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Populaire Trigonométrie >

(8.87)/(9.8)=sin(α)-0.1sqrt(1-sin^2(α))

Solution

\frac{8.87}{9.8} = sin (α) - 0.1 1 - sin^{2} (α)

Solution

α = 1.22084 \dots + 2 πn, α = π - 1.22084 \dots + 2 πn

Degrés

α = 69.94898 \dots^{\circ} + 36 0^{\circ} n, α = 110.05101 \dots^{\circ} + 36 0^{\circ} n

étapes des solutions

\frac{8.87}{9.8} = sin (α) - 0.1 1 - sin^{2} (α)

Transposer les termes des côtés

sin (α) - 0.1 1 - sin^{2} (α) = \frac{8.87}{9.8}

Résoudre par substitution

sin (α) - 0.1 1 - sin^{2} (α) = \frac{8.87}{9.8}

Soit :

sin (α) = u

u - 0.1 1 - u^{2} = \frac{8.87}{9.8}

u - 0.1 1 - u^{2} = \frac{8.87}{9.8} : u = \frac{173.852 + 70.3915576}{194.0008}

u - 0.1 1 - u^{2} = \frac{8.87}{9.8}

Multiplier les deux côtés par

9.8

u \cdot 9.8 - 0.1 1 - u^{2} \cdot 9.8 = \frac{8.87}{9.8} \cdot 9.8

Simplifier

9.8 u - 0.98 1 - u^{2} = 8.87

Supprimer les racines carrées

9.8 u - 0.98 1 - u^{2} = 8.87

Soustraire

9.8 u

des deux côtés

9.8 u - 0.98 1 - u^{2} - 9.8 u = 8.87 - 9.8 u

Simplifier

- 0.98 1 - u^{2} = 8.87 - 9.8 u

Mettre les deux côtés au carré:

0.9604 - 0.9604 u^{2} = 78.6769 - 173.852 u + 96.04 u^{2}

9.8 u - 0.98 1 - u^{2} = 8.87

(- 0.98 1 - u^{2})^{2} = (8.87 - 9.8 u)^{2}

Développer

(- 0.98 1 - u^{2})^{2} : 0.9604 - 0.9604 u^{2}

(- 0.98 1 - u^{2})^{2}

Appliquer la règle de l'exposant:

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

n

pair

(- 0.98 1 - u^{2})^{2} = (0.98 1 - u^{2})^{2}

= (0.98 1 - u^{2})^{2}

Appliquer la règle de l'exposant:

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

= 0.9 8^{2} (1 - u^{2})^{2}

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

= 0.9 8^{2} (1 - u^{2})

0.9 8^{2} = 0.9604

= 0.9604 (1 - u^{2})

Développer

0.9604 (1 - u^{2}) : 0.9604 - 0.9604 u^{2}

0.9604 (1 - u^{2})

Appliquer la loi de la distribution:

a (b - c) = ab - a c

a = 0.9604, b = 1, c = u^{2}

= 0.9604 \cdot 1 - 0.9604 u^{2}

= 1 \cdot 0.9604 - 0.9604 u^{2}

Multiplier les nombres :

1 \cdot 0.9604 = 0.9604

= 0.9604 - 0.9604 u^{2}

= 0.9604 - 0.9604 u^{2}

Développer

(8.87 - 9.8 u)^{2} : 78.6769 - 173.852 u + 96.04 u^{2}

(8.87 - 9.8 u)^{2}

Appliquer la formule du carré parfait:

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

a = 8.87, b = 9.8 u

= 8.8 7^{2} - 2 \cdot 8.87 \cdot 9.8 u + (9.8 u)^{2}

Simplifier

8.8 7^{2} - 2 \cdot 8.87 \cdot 9.8 u + (9.8 u)^{2} : 78.6769 - 173.852 u + 96.04 u^{2}

8.8 7^{2} - 2 \cdot 8.87 \cdot 9.8 u + (9.8 u)^{2}

8.8 7^{2} = 78.6769

8.8 7^{2}

8.8 7^{2} = 78.6769

= 78.6769

2 \cdot 8.87 \cdot 9.8 u = 173.852 u

2 \cdot 8.87 \cdot 9.8 u

Multiplier les nombres :

2 \cdot 8.87 \cdot 9.8 = 173.852

= 173.852 u

(9.8 u)^{2} = 96.04 u^{2}

(9.8 u)^{2}

Appliquer la règle de l'exposant:

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

= 9. 8^{2} u^{2}

9. 8^{2} = 96.04

= 96.04 u^{2}

= 78.6769 - 173.852 u + 96.04 u^{2}

= 78.6769 - 173.852 u + 96.04 u^{2}

0.9604 - 0.9604 u^{2} = 78.6769 - 173.852 u + 96.04 u^{2}

0.9604 - 0.9604 u^{2} = 78.6769 - 173.852 u + 96.04 u^{2}

0.9604 - 0.9604 u^{2} = 78.6769 - 173.852 u + 96.04 u^{2}

Résoudre

0.9604 - 0.9604 u^{2} = 78.6769 - 173.852 u + 96.04 u^{2} : u = \frac{173.852 + 70.3915576}{194.0008}, u = \frac{173.852 - 70.3915576}{194.0008}

0.9604 - 0.9604 u^{2} = 78.6769 - 173.852 u + 96.04 u^{2}

Transposer les termes des côtés

78.6769 - 173.852 u + 96.04 u^{2} = 0.9604 - 0.9604 u^{2}

Déplacer

0.9604 u^{2}

vers la gauche

78.6769 - 173.852 u + 96.04 u^{2} = 0.9604 - 0.9604 u^{2}

Ajouter

0.9604 u^{2}

aux deux côtés

78.6769 - 173.852 u + 96.04 u^{2} + 0.9604 u^{2} = 0.9604 - 0.9604 u^{2} + 0.9604 u^{2}

Simplifier

78.6769 - 173.852 u + 97.0004 u^{2} = 0.9604

78.6769 - 173.852 u + 97.0004 u^{2} = 0.9604

Déplacer

0.9604

vers la gauche

78.6769 - 173.852 u + 97.0004 u^{2} = 0.9604

Soustraire

0.9604

des deux côtés

78.6769 - 173.852 u + 97.0004 u^{2} - 0.9604 = 0.9604 - 0.9604

Simplifier

97.0004 u^{2} - 173.852 u + 77.7165 = 0

97.0004 u^{2} - 173.852 u + 77.7165 = 0

Résoudre par la formule quadratique

97.0004 u^{2} - 173.852 u + 77.7165 = 0

Formule de l'équation quadratique:

Pour

a = 97.0004, b = - 173.852, c = 77.7165

u_{1, 2} = \frac{- ( - 173.852 ) \pm ( - 173.852 ) ^{2} - 4 \cdot 97.0004 \cdot 77.7165}{2 \cdot 97.0004}

u_{1, 2} = \frac{- ( - 173.852 ) \pm ( - 173.852 ) ^{2} - 4 \cdot 97.0004 \cdot 77.7165}{2 \cdot 97.0004}

(- 173.852)^{2} - 4 \cdot 97.0004 \cdot 77.7165 = 70.3915576

(- 173.852)^{2} - 4 \cdot 97.0004 \cdot 77.7165

Appliquer la règle de l'exposant:

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

n

pair

(- 173.852)^{2} = 173.85 2^{2}

= 173.85 2^{2} - 4 \cdot 77.7165 \cdot 97.0004

Multiplier les nombres :

4 \cdot 97.0004 \cdot 77.7165 = 30154.12634 \dots

= 173.85 2^{2} - 30154.12634 \dots

173.85 2^{2} = 30224.517904

= 30224.517904 - 30154.12634 \dots

Soustraire les nombres :

30224.517904 - 30154.12634 \dots = 70.3915576

= 70.3915576

u_{1, 2} = \frac{- ( - 173.852 ) \pm 70.3915576}{2 \cdot 97.0004}

Séparer les solutions

u_{1} = \frac{- ( - 173.852 ) + 70.3915576}{2 \cdot 97.0004}, u_{2} = \frac{- ( - 173.852 ) - 70.3915576}{2 \cdot 97.0004}

u = \frac{- ( - 173.852 ) + 70.3915576}{2 \cdot 97.0004} : \frac{173.852 + 70.3915576}{194.0008}

\frac{- ( - 173.852 ) + 70.3915576}{2 \cdot 97.0004}

Appliquer la règle

- (- a) = a

= \frac{173.852 + 70.3915576}{2 \cdot 97.0004}

Multiplier les nombres :

2 \cdot 97.0004 = 194.0008

= \frac{173.852 + 70.3915576}{194.0008}

u = \frac{- ( - 173.852 ) - 70.3915576}{2 \cdot 97.0004} : \frac{173.852 - 70.3915576}{194.0008}

\frac{- ( - 173.852 ) - 70.3915576}{2 \cdot 97.0004}

Appliquer la règle

- (- a) = a

= \frac{173.852 - 70.3915576}{2 \cdot 97.0004}

Multiplier les nombres :

2 \cdot 97.0004 = 194.0008

= \frac{173.852 - 70.3915576}{194.0008}

Les solutions de l'équation de forme quadratique sont :

u = \frac{173.852 + 70.3915576}{194.0008}, u = \frac{173.852 - 70.3915576}{194.0008}

u = \frac{173.852 + 70.3915576}{194.0008}, u = \frac{173.852 - 70.3915576}{194.0008}

Vérifier les solutions:

u = \frac{173.852 + 70.3915576}{194.0008}

vrai

, u = \frac{173.852 - 70.3915576}{194.0008}

Faux

Vérifier des solutions en les intégrant dans

u - 0.1 1 - u^{2} = \frac{8.87}{9.8}

Retirer celles qui ne répondent pas à l'équation.

Insérer

u = \frac{173.852 + 70.3915576}{194.0008} :

vrai

(\frac{173.852 + 70.3915576}{194.0008}) - 0.1 1 - (\frac{173.852 + 70.3915576}{194.0008})^{2} = \frac{8.87}{9.8}

(\frac{173.852 + 70.3915576}{194.0008}) - 0.1 1 - (\frac{173.852 + 70.3915576}{194.0008})^{2} = 0.90510 \dots

(\frac{173.852 + 70.3915576}{194.0008}) - 0.1 1 - (\frac{173.852 + 70.3915576}{194.0008})^{2}

Retirer les parenthèses:

(a) = a

= \frac{173.852 + 70.3915576}{194.0008} - 0.1 1 - (\frac{173.852 + 70.3915576}{194.0008})^{2}

\frac{173.852 + 70.3915576}{194.0008} = \frac{182.24196 \dots}{194.0008}

\frac{173.852 + 70.3915576}{194.0008}

70.3915576 = 8.38996 \dots

= \frac{173.852 + 8.38996 \dots}{194.0008}

Additionner les nombres :

173.852 + 8.38996 \dots = 182.24196 \dots

= \frac{182.24196 \dots}{194.0008}

0.1 1 - (\frac{173.852 + 70.3915576}{194.0008})^{2} = 0.1 0.11755 \dots

0.1 1 - (\frac{173.852 + 70.3915576}{194.0008})^{2}

1 - (\frac{173.852 + 70.3915576}{194.0008})^{2} = 0.11755 \dots

1 - (\frac{173.852 + 70.3915576}{194.0008})^{2}

(\frac{173.852 + 70.3915576}{194.0008})^{2} = 0.88244 \dots

(\frac{173.852 + 70.3915576}{194.0008})^{2}

\frac{173.852 + 70.3915576}{194.0008} = \frac{182.24196 \dots}{194.0008}

\frac{173.852 + 70.3915576}{194.0008}

70.3915576 = 8.38996 \dots

= \frac{173.852 + 8.38996 \dots}{194.0008}

Additionner les nombres :

173.852 + 8.38996 \dots = 182.24196 \dots

= \frac{182.24196 \dots}{194.0008}

= (\frac{182.24196 \dots}{194.0008})^{2}

Appliquer la règle de l'exposant:

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

= \frac{182.24196 \dots ^{2}}{194.000 8 ^{2}}

182.24196 \dots^{2} = 33212.13478 \dots

= \frac{33212.13478 \dots}{194.000 8 ^{2}}

194.000 8^{2} = 37636.31040 \dots

= \frac{33212.13478 \dots}{37636.31040 \dots}

Diviser les nombres :

\frac{33212.13478 \dots}{37636.31040 \dots} = 0.88244 \dots

= 0.88244 \dots

= 1 - 0.88244 \dots

Soustraire les nombres :

1 - 0.88244 \dots = 0.11755 \dots

= 0.11755 \dots

= 0.1 0.11755 \dots

= \frac{182.24196 \dots}{194.0008} - 0.1 0.11755 \dots

\frac{182.24196 \dots}{194.0008} = 0.93938 \dots

\frac{182.24196 \dots}{194.0008}

Diviser les nombres :

\frac{182.24196 \dots}{194.0008} = 0.93938 \dots

= 0.93938 \dots

0.1 0.11755 \dots = 0.03428 \dots

0.1 0.11755 \dots

0.11755 \dots = 0.34285 \dots

= 0.1 \cdot 0.34285 \dots

Multiplier les nombres :

0.1 \cdot 0.34285 \dots = 0.03428 \dots

= 0.03428 \dots

= 0.93938 \dots - 0.03428 \dots

Soustraire les nombres :

0.93938 \dots - 0.03428 \dots = 0.90510 \dots

= 0.90510 \dots

\frac{8.87}{9.8} = 0.90510 \dots

\frac{8.87}{9.8}

Diviser les nombres :

\frac{8.87}{9.8} = 0.90510 \dots

= 0.90510 \dots

0.90510 \dots = 0.90510 \dots

vrai

Insérer

u = \frac{173.852 - 70.3915576}{194.0008} :

Faux

(\frac{173.852 - 70.3915576}{194.0008}) - 0.1 1 - (\frac{173.852 - 70.3915576}{194.0008})^{2} = \frac{8.87}{9.8}

(\frac{173.852 - 70.3915576}{194.0008}) - 0.1 1 - (\frac{173.852 - 70.3915576}{194.0008})^{2} = 0.80068 \dots

(\frac{173.852 - 70.3915576}{194.0008}) - 0.1 1 - (\frac{173.852 - 70.3915576}{194.0008})^{2}

Retirer les parenthèses:

(a) = a

= \frac{173.852 - 70.3915576}{194.0008} - 0.1 1 - (\frac{173.852 - 70.3915576}{194.0008})^{2}

\frac{173.852 - 70.3915576}{194.0008} = \frac{165.46203 \dots}{194.0008}

\frac{173.852 - 70.3915576}{194.0008}

70.3915576 = 8.38996 \dots

= \frac{173.852 - 8.38996 \dots}{194.0008}

Soustraire les nombres :

173.852 - 8.38996 \dots = 165.46203 \dots

= \frac{165.46203 \dots}{194.0008}

0.1 1 - (\frac{173.852 - 70.3915576}{194.0008})^{2} = 0.1 0.27257 \dots

0.1 1 - (\frac{173.852 - 70.3915576}{194.0008})^{2}

1 - (\frac{173.852 - 70.3915576}{194.0008})^{2} = 0.27257 \dots

1 - (\frac{173.852 - 70.3915576}{194.0008})^{2}

(\frac{173.852 - 70.3915576}{194.0008})^{2} = 0.72742 \dots

(\frac{173.852 - 70.3915576}{194.0008})^{2}

\frac{173.852 - 70.3915576}{194.0008} = \frac{165.46203 \dots}{194.0008}

\frac{173.852 - 70.3915576}{194.0008}

70.3915576 = 8.38996 \dots

= \frac{173.852 - 8.38996 \dots}{194.0008}

Soustraire les nombres :

173.852 - 8.38996 \dots = 165.46203 \dots

= \frac{165.46203 \dots}{194.0008}

= (\frac{165.46203 \dots}{194.0008})^{2}

Appliquer la règle de l'exposant:

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

= \frac{165.46203 \dots ^{2}}{194.000 8 ^{2}}

165.46203 \dots^{2} = 27377.68414 \dots

= \frac{27377.68414 \dots}{194.000 8 ^{2}}

194.000 8^{2} = 37636.31040 \dots

= \frac{27377.68414 \dots}{37636.31040 \dots}

Diviser les nombres :

\frac{27377.68414 \dots}{37636.31040 \dots} = 0.72742 \dots

= 0.72742 \dots

= 1 - 0.72742 \dots

Soustraire les nombres :

1 - 0.72742 \dots = 0.27257 \dots

= 0.27257 \dots

= 0.1 0.27257 \dots

= \frac{165.46203 \dots}{194.0008} - 0.1 0.27257 \dots

\frac{165.46203 \dots}{194.0008} = 0.85289 \dots

\frac{165.46203 \dots}{194.0008}

Diviser les nombres :

\frac{165.46203 \dots}{194.0008} = 0.85289 \dots

= 0.85289 \dots

0.1 0.27257 \dots = 0.05220 \dots

0.1 0.27257 \dots

0.27257 \dots = 0.52208 \dots

= 0.1 \cdot 0.52208 \dots

Multiplier les nombres :

0.1 \cdot 0.52208 \dots = 0.05220 \dots

= 0.05220 \dots

= 0.85289 \dots - 0.05220 \dots

Soustraire les nombres :

0.85289 \dots - 0.05220 \dots = 0.80068 \dots

= 0.80068 \dots

\frac{8.87}{9.8} = 0.90510 \dots

\frac{8.87}{9.8}

Diviser les nombres :

\frac{8.87}{9.8} = 0.90510 \dots

= 0.90510 \dots

0.80068 \dots = 0.90510 \dots

Faux

La solution est

u = \frac{173.852 + 70.3915576}{194.0008}

Remplacer

u = sin (α)

sin (α) = \frac{173.852 + 70.3915576}{194.0008}

sin (α) = \frac{173.852 + 70.3915576}{194.0008}

sin (α) = \frac{173.852 + 70.3915576}{194.0008} : α = arcsin (\frac{173.852 + 70.3915576}{194.0008}) + 2 πn, α = π - arcsin (\frac{173.852 + 70.3915576}{194.0008}) + 2 πn

sin (α) = \frac{173.852 + 70.3915576}{194.0008}

Appliquer les propriétés trigonométriques inverses

sin (α) = \frac{173.852 + 70.3915576}{194.0008}

Solutions générales pour

sin (α) = \frac{173.852 + 70.3915576}{194.0008}

sin (x) = a \Rightarrow x = arcsin (a) + 2 πn, x = π - arcsin (a) + 2 πn

α = arcsin (\frac{173.852 + 70.3915576}{194.0008}) + 2 πn, α = π - arcsin (\frac{173.852 + 70.3915576}{194.0008}) + 2 πn

α = arcsin (\frac{173.852 + 70.3915576}{194.0008}) + 2 πn, α = π - arcsin (\frac{173.852 + 70.3915576}{194.0008}) + 2 πn

Combiner toutes les solutions

α = arcsin (\frac{173.852 + 70.3915576}{194.0008}) + 2 πn, α = π - arcsin (\frac{173.852 + 70.3915576}{194.0008}) + 2 πn

Montrer les solutions sous la forme décimale

α = 1.22084 \dots + 2 πn, α = π - 1.22084 \dots + 2 πn

Graphe

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