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Populaire Trigonométrie >

cot^2(x)=tan(x/2)

Solution

cot^{2} (x) = tan (\frac{x}{2})

Solution

x = 2 \cdot 0.47448 \dots + 2 πn, x = 2 \cdot 1.34922 \dots + 2 πn

Degrés

x = 54.37186 \dots^{\circ} + 36 0^{\circ} n, x = 154.60973 \dots^{\circ} + 36 0^{\circ} n

étapes des solutions

cot^{2} (x) = tan (\frac{x}{2})

Soustraire

tan (\frac{x}{2})

des deux côtés

cot^{2} (x) - tan (\frac{x}{2}) = 0

Soit :

u = \frac{x}{2}

cot^{2} (2 u) - tan (u) = 0

Récrire en utilisant des identités trigonométriques

cot^{2} (2 u) - tan (u)

Utiliser l'identité trigonométrique de base:

cot (x) = \frac{1}{tan ( x )}

= (\frac{1}{tan ( 2 u )})^{2} - tan (u)

(\frac{1}{tan ( 2 u )})^{2} = \frac{1}{tan ^{2} ( 2 u )}

(\frac{1}{tan ( 2 u )})^{2}

Appliquer la règle de l'exposant:

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

= \frac{1 ^{2}}{tan ^{2} ( 2 u )}

Appliquer la règle

1^{a} = 1

1^{2} = 1

= \frac{1}{tan ^{2} ( 2 u )}

= \frac{1}{tan ^{2} ( 2 u )} - tan (u)

Utiliser l'identité d'angle double:

tan (2 x) = \frac{2 tan ( x )}{1 - tan ^{2} ( x )}

= \frac{1}{( \frac{2 t a n ( u )}{1 - t a n ^{2} ( u )} ) ^{2}} - tan (u)

Simplifier

\frac{1}{( \frac{2 t a n ( u )}{1 - t a n ^{2} ( u )} ) ^{2}} - tan (u) : \frac{( 1 - tan ^{2} ( u ) ) ^{2}}{4 tan ^{2} ( u )} - tan (u)

\frac{1}{( \frac{2 t a n ( u )}{1 - t a n ^{2} ( u )} ) ^{2}} - tan (u)

\frac{1}{( \frac{2 t a n ( u )}{1 - t a n ^{2} ( u )} ) ^{2}} = \frac{( 1 - tan ^{2} ( u ) ) ^{2}}{2 ^{2} tan ^{2} ( u )}

\frac{1}{( \frac{2 t a n ( u )}{1 - t a n ^{2} ( u )} ) ^{2}}

(\frac{2 tan ( u )}{1 - tan ^{2} ( u )})^{2} = \frac{2 ^{2} tan ^{2} ( u )}{( 1 - tan ^{2} ( u ) ) ^{2}}

(\frac{2 tan ( u )}{1 - tan ^{2} ( u )})^{2}

Appliquer la règle de l'exposant:

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

= \frac{( 2 tan ( u ) ) ^{2}}{( 1 - tan ^{2} ( u ) ) ^{2}}

Appliquer la règle de l'exposant:

(a \cdot b)^{n} = a^{n} b^{n}

(2 tan (u))^{2} = 2^{2} tan^{2} (u)

= \frac{2 ^{2} tan ^{2} ( u )}{( 1 - tan ^{2} ( u ) ) ^{2}}

= \frac{1}{\frac{2 ^{2} t a n ^{2} ( u )}{( 1 - t a n ^{2} ( u ) ) ^{2}}}

Appliquer la règle des fractions:

\frac{1}{\frac{b}{c}} = \frac{c}{b}

= \frac{( 1 - tan ^{2} ( u ) ) ^{2}}{2 ^{2} tan ^{2} ( u )}

= \frac{( - tan ^{2} ( u ) + 1 ) ^{2}}{2 ^{2} tan ^{2} ( u )} - tan (u)

2^{2} = 4

= \frac{( - tan ^{2} ( u ) + 1 ) ^{2}}{4 tan ^{2} ( u )} - tan (u)

= \frac{( 1 - tan ^{2} ( u ) ) ^{2}}{4 tan ^{2} ( u )} - tan (u)

\frac{( 1 - tan ^{2} ( u ) ) ^{2}}{4 tan ^{2} ( u )} - tan (u) = 0

Résoudre par substitution

\frac{( 1 - tan ^{2} ( u ) ) ^{2}}{4 tan ^{2} ( u )} - tan (u) = 0

Soit :

tan (u) = u

\frac{( 1 - u ^{2} ) ^{2}}{4 u ^{2}} - u = 0

\frac{( 1 - u ^{2} ) ^{2}}{4 u ^{2}} - u = 0 : u \approx 0.51361 \dots, u \approx 4.43910 \dots

\frac{( 1 - u ^{2} ) ^{2}}{4 u ^{2}} - u = 0

Multiplier les deux côtés par

4 u^{2}

\frac{( 1 - u ^{2} ) ^{2}}{4 u ^{2}} - u = 0

Multiplier les deux côtés par

4 u^{2}

\frac{( 1 - u ^{2} ) ^{2}}{4 u ^{2}} \cdot 4 u^{2} - u \cdot 4 u^{2} = 0 \cdot 4 u^{2}

Simplifier

\frac{( 1 - u ^{2} ) ^{2}}{4 u ^{2}} \cdot 4 u^{2} - u \cdot 4 u^{2} = 0 \cdot 4 u^{2}

Simplifier

\frac{( 1 - u ^{2} ) ^{2}}{4 u ^{2}} \cdot 4 u^{2} : (1 - u^{2})^{2}

\frac{( 1 - u ^{2} ) ^{2}}{4 u ^{2}} \cdot 4 u^{2}

Multiplier des fractions:

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

= \frac{( 1 - u ^{2} ) ^{2} \cdot 4 u ^{2}}{4 u ^{2}}

Annuler le facteur commun :

4

= \frac{( 1 - u ^{2} ) ^{2} u ^{2}}{u ^{2}}

Annuler le facteur commun :

u^{2}

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

Simplifier

- u \cdot 4 u^{2} : - 4 u^{3}

- u \cdot 4 u^{2}

Appliquer la règle de l'exposant:

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

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

= - 4 u^{1 + 2}

Additionner les nombres :

1 + 2 = 3

= - 4 u^{3}

Simplifier

0 \cdot 4 u^{2} : 0

0 \cdot 4 u^{2}

Appliquer la règle

0 \cdot a = 0

= 0

(1 - u^{2})^{2} - 4 u^{3} = 0

(1 - u^{2})^{2} - 4 u^{3} = 0

(1 - u^{2})^{2} - 4 u^{3} = 0

Résoudre

(1 - u^{2})^{2} - 4 u^{3} = 0 : u \approx 0.51361 \dots, u \approx 4.43910 \dots

(1 - u^{2})^{2} - 4 u^{3} = 0

Développer

(1 - u^{2})^{2} - 4 u^{3} : 1 - 2 u^{2} + u^{4} - 4 u^{3}

(1 - u^{2})^{2} - 4 u^{3}

(1 - u^{2})^{2} : 1 - 2 u^{2} + u^{4}

Appliquer la formule du carré parfait:

(a - b)^{2} = a^{2} - 2 ab + b^{2}

a = 1, b = u^{2}

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

Simplifier

1^{2} - 2 \cdot 1 \cdot u^{2} + (u^{2})^{2} : 1 - 2 u^{2} + u^{4}

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

Appliquer la règle

1^{a} = 1

1^{2} = 1

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

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

2 \cdot 1 \cdot u^{2}

Multiplier les nombres :

2 \cdot 1 = 2

= 2 u^{2}

(u^{2})^{2} = u^{4}

(u^{2})^{2}

Appliquer la règle de l'exposant:

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

= u^{2 \cdot 2}

Multiplier les nombres :

2 \cdot 2 = 4

= u^{4}

= 1 - 2 u^{2} + u^{4}

= 1 - 2 u^{2} + u^{4}

= 1 - 2 u^{2} + u^{4} - 4 u^{3}

1 - 2 u^{2} + u^{4} - 4 u^{3} = 0

Ecrire sous la forme standard

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

u^{4} - 4 u^{3} - 2 u^{2} + 1 = 0

Trouver une solution pour

u^{4} - 4 u^{3} - 2 u^{2} + 1 = 0

par la méthode de Newton-Raphson:

u \approx 0.51361 \dots

u^{4} - 4 u^{3} - 2 u^{2} + 1 = 0

Définition de l'approximation de Newton-Raphson

f (u) = u^{4} - 4 u^{3} - 2 u^{2} + 1

Trouver

f^{^{'}} (u) : 4 u^{3} - 12 u^{2} - 4 u

\frac{d}{d u} (u^{4} - 4 u^{3} - 2 u^{2} + 1)

Appliquer la règle de l'addition/soustraction:

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

= \frac{d}{d u} (u^{4}) - \frac{d}{d u} (4 u^{3}) - \frac{d}{d u} (2 u^{2}) + \frac{d}{d u} (1)

\frac{d}{d u} (u^{4}) = 4 u^{3}

\frac{d}{d u} (u^{4})

Appliquer la règle de la puissance:

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

= 4 u^{4 - 1}

Simplifier

= 4 u^{3}

\frac{d}{d u} (4 u^{3}) = 12 u^{2}

\frac{d}{d u} (4 u^{3})

Retirer la constante:

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

= 4 \frac{d}{d u} (u^{3})

Appliquer la règle de la puissance:

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

= 4 \cdot 3 u^{3 - 1}

Simplifier

= 12 u^{2}

\frac{d}{d u} (2 u^{2}) = 4 u

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

Retirer la constante:

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

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

Appliquer la règle de la puissance:

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

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

Simplifier

= 4 u

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

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

Dérivée d'une constante:

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

= 0

= 4 u^{3} - 12 u^{2} - 4 u + 0

Simplifier

= 4 u^{3} - 12 u^{2} - 4 u

Soit

u_{0} = 1

Calculer

u_{n + 1}

jusqu'à

Δ u_{n + 1} < 0.000001

u_{1} = 0.66666 \dots : Δ u_{1} = 0.33333 \dots

f (u_{0}) = 1^{4} - 4 \cdot 1^{3} - 2 \cdot 1^{2} + 1 = - 4

f^{^{'}} (u_{0}) = 4 \cdot 1^{3} - 12 \cdot 1^{2} - 4 \cdot 1 = - 12

u_{1} = 0.66666 \dots

Δ u_{1} = ∣ 0.66666 \dots - 1 ∣ = 0.33333 \dots

Δ u_{1} = 0.33333 \dots

u_{2} = 0.53804 \dots : Δ u_{2} = 0.12862 \dots

f (u_{1}) = 0.66666 \dots^{4} - 4 \cdot 0.66666 \dots^{3} - 2 \cdot 0.66666 \dots^{2} + 1 = - 0.87654 \dots

f^{^{'}} (u_{1}) = 4 \cdot 0.66666 \dots^{3} - 12 \cdot 0.66666 \dots^{2} - 4 \cdot 0.66666 \dots = - 6.81481 \dots

u_{2} = 0.53804 \dots

Δ u_{2} = ∣ 0.53804 \dots - 0.66666 \dots ∣ = 0.12862 \dots

Δ u_{2} = 0.12862 \dots

u_{3} = 0.51441 \dots : Δ u_{3} = 0.02362 \dots

f (u_{2}) = 0.53804 \dots^{4} - 4 \cdot 0.53804 \dots^{3} - 2 \cdot 0.53804 \dots^{2} + 1 = - 0.11821 \dots

f^{^{'}} (u_{2}) = 4 \cdot 0.53804 \dots^{3} - 12 \cdot 0.53804 \dots^{2} - 4 \cdot 0.53804 \dots = - 5.00302 \dots

u_{3} = 0.51441 \dots

Δ u_{3} = ∣ 0.51441 \dots - 0.53804 \dots ∣ = 0.02362 \dots

Δ u_{3} = 0.02362 \dots

u_{4} = 0.51362 \dots : Δ u_{4} = 0.00079 \dots

f (u_{3}) = 0.51441 \dots^{4} - 4 \cdot 0.51441 \dots^{3} - 2 \cdot 0.51441 \dots^{2} + 1 = - 0.00372 \dots

f^{^{'}} (u_{3}) = 4 \cdot 0.51441 \dots^{3} - 12 \cdot 0.51441 \dots^{2} - 4 \cdot 0.51441 \dots = - 4.68863 \dots

u_{4} = 0.51362 \dots

Δ u_{4} = ∣ 0.51362 \dots - 0.51441 \dots ∣ = 0.00079 \dots

Δ u_{4} = 0.00079 \dots

u_{5} = 0.51361 \dots : Δ u_{5} = 8.89103 E - 7

f (u_{4}) = 0.51362 \dots^{4} - 4 \cdot 0.51362 \dots^{3} - 2 \cdot 0.51362 \dots^{2} + 1 = - 4.15938 E - 6

f^{^{'}} (u_{4}) = 4 \cdot 0.51362 \dots^{3} - 12 \cdot 0.51362 \dots^{2} - 4 \cdot 0.51362 \dots = - 4.67817 \dots

u_{5} = 0.51361 \dots

Δ u_{5} = ∣ 0.51361 \dots - 0.51362 \dots ∣ = 8.89103 E - 7

Δ u_{5} = 8.89103 E - 7

u \approx 0.51361 \dots

Appliquer une division longue:

\frac{u ^{4} - 4 u ^{3} - 2 u ^{2} + 1}{u - 0.51361 \dots} = u^{3} - 3.48638 \dots u^{2} - 3.79067 \dots u - 1.94696 \dots

u^{3} - 3.48638 \dots u^{2} - 3.79067 \dots u - 1.94696 \dots \approx 0

Trouver une solution pour

u^{3} - 3.48638 \dots u^{2} - 3.79067 \dots u - 1.94696 \dots = 0

par la méthode de Newton-Raphson:

u \approx 4.43910 \dots

u^{3} - 3.48638 \dots u^{2} - 3.79067 \dots u - 1.94696 \dots = 0

Définition de l'approximation de Newton-Raphson

f (u) = u^{3} - 3.48638 \dots u^{2} - 3.79067 \dots u - 1.94696 \dots

Trouver

f^{^{'}} (u) : 3 u^{2} - 6.97276 \dots u - 3.79067 \dots

\frac{d}{d u} (u^{3} - 3.48638 \dots u^{2} - 3.79067 \dots u - 1.94696 \dots)

Appliquer la règle de l'addition/soustraction:

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

= \frac{d}{d u} (u^{3}) - \frac{d}{d u} (3.48638 \dots u^{2}) - \frac{d}{d u} (3.79067 \dots u) - \frac{d}{d u} (1.94696 \dots)

\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}{d u} (3.48638 \dots u^{2}) = 6.97276 \dots u

\frac{d}{d u} (3.48638 \dots u^{2})

Retirer la constante:

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

= 3.48638 \dots \frac{d}{d u} (u^{2})

Appliquer la règle de la puissance:

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

= 3.48638 \dots \cdot 2 u^{2 - 1}

Simplifier

= 6.97276 \dots u

\frac{d}{d u} (3.79067 \dots u) = 3.79067 \dots

\frac{d}{d u} (3.79067 \dots u)

Retirer la constante:

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

= 3.79067 \dots \frac{d u}{d u}

Appliquer la dérivée commune:

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

= 3.79067 \dots \cdot 1

Simplifier

= 3.79067 \dots

\frac{d}{d u} (1.94696 \dots) = 0

\frac{d}{d u} (1.94696 \dots)

Dérivée d'une constante:

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

= 0

= 3 u^{2} - 6.97276 \dots u - 3.79067 \dots - 0

Simplifier

= 3 u^{2} - 6.97276 \dots u - 3.79067 \dots

Soit

u_{0} = 4

Calculer

u_{n + 1}

jusqu'à

Δ u_{n + 1} < 0.000001

u_{1} = 4.54489 \dots : Δ u_{1} = 0.54489 \dots

f (u_{0}) = 4^{3} - 3.48638 \dots \cdot 4^{2} - 3.79067 \dots \cdot 4 - 1.94696 \dots = - 8.89174 \dots

f^{^{'}} (u_{0}) = 3 \cdot 4^{2} - 6.97276 \dots \cdot 4 - 3.79067 \dots = 16.31828 \dots

u_{1} = 4.54489 \dots

Δ u_{1} = ∣ 4.54489 \dots - 4 ∣ = 0.54489 \dots

Δ u_{1} = 0.54489 \dots

u_{2} = 4.44335 \dots : Δ u_{2} = 0.10154 \dots

f (u_{1}) = 4.54489 \dots^{3} - 3.48638 \dots \cdot 4.54489 \dots^{2} - 3.79067 \dots \cdot 4.54489 \dots - 1.94696 \dots = 2.68956 \dots

f^{^{'}} (u_{1}) = 3 \cdot 4.54489 \dots^{2} - 6.97276 \dots \cdot 4.54489 \dots - 3.79067 \dots = 26.48706 \dots

u_{2} = 4.44335 \dots

Δ u_{2} = ∣ 4.44335 \dots - 4.54489 \dots ∣ = 0.10154 \dots

Δ u_{2} = 0.10154 \dots

u_{3} = 4.43911 \dots : Δ u_{3} = 0.00423 \dots

f (u_{2}) = 4.44335 \dots^{3} - 3.48638 \dots \cdot 4.44335 \dots^{2} - 3.79067 \dots \cdot 4.44335 \dots - 1.94696 \dots = 0.10359 \dots

f^{^{'}} (u_{2}) = 3 \cdot 4.44335 \dots^{2} - 6.97276 \dots \cdot 4.44335 \dots - 3.79067 \dots = 24.45702 \dots

u_{3} = 4.43911 \dots

Δ u_{3} = ∣ 4.43911 \dots - 4.44335 \dots ∣ = 0.00423 \dots

Δ u_{3} = 0.00423 \dots

u_{4} = 4.43910 \dots : Δ u_{4} = 7.24246 E - 6

f (u_{3}) = 4.43911 \dots^{3} - 3.48638 \dots \cdot 4.43911 \dots^{2} - 3.79067 \dots \cdot 4.43911 \dots - 1.94696 \dots = 0.00017 \dots

f^{^{'}} (u_{3}) = 3 \cdot 4.43911 \dots^{2} - 6.97276 \dots \cdot 4.43911 \dots - 3.79067 \dots = 24.37369 \dots

u_{4} = 4.43910 \dots

Δ u_{4} = ∣ 4.43910 \dots - 4.43911 \dots ∣ = 7.24246 E - 6

Δ u_{4} = 7.24246 E - 6

u_{5} = 4.43910 \dots : Δ u_{5} = 2.11568 E - 11

f (u_{4}) = 4.43910 \dots^{3} - 3.48638 \dots \cdot 4.43910 \dots^{2} - 3.79067 \dots \cdot 4.43910 \dots - 1.94696 \dots = 5.15667 E - 10

f^{^{'}} (u_{4}) = 3 \cdot 4.43910 \dots^{2} - 6.97276 \dots \cdot 4.43910 \dots - 3.79067 \dots = 24.37355 \dots

u_{5} = 4.43910 \dots

Δ u_{5} = ∣ 4.43910 \dots - 4.43910 \dots ∣ = 2.11568 E - 11

Δ u_{5} = 2.11568 E - 11

u \approx 4.43910 \dots

Appliquer une division longue:

\frac{u ^{3} - 3.48638 \dots u ^{2} - 3.79067 \dots u - 1.94696 \dots}{u - 4.43910 \dots} = u^{2} + 0.95272 \dots u + 0.43859 \dots

u^{2} + 0.95272 \dots u + 0.43859 \dots \approx 0

Trouver une solution pour

u^{2} + 0.95272 \dots u + 0.43859 \dots = 0

par la méthode de Newton-Raphson:

Aucune solution pour

u \in R

u^{2} + 0.95272 \dots u + 0.43859 \dots = 0

Définition de l'approximation de Newton-Raphson

f (u) = u^{2} + 0.95272 \dots u + 0.43859 \dots

Trouver

f^{^{'}} (u) : 2 u + 0.95272 \dots

\frac{d}{d u} (u^{2} + 0.95272 \dots u + 0.43859 \dots)

Appliquer la règle de l'addition/soustraction:

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

= \frac{d}{d u} (u^{2}) + \frac{d}{d u} (0.95272 \dots u) + \frac{d}{d u} (0.43859 \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} (0.95272 \dots u) = 0.95272 \dots

\frac{d}{d u} (0.95272 \dots u)

Retirer la constante:

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

= 0.95272 \dots \frac{d u}{d u}

Appliquer la dérivée commune:

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

= 0.95272 \dots \cdot 1

Simplifier

= 0.95272 \dots

\frac{d}{d u} (0.43859 \dots) = 0

\frac{d}{d u} (0.43859 \dots)

Dérivée d'une constante:

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

= 0

= 2 u + 0.95272 \dots + 0

Simplifier

= 2 u + 0.95272 \dots

Soit

u_{0} = 0

Calculer

u_{n + 1}

jusqu'à

Δ u_{n + 1} < 0.000001

u_{1} = - 0.46035 \dots : Δ u_{1} = 0.46035 \dots

f (u_{0}) = 0^{2} + 0.95272 \dots \cdot 0 + 0.43859 \dots = 0.43859 \dots

f^{^{'}} (u_{0}) = 2 \cdot 0 + 0.95272 \dots = 0.95272 \dots

u_{1} = - 0.46035 \dots

Δ u_{1} = ∣ - 0.46035 \dots - 0 ∣ = 0.46035 \dots

Δ u_{1} = 0.46035 \dots

u_{2} = - 7.07923 \dots : Δ u_{2} = 6.61887 \dots

f (u_{1}) = (- 0.46035 \dots)^{2} + 0.95272 \dots (- 0.46035 \dots) + 0.43859 \dots = 0.21192 \dots

f^{^{'}} (u_{1}) = 2 (- 0.46035 \dots) + 0.95272 \dots = 0.03201 \dots

u_{2} = - 7.07923 \dots

Δ u_{2} = ∣ - 7.07923 \dots - (- 0.46035 \dots) ∣ = 6.61887 \dots

Δ u_{2} = 6.61887 \dots

u_{3} = - 3.76176 \dots : Δ u_{3} = 3.31746 \dots

f (u_{2}) = (- 7.07923 \dots)^{2} + 0.95272 \dots (- 7.07923 \dots) + 0.43859 \dots = 43.80952 \dots

f^{^{'}} (u_{2}) = 2 (- 7.07923 \dots) + 0.95272 \dots = - 13.20573 \dots

u_{3} = - 3.76176 \dots

Δ u_{3} = ∣ - 3.76176 \dots - (- 7.07923 \dots) ∣ = 3.31746 \dots

Δ u_{3} = 3.31746 \dots

u_{4} = - 2.08685 \dots : Δ u_{4} = 1.67491 \dots

f (u_{3}) = (- 3.76176 \dots)^{2} + 0.95272 \dots (- 3.76176 \dots) + 0.43859 \dots = 11.00555 \dots

f^{^{'}} (u_{3}) = 2 (- 3.76176 \dots) + 0.95272 \dots = - 6.57081 \dots

u_{4} = - 2.08685 \dots

Δ u_{4} = ∣ - 2.08685 \dots - (- 3.76176 \dots) ∣ = 1.67491 \dots

Δ u_{4} = 1.67491 \dots

u_{5} = - 1.21589 \dots : Δ u_{5} = 0.87096 \dots

f (u_{4}) = (- 2.08685 \dots)^{2} + 0.95272 \dots (- 2.08685 \dots) + 0.43859 \dots = 2.80534 \dots

f^{^{'}} (u_{4}) = 2 (- 2.08685 \dots) + 0.95272 \dots = - 3.22097 \dots

u_{5} = - 1.21589 \dots

Δ u_{5} = ∣ - 1.21589 \dots - (- 2.08685 \dots) ∣ = 0.87096 \dots

Δ u_{5} = 0.87096 \dots

u_{6} = - 0.70301 \dots : Δ u_{6} = 0.51287 \dots

f (u_{5}) = (- 1.21589 \dots)^{2} + 0.95272 \dots (- 1.21589 \dots) + 0.43859 \dots = 0.75857 \dots

f^{^{'}} (u_{5}) = 2 (- 1.21589 \dots) + 0.95272 \dots = - 1.47905 \dots

u_{6} = - 0.70301 \dots

Δ u_{6} = ∣ - 0.70301 \dots - (- 1.21589 \dots) ∣ = 0.51287 \dots

Δ u_{6} = 0.51287 \dots

u_{7} = - 0.12274 \dots : Δ u_{7} = 0.58027 \dots

f (u_{6}) = (- 0.70301 \dots)^{2} + 0.95272 \dots (- 0.70301 \dots) + 0.43859 \dots = 0.26304 \dots

f^{^{'}} (u_{6}) = 2 (- 0.70301 \dots) + 0.95272 \dots = - 0.45330 \dots

u_{7} = - 0.12274 \dots

Δ u_{7} = ∣ - 0.12274 \dots - (- 0.70301 \dots) ∣ = 0.58027 \dots

Δ u_{7} = 0.58027 \dots

u_{8} = - 0.59883 \dots : Δ u_{8} = 0.47609 \dots

f (u_{7}) = (- 0.12274 \dots)^{2} + 0.95272 \dots (- 0.12274 \dots) + 0.43859 \dots = 0.33672 \dots

f^{^{'}} (u_{7}) = 2 (- 0.12274 \dots) + 0.95272 \dots = 0.70724 \dots

u_{8} = - 0.59883 \dots

Δ u_{8} = ∣ - 0.59883 \dots - (- 0.12274 \dots) ∣ = 0.47609 \dots

Δ u_{8} = 0.47609 \dots

u_{9} = 0.32653 \dots : Δ u_{9} = 0.92537 \dots

f (u_{8}) = (- 0.59883 \dots)^{2} + 0.95272 \dots (- 0.59883 \dots) + 0.43859 \dots = 0.22667 \dots

f^{^{'}} (u_{8}) = 2 (- 0.59883 \dots) + 0.95272 \dots = - 0.24495 \dots

u_{9} = 0.32653 \dots

Δ u_{9} = ∣ 0.32653 \dots - (- 0.59883 \dots) ∣ = 0.92537 \dots

Δ u_{9} = 0.92537 \dots

u_{10} = - 0.20673 \dots : Δ u_{10} = 0.53326 \dots

f (u_{9}) = 0.32653 \dots^{2} + 0.95272 \dots \cdot 0.32653 \dots + 0.43859 \dots = 0.85631 \dots

f^{^{'}} (u_{9}) = 2 \cdot 0.32653 \dots + 0.95272 \dots = 1.60579 \dots

u_{10} = - 0.20673 \dots

Δ u_{10} = ∣ - 0.20673 \dots - 0.32653 \dots ∣ = 0.53326 \dots

Δ u_{10} = 0.53326 \dots

u_{11} = - 0.73406 \dots : Δ u_{11} = 0.52733 \dots

f (u_{10}) = (- 0.20673 \dots)^{2} + 0.95272 \dots (- 0.20673 \dots) + 0.43859 \dots = 0.28437 \dots

f^{^{'}} (u_{10}) = 2 (- 0.20673 \dots) + 0.95272 \dots = 0.53926 \dots

u_{11} = - 0.73406 \dots

Δ u_{11} = ∣ - 0.73406 \dots - (- 0.20673 \dots) ∣ = 0.52733 \dots

Δ u_{11} = 0.52733 \dots

u_{12} = - 0.19452 \dots : Δ u_{12} = 0.53954 \dots

f (u_{11}) = (- 0.73406 \dots)^{2} + 0.95272 \dots (- 0.73406 \dots) + 0.43859 \dots = 0.27807 \dots

f^{^{'}} (u_{11}) = 2 (- 0.73406 \dots) + 0.95272 \dots = - 0.51539 \dots

u_{12} = - 0.19452 \dots

Δ u_{12} = ∣ - 0.19452 \dots - (- 0.73406 \dots) ∣ = 0.53954 \dots

Δ u_{12} = 0.53954 \dots

Impossible de trouver une solution

Les solutions sont

u \approx 0.51361 \dots, u \approx 4.43910 \dots

u \approx 0.51361 \dots, u \approx 4.43910 \dots

Vérifier les solutions

Trouver les points non définis (singularité):

u = 0

Prendre le(s) dénominateur(s) de

\frac{( 1 - u ^{2} ) ^{2}}{4 u ^{2}} - u

et le comparer à zéro

Résoudre

4 u^{2} = 0 : u = 0

4 u^{2} = 0

Diviser les deux côtés par

4

4 u^{2} = 0

Diviser les deux côtés par

4

4 u^{2} = 0

Diviser les deux côtés par

4

\frac{4 u ^{2}}{4} = \frac{0}{4}

Simplifier

u^{2} = 0

u^{2} = 0

Appliquer la règle

x^{n} = 0 \Rightarrow x = 0

u = 0

Les points suivants ne sont pas définis

u = 0

Combiner des points indéfinis avec des solutions :

u \approx 0.51361 \dots, u \approx 4.43910 \dots

Remplacer

u = tan (u)

tan (u) \approx 0.51361 \dots, tan (u) \approx 4.43910 \dots

tan (u) \approx 0.51361 \dots, tan (u) \approx 4.43910 \dots

tan (u) = 0.51361 \dots : u = arctan (0.51361 \dots) + πn

tan (u) = 0.51361 \dots

Appliquer les propriétés trigonométriques inverses

tan (u) = 0.51361 \dots

Solutions générales pour

tan (u) = 0.51361 \dots

tan (x) = a \Rightarrow x = arctan (a) + πn

u = arctan (0.51361 \dots) + πn

u = arctan (0.51361 \dots) + πn

tan (u) = 4.43910 \dots : u = arctan (4.43910 \dots) + πn

tan (u) = 4.43910 \dots

Appliquer les propriétés trigonométriques inverses

tan (u) = 4.43910 \dots

Solutions générales pour

tan (u) = 4.43910 \dots

tan (x) = a \Rightarrow x = arctan (a) + πn

u = arctan (4.43910 \dots) + πn

u = arctan (4.43910 \dots) + πn

Combiner toutes les solutions

u = arctan (0.51361 \dots) + πn, u = arctan (4.43910 \dots) + πn

Remplacer

u = \frac{x}{2}

\frac{x}{2} = arctan (0.51361 \dots) + πn : x = 2 arctan (0.51361 \dots) + 2 πn

\frac{x}{2} = arctan (0.51361 \dots) + πn

Multiplier les deux côtés par

2

\frac{x}{2} = arctan (0.51361 \dots) + πn

Multiplier les deux côtés par

2

\frac{2 x}{2} = 2 arctan (0.51361 \dots) + 2 πn

Simplifier

x = 2 arctan (0.51361 \dots) + 2 πn

x = 2 arctan (0.51361 \dots) + 2 πn

\frac{x}{2} = arctan (4.43910 \dots) + πn : x = 2 arctan (4.43910 \dots) + 2 πn

\frac{x}{2} = arctan (4.43910 \dots) + πn

Multiplier les deux côtés par

2

\frac{x}{2} = arctan (4.43910 \dots) + πn

Multiplier les deux côtés par

2

\frac{2 x}{2} = 2 arctan (4.43910 \dots) + 2 πn

Simplifier

x = 2 arctan (4.43910 \dots) + 2 πn

x = 2 arctan (4.43910 \dots) + 2 πn

x = 2 arctan (0.51361 \dots) + 2 πn, x = 2 arctan (4.43910 \dots) + 2 πn

Montrer les solutions sous la forme décimale

x = 2 \cdot 0.47448 \dots + 2 πn, x = 2 \cdot 1.34922 \dots + 2 πn

Graphe

Exemples populaires

cos(x)= 15/21

cos (x) = \frac{15}{21}

cos(t)= 3/5

cos (t) = \frac{3}{5}

3+4cos(2x)+2cos(4x)=0

3 + 4 cos (2 x) + 2 cos (4 x) = 0

cos(θ)=-4/5 ,sin(2θ),180<θ<270

cos (θ) = - \frac{4}{5}, sin (2 θ), 18 0^{\circ} < θ < 27 0^{\circ}

2-2sin^2(3x)=1

2 - 2 sin^{2} (3 x) = 1