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Populaire >

sin(θ)-0.2cos(θ)=(7.51)/(9.8)

Solution

sin (θ) - 0.2 cos (θ) = \frac{7.51}{9.8}

Solution

θ = 2.48873 \dots + 2 πn, θ = 1.04764 \dots + 2 πn

Degrés

θ = 142.59422 \dots^{\circ} + 36 0^{\circ} n, θ = 60.02563 \dots^{\circ} + 36 0^{\circ} n

étapes des solutions

sin (θ) - 0.2 cos (θ) = \frac{7.51}{9.8}

Ajouter

0.2 cos (θ)

aux deux côtés

sin (θ) = 0.76632 \dots + 0.2 cos (θ)

Mettre les deux côtés au carré

sin^{2} (θ) = (0.76632 \dots + 0.2 cos (θ))^{2}

Soustraire

(0.76632 \dots + 0.2 cos (θ))^{2}

des deux côtés

sin^{2} (θ) - 0.58725 \dots - 0.30653 \dots cos (θ) - 0.04 cos^{2} (θ) = 0

Récrire en utilisant des identités trigonométriques

- 0.58725 \dots + sin^{2} (θ) - 0.04 cos^{2} (θ) - 0.30653 \dots cos (θ)

Utiliser l'identité hyperbolique:

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

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

= - 0.58725 \dots + 1 - cos^{2} (θ) - 0.04 cos^{2} (θ) - 0.30653 \dots cos (θ)

Simplifier

- 0.58725 \dots + 1 - cos^{2} (θ) - 0.04 cos^{2} (θ) - 0.30653 \dots cos (θ) : - 1.04 cos^{2} (θ) - 0.30653 \dots cos (θ) + 0.41274 \dots

- 0.58725 \dots + 1 - cos^{2} (θ) - 0.04 cos^{2} (θ) - 0.30653 \dots cos (θ)

Additionner les éléments similaires :

- cos^{2} (θ) - 0.04 cos^{2} (θ) = - 1.04 cos^{2} (θ)

= - 0.58725 \dots + 1 - 1.04 cos^{2} (θ) - 0.30653 \dots cos (θ)

Additionner/Soustraire les nombres :

- 0.58725 \dots + 1 = 0.41274 \dots

= - 1.04 cos^{2} (θ) - 0.30653 \dots cos (θ) + 0.41274 \dots

= - 1.04 cos^{2} (θ) - 0.30653 \dots cos (θ) + 0.41274 \dots

0.41274 \dots - 0.30653 \dots cos (θ) - 1.04 cos^{2} (θ) = 0

Résoudre par substitution

0.41274 \dots - 0.30653 \dots cos (θ) - 1.04 cos^{2} (θ) = 0

Soit :

cos (θ) = u

0.41274 \dots - 0.30653 \dots u - 1.04 u^{2} = 0

0.41274 \dots - 0.30653 \dots u - 1.04 u^{2} = 0 : u = - \frac{0.30653 \dots + 1.81097 \dots}{2.08}, u = \frac{1.81097 \dots - 0.30653 \dots}{2.08}

0.41274 \dots - 0.30653 \dots u - 1.04 u^{2} = 0

Ecrire sous la forme standard

a x^{2} + b x + c = 0

- 1.04 u^{2} - 0.30653 \dots u + 0.41274 \dots = 0

Résoudre par la formule quadratique

- 1.04 u^{2} - 0.30653 \dots u + 0.41274 \dots = 0

Formule de l'équation quadratique:

Pour

a = - 1.04, b = - 0.30653 \dots, c = 0.41274 \dots

u_{1, 2} = \frac{- ( - 0.30653 \dots ) \pm ( - 0.30653 \dots ) ^{2} - 4 ( - 1.04 ) \cdot 0.41274 \dots}{2 ( - 1.04 )}

u_{1, 2} = \frac{- ( - 0.30653 \dots ) \pm ( - 0.30653 \dots ) ^{2} - 4 ( - 1.04 ) \cdot 0.41274 \dots}{2 ( - 1.04 )}

(- 0.30653 \dots)^{2} - 4 (- 1.04) \cdot 0.41274 \dots = 1.81097 \dots

(- 0.30653 \dots)^{2} - 4 (- 1.04) \cdot 0.41274 \dots

Appliquer la règle

- (- a) = a

= (- 0.30653 \dots)^{2} + 4 \cdot 1.04 \cdot 0.41274 \dots

Appliquer la règle de l'exposant:

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

n

pair

(- 0.30653 \dots)^{2} = 0.30653 \dots^{2}

= 0.30653 \dots^{2} + 4 \cdot 0.41274 \dots \cdot 1.04

Multiplier les nombres :

4 \cdot 1.04 \cdot 0.41274 \dots = 1.71701 \dots

= 0.30653 \dots^{2} + 1.71701 \dots

0.30653 \dots^{2} = 0.09396 \dots

= 0.09396 \dots + 1.71701 \dots

Additionner les nombres :

0.09396 \dots + 1.71701 \dots = 1.81097 \dots

= 1.81097 \dots

u_{1, 2} = \frac{- ( - 0.30653 \dots ) \pm 1.81097 \dots}{2 ( - 1.04 )}

Séparer les solutions

u_{1} = \frac{- ( - 0.30653 \dots ) + 1.81097 \dots}{2 ( - 1.04 )}, u_{2} = \frac{- ( - 0.30653 \dots ) - 1.81097 \dots}{2 ( - 1.04 )}

u = \frac{- ( - 0.30653 \dots ) + 1.81097 \dots}{2 ( - 1.04 )} : - \frac{0.30653 \dots + 1.81097 \dots}{2.08}

\frac{- ( - 0.30653 \dots ) + 1.81097 \dots}{2 ( - 1.04 )}

Retirer les parenthèses:

(- a) = - a, - (- a) = a

= \frac{0.30653 \dots + 1.81097 \dots}{- 2 \cdot 1.04}

Multiplier les nombres :

2 \cdot 1.04 = 2.08

= \frac{0.30653 \dots + 1.81097 \dots}{- 2.08}

Appliquer la règle des fractions:

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

= - \frac{0.30653 \dots + 1.81097 \dots}{2.08}

u = \frac{- ( - 0.30653 \dots ) - 1.81097 \dots}{2 ( - 1.04 )} : \frac{1.81097 \dots - 0.30653 \dots}{2.08}

\frac{- ( - 0.30653 \dots ) - 1.81097 \dots}{2 ( - 1.04 )}

Retirer les parenthèses:

(- a) = - a, - (- a) = a

= \frac{0.30653 \dots - 1.81097 \dots}{- 2 \cdot 1.04}

Multiplier les nombres :

2 \cdot 1.04 = 2.08

= \frac{0.30653 \dots - 1.81097 \dots}{- 2.08}

Appliquer la règle des fractions:

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

0.30653 \dots - 1.81097 \dots = - (1.81097 \dots - 0.30653 \dots)

= \frac{1.81097 \dots - 0.30653 \dots}{2.08}

Les solutions de l'équation de forme quadratique sont :

u = - \frac{0.30653 \dots + 1.81097 \dots}{2.08}, u = \frac{1.81097 \dots - 0.30653 \dots}{2.08}

Remplacer

u = cos (θ)

cos (θ) = - \frac{0.30653 \dots + 1.81097 \dots}{2.08}, cos (θ) = \frac{1.81097 \dots - 0.30653 \dots}{2.08}

cos (θ) = - \frac{0.30653 \dots + 1.81097 \dots}{2.08}, cos (θ) = \frac{1.81097 \dots - 0.30653 \dots}{2.08}

cos (θ) = - \frac{0.30653 \dots + 1.81097 \dots}{2.08} : θ = arccos (- \frac{0.30653 \dots + 1.81097 \dots}{2.08}) + 2 πn, θ = - arccos (- \frac{0.30653 \dots + 1.81097 \dots}{2.08}) + 2 πn

cos (θ) = - \frac{0.30653 \dots + 1.81097 \dots}{2.08}

Appliquer les propriétés trigonométriques inverses

cos (θ) = - \frac{0.30653 \dots + 1.81097 \dots}{2.08}

Solutions générales pour

cos (θ) = - \frac{0.30653 \dots + 1.81097 \dots}{2.08}

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

θ = arccos (- \frac{0.30653 \dots + 1.81097 \dots}{2.08}) + 2 πn, θ = - arccos (- \frac{0.30653 \dots + 1.81097 \dots}{2.08}) + 2 πn

θ = arccos (- \frac{0.30653 \dots + 1.81097 \dots}{2.08}) + 2 πn, θ = - arccos (- \frac{0.30653 \dots + 1.81097 \dots}{2.08}) + 2 πn

cos (θ) = \frac{1.81097 \dots - 0.30653 \dots}{2.08} : θ = arccos (\frac{1.81097 \dots - 0.30653 \dots}{2.08}) + 2 πn, θ = 2 π - arccos (\frac{1.81097 \dots - 0.30653 \dots}{2.08}) + 2 πn

cos (θ) = \frac{1.81097 \dots - 0.30653 \dots}{2.08}

Appliquer les propriétés trigonométriques inverses

cos (θ) = \frac{1.81097 \dots - 0.30653 \dots}{2.08}

Solutions générales pour

cos (θ) = \frac{1.81097 \dots - 0.30653 \dots}{2.08}

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

θ = arccos (\frac{1.81097 \dots - 0.30653 \dots}{2.08}) + 2 πn, θ = 2 π - arccos (\frac{1.81097 \dots - 0.30653 \dots}{2.08}) + 2 πn

θ = arccos (\frac{1.81097 \dots - 0.30653 \dots}{2.08}) + 2 πn, θ = 2 π - arccos (\frac{1.81097 \dots - 0.30653 \dots}{2.08}) + 2 πn

Combiner toutes les solutions

θ = arccos (- \frac{0.30653 \dots + 1.81097 \dots}{2.08}) + 2 πn, θ = - arccos (- \frac{0.30653 \dots + 1.81097 \dots}{2.08}) + 2 πn, θ = arccos (\frac{1.81097 \dots - 0.30653 \dots}{2.08}) + 2 πn, θ = 2 π - arccos (\frac{1.81097 \dots - 0.30653 \dots}{2.08}) + 2 πn

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

Vérifier des solutions en les intégrant dans

sin (θ) - 0.2 cos (θ) = \frac{7.51}{9.8}

Retirer celles qui ne répondent pas à l'équation.

Vérifier la solution

arccos (- \frac{0.30653 \dots + 1.81097 \dots}{2.08}) + 2 πn :

vrai

arccos (- \frac{0.30653 \dots + 1.81097 \dots}{2.08}) + 2 πn

Insérer

n = 1

arccos (- \frac{0.30653 \dots + 1.81097 \dots}{2.08}) + 2 π 1

Pour

sin (θ) - 0.2 cos (θ) = \frac{7.51}{9.8}

insérer

θ = arccos (- \frac{0.30653 \dots + 1.81097 \dots}{2.08}) + 2 π 1

sin (arccos (- \frac{0.30653 \dots + 1.81097 \dots}{2.08}) + 2 π 1) - 0.2 cos (arccos (- \frac{0.30653 \dots + 1.81097 \dots}{2.08}) + 2 π 1) = \frac{7.51}{9.8}

Redéfinir

0.76632 \dots = 0.76632 \dots

\Rightarrow vrai

Vérifier la solution

- arccos (- \frac{0.30653 \dots + 1.81097 \dots}{2.08}) + 2 πn :

Faux

- arccos (- \frac{0.30653 \dots + 1.81097 \dots}{2.08}) + 2 πn

Insérer

n = 1

- arccos (- \frac{0.30653 \dots + 1.81097 \dots}{2.08}) + 2 π 1

Pour

sin (θ) - 0.2 cos (θ) = \frac{7.51}{9.8}

insérer

θ = - arccos (- \frac{0.30653 \dots + 1.81097 \dots}{2.08}) + 2 π 1

sin (- arccos (- \frac{0.30653 \dots + 1.81097 \dots}{2.08}) + 2 π 1) - 0.2 cos (- arccos (- \frac{0.30653 \dots + 1.81097 \dots}{2.08}) + 2 π 1) = \frac{7.51}{9.8}

Redéfinir

- 0.44858 \dots = 0.76632 \dots

\Rightarrow Faux

Vérifier la solution

arccos (\frac{1.81097 \dots - 0.30653 \dots}{2.08}) + 2 πn :

vrai

arccos (\frac{1.81097 \dots - 0.30653 \dots}{2.08}) + 2 πn

Insérer

n = 1

arccos (\frac{1.81097 \dots - 0.30653 \dots}{2.08}) + 2 π 1

Pour

sin (θ) - 0.2 cos (θ) = \frac{7.51}{9.8}

insérer

θ = arccos (\frac{1.81097 \dots - 0.30653 \dots}{2.08}) + 2 π 1

sin (arccos (\frac{1.81097 \dots - 0.30653 \dots}{2.08}) + 2 π 1) - 0.2 cos (arccos (\frac{1.81097 \dots - 0.30653 \dots}{2.08}) + 2 π 1) = \frac{7.51}{9.8}

Redéfinir

0.76632 \dots = 0.76632 \dots

\Rightarrow vrai

Vérifier la solution

2 π - arccos (\frac{1.81097 \dots - 0.30653 \dots}{2.08}) + 2 πn :

Faux

2 π - arccos (\frac{1.81097 \dots - 0.30653 \dots}{2.08}) + 2 πn

Insérer

n = 1

2 π - arccos (\frac{1.81097 \dots - 0.30653 \dots}{2.08}) + 2 π 1

Pour

sin (θ) - 0.2 cos (θ) = \frac{7.51}{9.8}

insérer

θ = 2 π - arccos (\frac{1.81097 \dots - 0.30653 \dots}{2.08}) + 2 π 1

sin (2 π - arccos (\frac{1.81097 \dots - 0.30653 \dots}{2.08}) + 2 π 1) - 0.2 cos (2 π - arccos (\frac{1.81097 \dots - 0.30653 \dots}{2.08}) + 2 π 1) = \frac{7.51}{9.8}

Redéfinir

- 0.96617 \dots = 0.76632 \dots

\Rightarrow Faux

θ = arccos (- \frac{0.30653 \dots + 1.81097 \dots}{2.08}) + 2 πn, θ = arccos (\frac{1.81097 \dots - 0.30653 \dots}{2.08}) + 2 πn

Montrer les solutions sous la forme décimale

θ = 2.48873 \dots + 2 πn, θ = 1.04764 \dots + 2 πn

Exemples populaires

2sqrt(3)=-6tan(θ)

23 = - 6 tan (θ)

-3-5/2*cos(θ)=-3-3cos(θ)

- 3 - \frac{5}{2} \cdot cos (θ) = - 3 - 3 cos (θ)

tan(θ)=-6/5

tan (θ) = - \frac{6}{5}

3sec(2θ)=-sqrt(18)

3 sec (2 θ) = - 18

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

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