+v) = sinucosv + sinvcosu
sin(u-v) = sinucosv – sinvcosu
sin(θ+θ) = sinθcosθ + sinθcosθ = 2sinθcosθ
sin(2θ) = 2sinθcosθ

cos(u+v) = cosucosv – sinusinv
cos(u-v) = cosucosv + sinusinv
cos(2θ) = cos(θ+θ) = cosθcosθ – sinθsinθ = cos²θ – sin²θ = cos2θ

tan(u+v) = (tanu+tanv)/(1-tanutanv)
tan(u-v) = (tanu-tanv)/1(1+tanutanv)
tanπ = sinπ/cosπ = 0/-1 = 0

sin75º = sin(45º+30º)
sin15º = sin(45º-30º)

Ángulos de referencia.

  1. θ = ángulo de referencia.
  2. θ = 180º o π – ángulo de referencia.
  3. θ – 180º = ángulo de referencia.
  4. 360º – θ = ángulo de referencia.

Cuadrantes de identidades trigonométricas inversas.

  • sin¯¹(x) = 1 y 4 cuadrante.
  • cos¯¹(x) = 1 y 2 cuadrante.
  • tan¯¹(x) = 1 y 4 cuadrante.
  • cot¯¹(x) = 1 o 2 cuadrante.
  • sec¯¹(x) = 1 o 2 cuadrante.
  • csc¯¹(x) = 1 o 4 cuadrante.

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Filed under: Civil Engineering, Estudio

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PIITster. Diploma in Economics and Business. 6th best Discus Thrower in Central America. 5x Discus National Winner (El Salvador). Civil Engineering Sophomore. Yogini. Small Product Lab Winner. Author of The Mini-Guide for Writing a Super Complete Post in 20 Minutes. 5x Shotput National Winner (El Salvador). Business Management Junior.