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Popular >

6cos^2(x)-7cos(x)+2=0

Solución

6 cos^{2} (x) - 7 cos (x) + 2 = 0

Solución

x = 0.84106 \dots + 2 πn, x = 2 π - 0.84106 \dots + 2 πn, x = \frac{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

Grados

x = 48.18968 \dots^{\circ} + 36 0^{\circ} n, x = 311.81031 \dots^{\circ} + 36 0^{\circ} n, x = 6 0^{\circ} + 36 0^{\circ} n, x = 30 0^{\circ} + 36 0^{\circ} n

Pasos de solución

6 cos^{2} (x) - 7 cos (x) + 2 = 0

Usando el método de sustitución

6 cos^{2} (x) - 7 cos (x) + 2 = 0

Sea:

cos (x) = u

6 u^{2} - 7 u + 2 = 0

6 u^{2} - 7 u + 2 = 0 : u = \frac{2}{3}, u = \frac{1}{2}

6 u^{2} - 7 u + 2 = 0

Resolver con la fórmula general para ecuaciones de segundo grado:

6 u^{2} - 7 u + 2 = 0

Formula general para ecuaciones de segundo grado:

Para

a = 6, b = - 7, c = 2

u_{1, 2} = \frac{- ( - 7 ) \pm ( - 7 ) ^{2} - 4 \cdot 6 \cdot 2}{2 \cdot 6}

u_{1, 2} = \frac{- ( - 7 ) \pm ( - 7 ) ^{2} - 4 \cdot 6 \cdot 2}{2 \cdot 6}

(- 7)^{2} - 4 \cdot 6 \cdot 2 = 1

(- 7)^{2} - 4 \cdot 6 \cdot 2

Aplicar las leyes de los exponentes:

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

n

es par

(- 7)^{2} = 7^{2}

= 7^{2} - 4 \cdot 6 \cdot 2

Multiplicar los numeros:

4 \cdot 6 \cdot 2 = 48

= 7^{2} - 48

7^{2} = 49

= 49 - 48

Restar:

49 - 48 = 1

= 1

Aplicar la regla

1 = 1

= 1

u_{1, 2} = \frac{- ( - 7 ) \pm 1}{2 \cdot 6}

Separar las soluciones

u_{1} = \frac{- ( - 7 ) + 1}{2 \cdot 6}, u_{2} = \frac{- ( - 7 ) - 1}{2 \cdot 6}

u = \frac{- ( - 7 ) + 1}{2 \cdot 6} : \frac{2}{3}

\frac{- ( - 7 ) + 1}{2 \cdot 6}

Aplicar la regla

- (- a) = a

= \frac{7 + 1}{2 \cdot 6}

Sumar:

7 + 1 = 8

= \frac{8}{2 \cdot 6}

Multiplicar los numeros:

2 \cdot 6 = 12

= \frac{8}{12}

Eliminar los terminos comunes:

4

= \frac{2}{3}

u = \frac{- ( - 7 ) - 1}{2 \cdot 6} : \frac{1}{2}

\frac{- ( - 7 ) - 1}{2 \cdot 6}

Aplicar la regla

- (- a) = a

= \frac{7 - 1}{2 \cdot 6}

Restar:

7 - 1 = 6

= \frac{6}{2 \cdot 6}

Multiplicar los numeros:

2 \cdot 6 = 12

= \frac{6}{12}

Eliminar los terminos comunes:

6

= \frac{1}{2}

Las soluciones a la ecuación de segundo grado son:

u = \frac{2}{3}, u = \frac{1}{2}

Sustituir en la ecuación

u = cos (x)

cos (x) = \frac{2}{3}, cos (x) = \frac{1}{2}

cos (x) = \frac{2}{3}, cos (x) = \frac{1}{2}

cos (x) = \frac{2}{3} : x = arccos (\frac{2}{3}) + 2 πn, x = 2 π - arccos (\frac{2}{3}) + 2 πn

cos (x) = \frac{2}{3}

Aplicar propiedades trigonométricas inversas

cos (x) = \frac{2}{3}

Soluciones generales para

cos (x) = \frac{2}{3}

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

x = arccos (\frac{2}{3}) + 2 πn, x = 2 π - arccos (\frac{2}{3}) + 2 πn

x = arccos (\frac{2}{3}) + 2 πn, x = 2 π - arccos (\frac{2}{3}) + 2 πn

cos (x) = \frac{1}{2} : x = \frac{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

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

Soluciones generales para

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

cos (x)

tabla de valores periódicos con

2 πn

intervalos:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

x = \frac{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

x = \frac{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

Combinar toda las soluciones

x = arccos (\frac{2}{3}) + 2 πn, x = 2 π - arccos (\frac{2}{3}) + 2 πn, x = \frac{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

Mostrar soluciones en forma decimal

x = 0.84106 \dots + 2 πn, x = 2 π - 0.84106 \dots + 2 πn, x = \frac{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

Ejemplos populares

csc(θ)=(sqrt(20))/4

csc (θ) = \frac{20}{4}

sin(x+10)=cos(x+20)

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

c^2=b^2+a^2-2(b)(a)cos(θ)

c^{2} = b^{2} + a^{2} - 2 (b) (a) cos (θ)

5cos(θ)=2+cos(θ)

5 cos (θ) = 2 + cos (θ)

cos(x+pi)=1

cos (x + π) = 1