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Populaire Trigonométrie >

sqrt(1-cos^2(x))=0.75sqrt(1-cos^2(y))

Solution

1 - cos^{2} (x) = 0.75 1 - cos^{2} (y)

Solution

x = arccos (0.5625 cos^{2} (y) + 0.4375) + 2 πn, x = - arccos (0.5625 cos^{2} (y) + 0.4375) + 2 πn, x = arccos (- 0.5625 cos^{2} (y) + 0.4375) + 2 πn, x = - arccos (- 0.5625 cos^{2} (y) + 0.4375) + 2 πn

étapes des solutions

1 - cos^{2} (x) = 0.75 1 - cos^{2} (y)

Résoudre par substitution

1 - cos^{2} (x) = 0.75 1 - cos^{2} (y)

Soit :

cos (x) = u

1 - u^{2} = 0.75 1 - cos^{2} (y)

1 - u^{2} = 0.75 1 - cos^{2} (y) : u = 0.5625 cos^{2} (y) + 0.4375, u = - 0.5625 cos^{2} (y) + 0.4375

1 - u^{2} = 0.75 1 - cos^{2} (y)

Mettre les deux côtés au carré:

1 - u^{2} = 0.5625 - 0.5625 cos^{2} (y)

1 - u^{2} = 0.75 1 - cos^{2} (y)

(1 - u^{2})^{2} = (0.75 1 - cos^{2} (y))^{2}

Développer

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

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

Appliquer la règle des radicaux:

a = a^{\frac{1}{2}}

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

Appliquer la règle de l'exposant:

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

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

\frac{1}{2} \cdot 2 = 1

\frac{1}{2} \cdot 2

Multiplier des fractions:

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

= \frac{1 \cdot 2}{2}

Annuler le facteur commun :

2

= 1

= 1 - u^{2}

Développer

(0.75 1 - cos^{2} (y))^{2} : 0.5625 - 0.5625 cos^{2} (y)

(0.75 1 - cos^{2} (y))^{2}

Appliquer la règle de l'exposant:

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

= 0.7 5^{2} (- cos^{2} (y) + 1)^{2}

(1 - cos^{2} (y))^{2} : 1 - cos^{2} (y)

Appliquer la règle des radicaux:

a = a^{\frac{1}{2}}

= ((1 - cos^{2} (y))^{\frac{1}{2}})^{2}

Appliquer la règle de l'exposant:

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

= (1 - cos^{2} (y))^{\frac{1}{2} \cdot 2}

\frac{1}{2} \cdot 2 = 1

\frac{1}{2} \cdot 2

Multiplier des fractions:

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

= \frac{1 \cdot 2}{2}

Annuler le facteur commun :

2

= 1

= 1 - cos^{2} (y)

= 0.7 5^{2} (1 - cos^{2} (y))

0.7 5^{2} = 0.5625

= 0.5625 (- cos^{2} (y) + 1)

Appliquer la loi de la distribution:

a (b - c) = ab - a c

a = 0.5625, b = 1, c = cos^{2} (y)

= 0.5625 \cdot 1 - 0.5625 cos^{2} (y)

= 1 \cdot 0.5625 - 0.5625 cos^{2} (y)

Multiplier les nombres :

1 \cdot 0.5625 = 0.5625

= 0.5625 - 0.5625 cos^{2} (y)

1 - u^{2} = 0.5625 - 0.5625 cos^{2} (y)

1 - u^{2} = 0.5625 - 0.5625 cos^{2} (y)

Résoudre

1 - u^{2} = 0.5625 - 0.5625 cos^{2} (y) : u = 0.5625 cos^{2} (y) + 0.4375, u = - 0.5625 cos^{2} (y) + 0.4375

1 - u^{2} = 0.5625 - 0.5625 cos^{2} (y)

Déplacer

1

vers la droite

1 - u^{2} = 0.5625 - 0.5625 cos^{2} (y)

Soustraire

1

des deux côtés

1 - u^{2} - 1 = 0.5625 - 0.5625 cos^{2} (y) - 1

Simplifier

- u^{2} = - 0.5625 cos^{2} (y) - 0.4375

- u^{2} = - 0.5625 cos^{2} (y) - 0.4375

Diviser les deux côtés par

- 1

- u^{2} = - 0.5625 cos^{2} (y) - 0.4375

Diviser les deux côtés par

- 1

\frac{- u ^{2}}{- 1} = - \frac{0.5625 cos ^{2} ( y )}{- 1} - \frac{0.4375}{- 1}

Simplifier

\frac{- u ^{2}}{- 1} = - \frac{0.5625 cos ^{2} ( y )}{- 1} - \frac{0.4375}{- 1}

Simplifier

\frac{- u ^{2}}{- 1} : u^{2}

\frac{- u ^{2}}{- 1}

Appliquer la règle des fractions:

\frac{- a}{- b} = \frac{a}{b}

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

Appliquer la règle

\frac{a}{1} = a

= u^{2}

Simplifier

- \frac{0.5625 cos ^{2} ( y )}{- 1} - \frac{0.4375}{- 1} : 0.5625 cos^{2} (y) + 0.4375

- \frac{0.5625 cos ^{2} ( y )}{- 1} - \frac{0.4375}{- 1}

Appliquer la règle

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

= \frac{- 0.5625 cos ^{2} ( y ) - 0.4375}{- 1}

Appliquer la règle des fractions:

\frac{a}{- b} = - \frac{a}{b}

= - \frac{- 0.5625 cos ^{2} ( y ) - 0.4375}{1}

Appliquer la règle

\frac{a}{1} = a

= - (- 0.5625 cos^{2} (y) - 0.4375)

Distribuer des parenthèses

= - (- 0.5625 cos^{2} (y)) - (- 0.4375)

Appliquer les règles des moins et des plus

- (- a) = a

= 0.5625 cos^{2} (y) + 0.4375

u^{2} = 0.5625 cos^{2} (y) + 0.4375

u^{2} = 0.5625 cos^{2} (y) + 0.4375

u^{2} = 0.5625 cos^{2} (y) + 0.4375

Pour

x^{2} = f (a)

les solutions sont

x = f (a), - f (a)

u = 0.5625 cos^{2} (y) + 0.4375, u = - 0.5625 cos^{2} (y) + 0.4375

u = 0.5625 cos^{2} (y) + 0.4375, u = - 0.5625 cos^{2} (y) + 0.4375

Vérifier les solutions:

u = 0.5625 cos^{2} (y) + 0.4375

vrai

, u = - 0.5625 cos^{2} (y) + 0.4375

vrai

Vérifier des solutions en les intégrant dans

1 - u^{2} = 0.75 1 - cos^{2} (y)

Retirer celles qui ne répondent pas à l'équation.

Insérer

u = 0.5625 cos^{2} (y) + 0.4375 :

vrai

1 - (0.5625 cos^{2} (y) + 0.4375)^{2} = 0.75 1 - cos^{2} (y)

Simplifier

1 - (0.5625 cos^{2} (y) + 0.4375)^{2} : - 0.5625 cos^{2} (y) + 0.5625

1 - (0.5625 cos^{2} (y) + 0.4375)^{2}

(0.5625 cos^{2} (y) + 0.4375)^{2} = 0.5625 cos^{2} (y) + 0.4375

(0.5625 cos^{2} (y) + 0.4375)^{2}

Appliquer la règle des radicaux:

a = a^{\frac{1}{2}}

= ((0.5625 cos^{2} (y) + 0.4375)^{\frac{1}{2}})^{2}

Appliquer la règle de l'exposant:

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

= (0.5625 cos^{2} (y) + 0.4375)^{\frac{1}{2} \cdot 2}

\frac{1}{2} \cdot 2 = 1

\frac{1}{2} \cdot 2

Multiplier des fractions:

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

= \frac{1 \cdot 2}{2}

Annuler le facteur commun :

2

= 1

= 0.5625 cos^{2} (y) + 0.4375

= 1 - (0.5625 cos^{2} (y) + 0.4375)

Développer

1 - (0.5625 cos^{2} (y) + 0.4375) : - 0.5625 cos^{2} (y) + 0.5625

1 - (0.5625 cos^{2} (y) + 0.4375)

- (0.5625 cos^{2} (y) + 0.4375) : - 0.5625 cos^{2} (y) - 0.4375

- (0.5625 cos^{2} (y) + 0.4375)

Distribuer des parenthèses

= - (0.5625 cos^{2} (y)) - (0.4375)

Appliquer les règles des moins et des plus

+ (- a) = - a

= - 0.5625 cos^{2} (y) - 0.4375

= 1 - 0.5625 cos^{2} (y) - 0.4375

Soustraire les nombres :

1 - 0.4375 = 0.5625

= - 0.5625 cos^{2} (y) + 0.5625

= - 0.5625 cos^{2} (y) + 0.5625

- 0.5625 cos^{2} (y) + 0.5625 = 0.75 1 - cos^{2} (y)

vrai

Insérer

u = - 0.5625 cos^{2} (y) + 0.4375 :

vrai

1 - (- 0.5625 cos^{2} (y) + 0.4375)^{2} = 0.75 1 - cos^{2} (y)

Simplifier

1 - (- 0.5625 cos^{2} (y) + 0.4375)^{2} : - 0.5625 cos^{2} (y) + 0.5625

1 - (- 0.5625 cos^{2} (y) + 0.4375)^{2}

(- 0.5625 cos^{2} (y) + 0.4375)^{2} = 0.5625 cos^{2} (y) + 0.4375

(- 0.5625 cos^{2} (y) + 0.4375)^{2}

Appliquer la règle de l'exposant:

(- a)^{n} = a^{n},

n

pair

(- 0.5625 cos^{2} (y) + 0.4375)^{2} = (0.5625 cos^{2} (y) + 0.4375)^{2}

= (0.5625 cos^{2} (y) + 0.4375)^{2}

Appliquer la règle des radicaux:

a = a^{\frac{1}{2}}

= ((0.5625 cos^{2} (y) + 0.4375)^{\frac{1}{2}})^{2}

Appliquer la règle de l'exposant:

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

= (0.5625 cos^{2} (y) + 0.4375)^{\frac{1}{2} \cdot 2}

\frac{1}{2} \cdot 2 = 1

\frac{1}{2} \cdot 2

Multiplier des fractions:

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

= \frac{1 \cdot 2}{2}

Annuler le facteur commun :

2

= 1

= 0.5625 cos^{2} (y) + 0.4375

= 1 - (0.5625 cos^{2} (y) + 0.4375)

Développer

1 - (0.5625 cos^{2} (y) + 0.4375) : - 0.5625 cos^{2} (y) + 0.5625

1 - (0.5625 cos^{2} (y) + 0.4375)

- (0.5625 cos^{2} (y) + 0.4375) : - 0.5625 cos^{2} (y) - 0.4375

- (0.5625 cos^{2} (y) + 0.4375)

Distribuer des parenthèses

= - (0.5625 cos^{2} (y)) - (0.4375)

Appliquer les règles des moins et des plus

+ (- a) = - a

= - 0.5625 cos^{2} (y) - 0.4375

= 1 - 0.5625 cos^{2} (y) - 0.4375

Soustraire les nombres :

1 - 0.4375 = 0.5625

= - 0.5625 cos^{2} (y) + 0.5625

= - 0.5625 cos^{2} (y) + 0.5625

- 0.5625 cos^{2} (y) + 0.5625 = 0.75 1 - cos^{2} (y)

vrai

Les solutions sont

u = 0.5625 cos^{2} (y) + 0.4375, u = - 0.5625 cos^{2} (y) + 0.4375

Remplacer

u = cos (x)

cos (x) = 0.5625 cos^{2} (y) + 0.4375, cos (x) = - 0.5625 cos^{2} (y) + 0.4375

cos (x) = 0.5625 cos^{2} (y) + 0.4375, cos (x) = - 0.5625 cos^{2} (y) + 0.4375

cos (x) = 0.5625 cos^{2} (y) + 0.4375 : x = arccos (0.5625 cos^{2} (y) + 0.4375) + 2 πn, x = - arccos (0.5625 cos^{2} (y) + 0.4375) + 2 πn

cos (x) = 0.5625 cos^{2} (y) + 0.4375

Appliquer les propriétés trigonométriques inverses

cos (x) = 0.5625 cos^{2} (y) + 0.4375

Solutions générales pour

cos (x) = 0.5625 cos^{2} (y) + 0.4375

cos (x) = a \Rightarrow x = arccos (a) + 2 πn, x = - arccos (a) + 2 πn

x = arccos (0.5625 cos^{2} (y) + 0.4375) + 2 πn, x = - arccos (0.5625 cos^{2} (y) + 0.4375) + 2 πn

x = arccos (0.5625 cos^{2} (y) + 0.4375) + 2 πn, x = - arccos (0.5625 cos^{2} (y) + 0.4375) + 2 πn

cos (x) = - 0.5625 cos^{2} (y) + 0.4375 : x = arccos (- 0.5625 cos^{2} (y) + 0.4375) + 2 πn, x = - arccos (- 0.5625 cos^{2} (y) + 0.4375) + 2 πn

cos (x) = - 0.5625 cos^{2} (y) + 0.4375

Appliquer les propriétés trigonométriques inverses

cos (x) = - 0.5625 cos^{2} (y) + 0.4375

Solutions générales pour

cos (x) = - 0.5625 cos^{2} (y) + 0.4375

cos (x) = a \Rightarrow x = arccos (a) + 2 πn, x = - arccos (a) + 2 πn

x = arccos (- 0.5625 cos^{2} (y) + 0.4375) + 2 πn, x = - arccos (- 0.5625 cos^{2} (y) + 0.4375) + 2 πn

x = arccos (- 0.5625 cos^{2} (y) + 0.4375) + 2 πn, x = - arccos (- 0.5625 cos^{2} (y) + 0.4375) + 2 πn

Combiner toutes les solutions

x = arccos (0.5625 cos^{2} (y) + 0.4375) + 2 πn, x = - arccos (0.5625 cos^{2} (y) + 0.4375) + 2 πn, x = arccos (- 0.5625 cos^{2} (y) + 0.4375) + 2 πn, x = - arccos (- 0.5625 cos^{2} (y) + 0.4375) + 2 πn

Graphe

Exemples populaires

sec^2(x)-6tan(x)+4=0

sec^{2} (x) - 6 tan (x) + 4 = 0

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

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

-5tan(x)-12=tan(x)-8

- 5 tan (x) - 12 = tan (x) - 8

5=sec^2(x)+3

5 = sec^{2} (x) + 3

sin(θ)= 24/25

sin (θ) = \frac{24}{25}