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Populaire Trigonométrie >

sin^2(x)=sec(x)

Solution

sin^{2} (x) = sec (x)

Solution

Aucune solution pour x \in R

étapes des solutions

sin^{2} (x) = sec (x)

Mettre les deux côtés au carré

(sin^{2} (x))^{2} = sec^{2} (x)

Soustraire

sec^{2} (x)

des deux côtés

sin^{4} (x) - sec^{2} (x) = 0

Exprimer avec sinus, cosinus

- sec^{2} (x) + sin^{4} (x)

Utiliser l'identité trigonométrique de base:

sec (x) = \frac{1}{cos ( x )}

= - (\frac{1}{cos ( x )})^{2} + sin^{4} (x)

Simplifier

- (\frac{1}{cos ( x )})^{2} + sin^{4} (x) : \frac{- 1 + sin ^{4} ( x ) cos ^{2} ( x )}{cos ^{2} ( x )}

- (\frac{1}{cos ( x )})^{2} + sin^{4} (x)

(\frac{1}{cos ( x )})^{2} = \frac{1}{cos ^{2} ( x )}

(\frac{1}{cos ( x )})^{2}

Appliquer la règle de l'exposant:

(\frac{a}{b})^{c} = \frac{a ^{c}}{b ^{c}}

= \frac{1 ^{2}}{cos ^{2} ( x )}

Appliquer la règle

1^{a} = 1

1^{2} = 1

= \frac{1}{cos ^{2} ( x )}

= - \frac{1}{cos ^{2} ( x )} + sin^{4} (x)

Convertir un élément en fraction:

sin^{4} (x) = \frac{sin ^{4} ( x ) cos ^{2} ( x )}{cos ^{2} ( x )}

= - \frac{1}{cos ^{2} ( x )} + \frac{sin ^{4} ( x ) cos ^{2} ( x )}{cos ^{2} ( x )}

Puisque les dénominateurs sont égaux, combiner les fractions:

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{- 1 + sin ^{4} ( x ) cos ^{2} ( x )}{cos ^{2} ( x )}

= \frac{- 1 + sin ^{4} ( x ) cos ^{2} ( x )}{cos ^{2} ( x )}

\frac{- 1 + cos ^{2} ( x ) sin ^{4} ( x )}{cos ^{2} ( x )} = 0

\frac{f ( x )}{g ( x )} = 0 \Rightarrow f (x) = 0

- 1 + cos^{2} (x) sin^{4} (x) = 0

Factoriser

- 1 + cos^{2} (x) sin^{4} (x) : (sin^{2} (x) cos (x) + 1) (sin^{2} (x) cos (x) - 1)

- 1 + cos^{2} (x) sin^{4} (x)

Récrire

- 1 + sin^{4} (x) cos^{2} (x)

comme

- 1 + (sin^{2} (x) cos (x))^{2}

- 1 + sin^{4} (x) cos^{2} (x)

Appliquer la règle de l'exposant:

a^{b c} = (a^{b})^{c}

sin^{4} (x) = (sin^{2} (x))^{2}

= - 1 + (sin^{2} (x))^{2} cos^{2} (x)

Appliquer la règle de l'exposant:

a^{m} b^{m} = (ab)^{m}

(sin^{2} (x))^{2} cos^{2} (x) = (sin^{2} (x) cos (x))^{2}

= - 1 + (sin^{2} (x) cos (x))^{2}

= - 1 + (sin^{2} (x) cos (x))^{2}

Appliquer la formule de différence de deux carrés :

x^{2} - y^{2} = (x + y) (x - y)

- 1 + (sin^{2} (x) cos (x))^{2} = (sin^{2} (x) cos (x) + 1) (sin^{2} (x) cos (x) - 1)

= (sin^{2} (x) cos (x) + 1) (sin^{2} (x) cos (x) - 1)

(sin^{2} (x) cos (x) + 1) (sin^{2} (x) cos (x) - 1) = 0

En solutionnant chaque partie séparément

sin^{2} (x) cos (x) + 1 = 0 or sin^{2} (x) cos (x) - 1 = 0

sin^{2} (x) cos (x) + 1 = 0 :

Aucune solution

sin^{2} (x) cos (x) + 1 = 0

Récrire en utilisant des identités trigonométriques

1 + cos (x) sin^{2} (x)

Utiliser l'identité hyperbolique:

cos^{2} (x) + sin^{2} (x) = 1

sin^{2} (x) = 1 - cos^{2} (x)

= 1 + cos (x) (1 - cos^{2} (x))

1 + (1 - cos^{2} (x)) cos (x) = 0

Résoudre par substitution

1 + (1 - cos^{2} (x)) cos (x) = 0

Soit :

cos (x) = u

1 + (1 - u^{2}) u = 0

1 + (1 - u^{2}) u = 0 : u \approx 1.32471 \dots

1 + (1 - u^{2}) u = 0

Développer

1 + (1 - u^{2}) u : 1 + u - u^{3}

1 + (1 - u^{2}) u

= 1 + u (1 - u^{2})

Développer

u (1 - u^{2}) : u - u^{3}

u (1 - u^{2})

Appliquer la loi de la distribution:

a (b - c) = ab - a c

a = u, b = 1, c = u^{2}

= u \cdot 1 - u u^{2}

= 1 \cdot u - u^{2} u

Simplifier

1 \cdot u - u^{2} u : u - u^{3}

1 \cdot u - u^{2} u

1 \cdot u = u

1 \cdot u

Multiplier:

1 \cdot u = u

= u

u^{2} u = u^{3}

u^{2} u

Appliquer la règle de l'exposant:

a^{b} \cdot a^{c} = a^{b + c}

u^{2} u = u^{2 + 1}

= u^{2 + 1}

Additionner les nombres :

2 + 1 = 3

= u^{3}

= u - u^{3}

= u - u^{3}

= 1 + u - u^{3}

1 + u - u^{3} = 0

Ecrire sous la forme standard

a_{n} x^{n} + \dots + a_{1} x + a_{0} = 0

- u^{3} + u + 1 = 0

Trouver une solution pour

- u^{3} + u + 1 = 0

par la méthode de Newton-Raphson:

u \approx 1.32471 \dots

- u^{3} + u + 1 = 0

Définition de l'approximation de Newton-Raphson

f (u) = - u^{3} + u + 1

Trouver

f^{^{'}} (u) : - 3 u^{2} + 1

\frac{d}{d u} (- u^{3} + u + 1)

Appliquer la règle de l'addition/soustraction:

(f \pm g)^{'} = f^{'} \pm g^{'}

= - \frac{d}{d u} (u^{3}) + \frac{d u}{d u} + \frac{d}{d u} (1)

\frac{d}{d u} (u^{3}) = 3 u^{2}

\frac{d}{d u} (u^{3})

Appliquer la règle de la puissance:

\frac{d}{d x} (x^{a}) = a \cdot x^{a - 1}

= 3 u^{3 - 1}

Simplifier

= 3 u^{2}

\frac{d u}{d u} = 1

\frac{d u}{d u}

Appliquer la dérivée commune:

\frac{d u}{d u} = 1

= 1

\frac{d}{d u} (1) = 0

\frac{d}{d u} (1)

Dérivée d'une constante:

\frac{d}{d x} (a) = 0

= 0

= - 3 u^{2} + 1 + 0

Simplifier

= - 3 u^{2} + 1

Soit

u_{0} = 1

Calculer

u_{n + 1}

jusqu'à

Δ u_{n + 1} < 0.000001

u_{1} = 1.5 : Δ u_{1} = 0.5

f (u_{0}) = - 1^{3} + 1 + 1 = 1

f^{^{'}} (u_{0}) = - 3 \cdot 1^{2} + 1 = - 2

u_{1} = 1.5

Δ u_{1} = ∣ 1.5 - 1 ∣ = 0.5

Δ u_{1} = 0.5

u_{2} = 1.34782 \dots : Δ u_{2} = 0.15217 \dots

f (u_{1}) = - 1. 5^{3} + 1.5 + 1 = - 0.875

f^{^{'}} (u_{1}) = - 3 \cdot 1. 5^{2} + 1 = - 5.75

u_{2} = 1.34782 \dots

Δ u_{2} = ∣ 1.34782 \dots - 1.5 ∣ = 0.15217 \dots

Δ u_{2} = 0.15217 \dots

u_{3} = 1.32520 \dots : Δ u_{3} = 0.02262 \dots

f (u_{2}) = - 1.34782 \dots^{3} + 1.34782 \dots + 1 = - 0.10068 \dots

f^{^{'}} (u_{2}) = - 3 \cdot 1.34782 \dots^{2} + 1 = - 4.44990 \dots

u_{3} = 1.32520 \dots

Δ u_{3} = ∣ 1.32520 \dots - 1.34782 \dots ∣ = 0.02262 \dots

Δ u_{3} = 0.02262 \dots

u_{4} = 1.32471 \dots : Δ u_{4} = 0.00048 \dots

f (u_{3}) = - 1.32520 \dots^{3} + 1.32520 \dots + 1 = - 0.00205 \dots

f^{^{'}} (u_{3}) = - 3 \cdot 1.32520 \dots^{2} + 1 = - 4.26846 \dots

u_{4} = 1.32471 \dots

Δ u_{4} = ∣ 1.32471 \dots - 1.32520 \dots ∣ = 0.00048 \dots

Δ u_{4} = 0.00048 \dots

u_{5} = 1.32471 \dots : Δ u_{5} = 2.16754 E - 7

f (u_{4}) = - 1.32471 \dots^{3} + 1.32471 \dots + 1 = - 9.24378 E - 7

f^{^{'}} (u_{4}) = - 3 \cdot 1.32471 \dots^{2} + 1 = - 4.26463 \dots

u_{5} = 1.32471 \dots

Δ u_{5} = ∣ 1.32471 \dots - 1.32471 \dots ∣ = 2.16754 E - 7

Δ u_{5} = 2.16754 E - 7

u \approx 1.32471 \dots

Appliquer une division longue:

\frac{- u ^{3} + u + 1}{u - 1.32471 \dots} = - u^{2} - 1.32471 \dots u - 0.75487 \dots

- u^{2} - 1.32471 \dots u - 0.75487 \dots \approx 0

Trouver une solution pour

- u^{2} - 1.32471 \dots u - 0.75487 \dots = 0

par la méthode de Newton-Raphson:

Aucune solution pour

u \in R

- u^{2} - 1.32471 \dots u - 0.75487 \dots = 0

Définition de l'approximation de Newton-Raphson

f (u) = - u^{2} - 1.32471 \dots u - 0.75487 \dots

Trouver

f^{^{'}} (u) : - 2 u - 1.32471 \dots

\frac{d}{d u} (- u^{2} - 1.32471 \dots u - 0.75487 \dots)

Appliquer la règle de l'addition/soustraction:

(f \pm g)^{'} = f^{'} \pm g^{'}

= - \frac{d}{d u} (u^{2}) - \frac{d}{d u} (1.32471 \dots u) - \frac{d}{d u} (0.75487 \dots)

\frac{d}{d u} (u^{2}) = 2 u

\frac{d}{d u} (u^{2})

Appliquer la règle de la puissance:

\frac{d}{d x} (x^{a}) = a \cdot x^{a - 1}

= 2 u^{2 - 1}

Simplifier

= 2 u

\frac{d}{d u} (1.32471 \dots u) = 1.32471 \dots

\frac{d}{d u} (1.32471 \dots u)

Retirer la constante:

(a \cdot f)^{'} = a \cdot f^{'}

= 1.32471 \dots \frac{d u}{d u}

Appliquer la dérivée commune:

\frac{d u}{d u} = 1

= 1.32471 \dots \cdot 1

Simplifier

= 1.32471 \dots

\frac{d}{d u} (0.75487 \dots) = 0

\frac{d}{d u} (0.75487 \dots)

Dérivée d'une constante:

\frac{d}{d x} (a) = 0

= 0

= - 2 u - 1.32471 \dots - 0

Simplifier

= - 2 u - 1.32471 \dots

Soit

u_{0} = - 1

Calculer

u_{n + 1}

jusqu'à

Δ u_{n + 1} < 0.000001

u_{1} = - 0.36299 \dots : Δ u_{1} = 0.63700 \dots

f (u_{0}) = - (- 1)^{2} - 1.32471 \dots (- 1) - 0.75487 \dots = - 0.43015 \dots

f^{^{'}} (u_{0}) = - 2 (- 1) - 1.32471 \dots = 0.67528 \dots

u_{1} = - 0.36299 \dots

Δ u_{1} = ∣ - 0.36299 \dots - (- 1) ∣ = 0.63700 \dots

Δ u_{1} = 0.63700 \dots

u_{2} = - 1.04072 \dots : Δ u_{2} = 0.67772 \dots

f (u_{1}) = - (- 0.36299 \dots)^{2} - 1.32471 \dots (- 0.36299 \dots) - 0.75487 \dots = - 0.40577 \dots

f^{^{'}} (u_{1}) = - 2 (- 0.36299 \dots) - 1.32471 \dots = - 0.59873 \dots

u_{2} = - 1.04072 \dots

Δ u_{2} = ∣ - 1.04072 \dots - (- 0.36299 \dots) ∣ = 0.67772 \dots

Δ u_{2} = 0.67772 \dots

u_{3} = - 0.43374 \dots : Δ u_{3} = 0.60697 \dots

f (u_{2}) = - (- 1.04072 \dots)^{2} - 1.32471 \dots (- 1.04072 \dots) - 0.75487 \dots = - 0.45931 \dots

f^{^{'}} (u_{2}) = - 2 (- 1.04072 \dots) - 1.32471 \dots = 0.75672 \dots

u_{3} = - 0.43374 \dots

Δ u_{3} = ∣ - 0.43374 \dots - (- 1.04072 \dots) ∣ = 0.60697 \dots

Δ u_{3} = 0.60697 \dots

u_{4} = - 1.23950 \dots : Δ u_{4} = 0.80576 \dots

f (u_{3}) = - (- 0.43374 \dots)^{2} - 1.32471 \dots (- 0.43374 \dots) - 0.75487 \dots = - 0.36842 \dots

f^{^{'}} (u_{3}) = - 2 (- 0.43374 \dots) - 1.32471 \dots = - 0.45723 \dots

u_{4} = - 1.23950 \dots

Δ u_{4} = ∣ - 1.23950 \dots - (- 0.43374 \dots) ∣ = 0.80576 \dots

Δ u_{4} = 0.80576 \dots

u_{5} = - 0.67703 \dots : Δ u_{5} = 0.56247 \dots

f (u_{4}) = - (- 1.23950 \dots)^{2} - 1.32471 \dots (- 1.23950 \dots) - 0.75487 \dots = - 0.64925 \dots

f^{^{'}} (u_{4}) = - 2 (- 1.23950 \dots) - 1.32471 \dots = 1.15429 \dots

u_{5} = - 0.67703 \dots

Δ u_{5} = ∣ - 0.67703 \dots - (- 1.23950 \dots) ∣ = 0.56247 \dots

Δ u_{5} = 0.56247 \dots

u_{6} = 10.09982 \dots : Δ u_{6} = 10.77686 \dots

f (u_{5}) = - (- 0.67703 \dots)^{2} - 1.32471 \dots (- 0.67703 \dots) - 0.75487 \dots = - 0.31637 \dots

f^{^{'}} (u_{5}) = - 2 (- 0.67703 \dots) - 1.32471 \dots = 0.02935 \dots

u_{6} = 10.09982 \dots

Δ u_{6} = ∣ 10.09982 \dots - (- 0.67703 \dots) ∣ = 10.77686 \dots

Δ u_{6} = 10.77686 \dots

u_{7} = 4.70404 \dots : Δ u_{7} = 5.39578 \dots

f (u_{6}) = - 10.09982 \dots^{2} - 1.32471 \dots \cdot 10.09982 \dots - 0.75487 \dots = - 116.14080 \dots

f^{^{'}} (u_{6}) = - 2 \cdot 10.09982 \dots - 1.32471 \dots = - 21.52437 \dots

u_{7} = 4.70404 \dots

Δ u_{7} = ∣ 4.70404 \dots - 10.09982 \dots ∣ = 5.39578 \dots

Δ u_{7} = 5.39578 \dots

u_{8} = 1.99138 \dots : Δ u_{8} = 2.71265 \dots

f (u_{7}) = - 4.70404 \dots^{2} - 1.32471 \dots \cdot 4.70404 \dots - 0.75487 \dots = - 29.11445 \dots

f^{^{'}} (u_{7}) = - 2 \cdot 4.70404 \dots - 1.32471 \dots = - 10.73280 \dots

u_{8} = 1.99138 \dots

Δ u_{8} = ∣ 1.99138 \dots - 4.70404 \dots ∣ = 2.71265 \dots

Δ u_{8} = 2.71265 \dots

u_{9} = 0.60494 \dots : Δ u_{9} = 1.38644 \dots

f (u_{8}) = - 1.99138 \dots^{2} - 1.32471 \dots \cdot 1.99138 \dots - 0.75487 \dots = - 7.35852 \dots

f^{^{'}} (u_{8}) = - 2 \cdot 1.99138 \dots - 1.32471 \dots = - 5.30749 \dots

u_{9} = 0.60494 \dots

Δ u_{9} = ∣ 0.60494 \dots - 1.99138 \dots ∣ = 1.38644 \dots

Δ u_{9} = 1.38644 \dots

u_{10} = - 0.15344 \dots : Δ u_{10} = 0.75838 \dots

f (u_{9}) = - 0.60494 \dots^{2} - 1.32471 \dots \cdot 0.60494 \dots - 0.75487 \dots = - 1.92221 \dots

f^{^{'}} (u_{9}) = - 2 \cdot 0.60494 \dots - 1.32471 \dots = - 2.53460 \dots

u_{10} = - 0.15344 \dots

Δ u_{10} = ∣ - 0.15344 \dots - 0.60494 \dots ∣ = 0.75838 \dots

Δ u_{10} = 0.75838 \dots

u_{11} = - 0.71852 \dots : Δ u_{11} = 0.56507 \dots

f (u_{10}) = - (- 0.15344 \dots)^{2} - 1.32471 \dots (- 0.15344 \dots) - 0.75487 \dots = - 0.57515 \dots

f^{^{'}} (u_{10}) = - 2 (- 0.15344 \dots) - 1.32471 \dots = - 1.01783 \dots

u_{11} = - 0.71852 \dots

Δ u_{11} = ∣ - 0.71852 \dots - (- 0.15344 \dots) ∣ = 0.56507 \dots

Δ u_{11} = 0.56507 \dots

u_{12} = 2.12427 \dots : Δ u_{12} = 2.84279 \dots

f (u_{11}) = - (- 0.71852 \dots)^{2} - 1.32471 \dots (- 0.71852 \dots) - 0.75487 \dots = - 0.31931 \dots

f^{^{'}} (u_{11}) = - 2 (- 0.71852 \dots) - 1.32471 \dots = 0.11232 \dots

u_{12} = 2.12427 \dots

Δ u_{12} = ∣ 2.12427 \dots - (- 0.71852 \dots) ∣ = 2.84279 \dots

Δ u_{12} = 2.84279 \dots

Impossible de trouver une solution

La solution est

u \approx 1.32471 \dots

Remplacer

u = cos (x)

cos (x) \approx 1.32471 \dots

cos (x) \approx 1.32471 \dots

cos (x) = 1.32471 \dots :

Aucune solution

cos (x) = 1.32471 \dots

- 1 \leq cos (x) \leq 1

Aucune solution

Combiner toutes les solutions

Aucune solution

sin^{2} (x) cos (x) - 1 = 0 :

Aucune solution

sin^{2} (x) cos (x) - 1 = 0

Récrire en utilisant des identités trigonométriques

- 1 + cos (x) sin^{2} (x)

Utiliser l'identité hyperbolique:

cos^{2} (x) + sin^{2} (x) = 1

sin^{2} (x) = 1 - cos^{2} (x)

= - 1 + cos (x) (1 - cos^{2} (x))

- 1 + (1 - cos^{2} (x)) cos (x) = 0

Résoudre par substitution

- 1 + (1 - cos^{2} (x)) cos (x) = 0

Soit :

cos (x) = u

- 1 + (1 - u^{2}) u = 0

- 1 + (1 - u^{2}) u = 0 : u \approx - 1.32471 \dots

- 1 + (1 - u^{2}) u = 0

Développer

- 1 + (1 - u^{2}) u : - 1 + u - u^{3}

- 1 + (1 - u^{2}) u

= - 1 + u (1 - u^{2})

Développer

u (1 - u^{2}) : u - u^{3}

u (1 - u^{2})

Appliquer la loi de la distribution:

a (b - c) = ab - a c

a = u, b = 1, c = u^{2}

= u \cdot 1 - u u^{2}

= 1 \cdot u - u^{2} u

Simplifier

1 \cdot u - u^{2} u : u - u^{3}

1 \cdot u - u^{2} u

1 \cdot u = u

1 \cdot u

Multiplier:

1 \cdot u = u

= u

u^{2} u = u^{3}

u^{2} u

Appliquer la règle de l'exposant:

a^{b} \cdot a^{c} = a^{b + c}

u^{2} u = u^{2 + 1}

= u^{2 + 1}

Additionner les nombres :

2 + 1 = 3

= u^{3}

= u - u^{3}

= u - u^{3}

= - 1 + u - u^{3}

- 1 + u - u^{3} = 0

Ecrire sous la forme standard

a_{n} x^{n} + \dots + a_{1} x + a_{0} = 0

- u^{3} + u - 1 = 0

Trouver une solution pour

- u^{3} + u - 1 = 0

par la méthode de Newton-Raphson:

u \approx - 1.32471 \dots

- u^{3} + u - 1 = 0

Définition de l'approximation de Newton-Raphson

f (u) = - u^{3} + u - 1

Trouver

f^{^{'}} (u) : - 3 u^{2} + 1

\frac{d}{d u} (- u^{3} + u - 1)

Appliquer la règle de l'addition/soustraction:

(f \pm g)^{'} = f^{'} \pm g^{'}

= - \frac{d}{d u} (u^{3}) + \frac{d u}{d u} - \frac{d}{d u} (1)

\frac{d}{d u} (u^{3}) = 3 u^{2}

\frac{d}{d u} (u^{3})

Appliquer la règle de la puissance:

\frac{d}{d x} (x^{a}) = a \cdot x^{a - 1}

= 3 u^{3 - 1}

Simplifier

= 3 u^{2}

\frac{d u}{d u} = 1

\frac{d u}{d u}

Appliquer la dérivée commune:

\frac{d u}{d u} = 1

= 1

\frac{d}{d u} (1) = 0

\frac{d}{d u} (1)

Dérivée d'une constante:

\frac{d}{d x} (a) = 0

= 0

= - 3 u^{2} + 1 - 0

Simplifier

= - 3 u^{2} + 1

Soit

u_{0} = - 1

Calculer

u_{n + 1}

jusqu'à

Δ u_{n + 1} < 0.000001

u_{1} = - 1.5 : Δ u_{1} = 0.5

f (u_{0}) = - (- 1)^{3} + (- 1) - 1 = - 1

f^{^{'}} (u_{0}) = - 3 (- 1)^{2} + 1 = - 2

u_{1} = - 1.5

Δ u_{1} = ∣ - 1.5 - (- 1) ∣ = 0.5

Δ u_{1} = 0.5

u_{2} = - 1.34782 \dots : Δ u_{2} = 0.15217 \dots

f (u_{1}) = - (- 1.5)^{3} + (- 1.5) - 1 = 0.875

f^{^{'}} (u_{1}) = - 3 (- 1.5)^{2} + 1 = - 5.75

u_{2} = - 1.34782 \dots

Δ u_{2} = ∣ - 1.34782 \dots - (- 1.5) ∣ = 0.15217 \dots

Δ u_{2} = 0.15217 \dots

u_{3} = - 1.32520 \dots : Δ u_{3} = 0.02262 \dots

f (u_{2}) = - (- 1.34782 \dots)^{3} + (- 1.34782 \dots) - 1 = 0.10068 \dots

f^{^{'}} (u_{2}) = - 3 (- 1.34782 \dots)^{2} + 1 = - 4.44990 \dots

u_{3} = - 1.32520 \dots

Δ u_{3} = ∣ - 1.32520 \dots - (- 1.34782 \dots) ∣ = 0.02262 \dots

Δ u_{3} = 0.02262 \dots

u_{4} = - 1.32471 \dots : Δ u_{4} = 0.00048 \dots

f (u_{3}) = - (- 1.32520 \dots)^{3} + (- 1.32520 \dots) - 1 = 0.00205 \dots

f^{^{'}} (u_{3}) = - 3 (- 1.32520 \dots)^{2} + 1 = - 4.26846 \dots

u_{4} = - 1.32471 \dots

Δ u_{4} = ∣ - 1.32471 \dots - (- 1.32520 \dots) ∣ = 0.00048 \dots

Δ u_{4} = 0.00048 \dots

u_{5} = - 1.32471 \dots : Δ u_{5} = 2.16754 E - 7

f (u_{4}) = - (- 1.32471 \dots)^{3} + (- 1.32471 \dots) - 1 = 9.24378 E - 7

f^{^{'}} (u_{4}) = - 3 (- 1.32471 \dots)^{2} + 1 = - 4.26463 \dots

u_{5} = - 1.32471 \dots

Δ u_{5} = ∣ - 1.32471 \dots - (- 1.32471 \dots) ∣ = 2.16754 E - 7

Δ u_{5} = 2.16754 E - 7

u \approx - 1.32471 \dots

Appliquer une division longue:

\frac{- u ^{3} + u - 1}{u + 1.32471 \dots} = - u^{2} + 1.32471 \dots u - 0.75487 \dots

- u^{2} + 1.32471 \dots u - 0.75487 \dots \approx 0

Trouver une solution pour

- u^{2} + 1.32471 \dots u - 0.75487 \dots = 0

par la méthode de Newton-Raphson:

Aucune solution pour

u \in R

- u^{2} + 1.32471 \dots u - 0.75487 \dots = 0

Définition de l'approximation de Newton-Raphson

f (u) = - u^{2} + 1.32471 \dots u - 0.75487 \dots

Trouver

f^{^{'}} (u) : - 2 u + 1.32471 \dots

\frac{d}{d u} (- u^{2} + 1.32471 \dots u - 0.75487 \dots)

Appliquer la règle de l'addition/soustraction:

(f \pm g)^{'} = f^{'} \pm g^{'}

= - \frac{d}{d u} (u^{2}) + \frac{d}{d u} (1.32471 \dots u) - \frac{d}{d u} (0.75487 \dots)

\frac{d}{d u} (u^{2}) = 2 u

\frac{d}{d u} (u^{2})

Appliquer la règle de la puissance:

\frac{d}{d x} (x^{a}) = a \cdot x^{a - 1}

= 2 u^{2 - 1}

Simplifier

= 2 u

\frac{d}{d u} (1.32471 \dots u) = 1.32471 \dots

\frac{d}{d u} (1.32471 \dots u)

Retirer la constante:

(a \cdot f)^{'} = a \cdot f^{'}

= 1.32471 \dots \frac{d u}{d u}

Appliquer la dérivée commune:

\frac{d u}{d u} = 1

= 1.32471 \dots \cdot 1

Simplifier

= 1.32471 \dots

\frac{d}{d u} (0.75487 \dots) = 0

\frac{d}{d u} (0.75487 \dots)

Dérivée d'une constante:

\frac{d}{d x} (a) = 0

= 0

= - 2 u + 1.32471 \dots - 0

Simplifier

= - 2 u + 1.32471 \dots

Soit

u_{0} = 1

Calculer

u_{n + 1}

jusqu'à

Δ u_{n + 1} < 0.000001

u_{1} = 0.36299 \dots : Δ u_{1} = 0.63700 \dots

f (u_{0}) = - 1^{2} + 1.32471 \dots \cdot 1 - 0.75487 \dots = - 0.43015 \dots

f^{^{'}} (u_{0}) = - 2 \cdot 1 + 1.32471 \dots = - 0.67528 \dots

u_{1} = 0.36299 \dots

Δ u_{1} = ∣ 0.36299 \dots - 1 ∣ = 0.63700 \dots

Δ u_{1} = 0.63700 \dots

u_{2} = 1.04072 \dots : Δ u_{2} = 0.67772 \dots

f (u_{1}) = - 0.36299 \dots^{2} + 1.32471 \dots \cdot 0.36299 \dots - 0.75487 \dots = - 0.40577 \dots

f^{^{'}} (u_{1}) = - 2 \cdot 0.36299 \dots + 1.32471 \dots = 0.59873 \dots

u_{2} = 1.04072 \dots

Δ u_{2} = ∣ 1.04072 \dots - 0.36299 \dots ∣ = 0.67772 \dots

Δ u_{2} = 0.67772 \dots

u_{3} = 0.43374 \dots : Δ u_{3} = 0.60697 \dots

f (u_{2}) = - 1.04072 \dots^{2} + 1.32471 \dots \cdot 1.04072 \dots - 0.75487 \dots = - 0.45931 \dots

f^{^{'}} (u_{2}) = - 2 \cdot 1.04072 \dots + 1.32471 \dots = - 0.75672 \dots

u_{3} = 0.43374 \dots

Δ u_{3} = ∣ 0.43374 \dots - 1.04072 \dots ∣ = 0.60697 \dots

Δ u_{3} = 0.60697 \dots

u_{4} = 1.23950 \dots : Δ u_{4} = 0.80576 \dots

f (u_{3}) = - 0.43374 \dots^{2} + 1.32471 \dots \cdot 0.43374 \dots - 0.75487 \dots = - 0.36842 \dots

f^{^{'}} (u_{3}) = - 2 \cdot 0.43374 \dots + 1.32471 \dots = 0.45723 \dots

u_{4} = 1.23950 \dots

Δ u_{4} = ∣ 1.23950 \dots - 0.43374 \dots ∣ = 0.80576 \dots

Δ u_{4} = 0.80576 \dots

u_{5} = 0.67703 \dots : Δ u_{5} = 0.56247 \dots

f (u_{4}) = - 1.23950 \dots^{2} + 1.32471 \dots \cdot 1.23950 \dots - 0.75487 \dots = - 0.64925 \dots

f^{^{'}} (u_{4}) = - 2 \cdot 1.23950 \dots + 1.32471 \dots = - 1.15429 \dots

u_{5} = 0.67703 \dots

Δ u_{5} = ∣ 0.67703 \dots - 1.23950 \dots ∣ = 0.56247 \dots

Δ u_{5} = 0.56247 \dots

u_{6} = - 10.09982 \dots : Δ u_{6} = 10.77686 \dots

f (u_{5}) = - 0.67703 \dots^{2} + 1.32471 \dots \cdot 0.67703 \dots - 0.75487 \dots = - 0.31637 \dots

f^{^{'}} (u_{5}) = - 2 \cdot 0.67703 \dots + 1.32471 \dots = - 0.02935 \dots

u_{6} = - 10.09982 \dots

Δ u_{6} = ∣ - 10.09982 \dots - 0.67703 \dots ∣ = 10.77686 \dots

Δ u_{6} = 10.77686 \dots

u_{7} = - 4.70404 \dots : Δ u_{7} = 5.39578 \dots

f (u_{6}) = - (- 10.09982 \dots)^{2} + 1.32471 \dots (- 10.09982 \dots) - 0.75487 \dots = - 116.14080 \dots

f^{^{'}} (u_{6}) = - 2 (- 10.09982 \dots) + 1.32471 \dots = 21.52437 \dots

u_{7} = - 4.70404 \dots

Δ u_{7} = ∣ - 4.70404 \dots - (- 10.09982 \dots) ∣ = 5.39578 \dots

Δ u_{7} = 5.39578 \dots

u_{8} = - 1.99138 \dots : Δ u_{8} = 2.71265 \dots

f (u_{7}) = - (- 4.70404 \dots)^{2} + 1.32471 \dots (- 4.70404 \dots) - 0.75487 \dots = - 29.11445 \dots

f^{^{'}} (u_{7}) = - 2 (- 4.70404 \dots) + 1.32471 \dots = 10.73280 \dots

u_{8} = - 1.99138 \dots

Δ u_{8} = ∣ - 1.99138 \dots - (- 4.70404 \dots) ∣ = 2.71265 \dots

Δ u_{8} = 2.71265 \dots

u_{9} = - 0.60494 \dots : Δ u_{9} = 1.38644 \dots

f (u_{8}) = - (- 1.99138 \dots)^{2} + 1.32471 \dots (- 1.99138 \dots) - 0.75487 \dots = - 7.35852 \dots

f^{^{'}} (u_{8}) = - 2 (- 1.99138 \dots) + 1.32471 \dots = 5.30749 \dots

u_{9} = - 0.60494 \dots

Δ u_{9} = ∣ - 0.60494 \dots - (- 1.99138 \dots) ∣ = 1.38644 \dots

Δ u_{9} = 1.38644 \dots

u_{10} = 0.15344 \dots : Δ u_{10} = 0.75838 \dots

f (u_{9}) = - (- 0.60494 \dots)^{2} + 1.32471 \dots (- 0.60494 \dots) - 0.75487 \dots = - 1.92221 \dots

f^{^{'}} (u_{9}) = - 2 (- 0.60494 \dots) + 1.32471 \dots = 2.53460 \dots

u_{10} = 0.15344 \dots

Δ u_{10} = ∣ 0.15344 \dots - (- 0.60494 \dots) ∣ = 0.75838 \dots

Δ u_{10} = 0.75838 \dots

u_{11} = 0.71852 \dots : Δ u_{11} = 0.56507 \dots

f (u_{10}) = - 0.15344 \dots^{2} + 1.32471 \dots \cdot 0.15344 \dots - 0.75487 \dots = - 0.57515 \dots

f^{^{'}} (u_{10}) = - 2 \cdot 0.15344 \dots + 1.32471 \dots = 1.01783 \dots

u_{11} = 0.71852 \dots

Δ u_{11} = ∣ 0.71852 \dots - 0.15344 \dots ∣ = 0.56507 \dots

Δ u_{11} = 0.56507 \dots

u_{12} = - 2.12427 \dots : Δ u_{12} = 2.84279 \dots

f (u_{11}) = - 0.71852 \dots^{2} + 1.32471 \dots \cdot 0.71852 \dots - 0.75487 \dots = - 0.31931 \dots

f^{^{'}} (u_{11}) = - 2 \cdot 0.71852 \dots + 1.32471 \dots = - 0.11232 \dots

u_{12} = - 2.12427 \dots

Δ u_{12} = ∣ - 2.12427 \dots - 0.71852 \dots ∣ = 2.84279 \dots

Δ u_{12} = 2.84279 \dots

Impossible de trouver une solution

La solution est

u \approx - 1.32471 \dots

Remplacer

u = cos (x)

cos (x) \approx - 1.32471 \dots

cos (x) \approx - 1.32471 \dots

cos (x) = - 1.32471 \dots :

Aucune solution

cos (x) = - 1.32471 \dots

- 1 \leq cos (x) \leq 1

Aucune solution

Combiner toutes les solutions

Aucune solution

Combiner toutes les solutions

Aucune solution

Vérifier les solutions en les intégrant dans l'équation d'origine

Vérifier des solutions en les intégrant dans

sin^{2} (x) = sec (x)

Retirer celles qui ne répondent pas à l'équation.

Aucune solution pour x \in R

Graphe

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