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Popular Trigonometría >

6cos^2(x/2)+cos(x)+1=0

Solución

6 cos^{2} (\frac{x}{2}) + cos (x) + 1 = 0

Solución

x = π + 2 πn

Grados

x = 18 0^{\circ} + 36 0^{\circ} n

Pasos de solución

6 cos^{2} (\frac{x}{2}) + cos (x) + 1 = 0

Re-escribir usando identidades trigonométricas

1 + cos (x) + 6 cos^{2} (\frac{x}{2})

Usar la siguiente identidad:

cos^{2} (θ) = \frac{1 + cos ( 2 θ )}{2}

Utilizar la identidad trigonométrica del ángulo doble

cos (2 θ) = 2 cos^{2} (θ) - 1

Intercambiar lados

2 cos^{2} (θ) - 1 = cos (2 θ)

Sumar

1

a ambos lados

2 sin^{2} (θ) = 1 + cos (2 θ)

Dividir ambos lados entre

2

cos^{2} (θ) = \frac{1 + cos ( 2 θ )}{2}

= 1 + cos (x) + 6 \cdot \frac{1 + cos ( 2 \cdot \frac{x}{2} )}{2}

Simplificar

1 + cos (x) + 6 \cdot \frac{1 + cos ( 2 \cdot \frac{x}{2} )}{2} : 4 cos (x) + 4

1 + cos (x) + 6 \cdot \frac{1 + cos ( 2 \cdot \frac{x}{2} )}{2}

6 \cdot \frac{1 + cos ( 2 \cdot \frac{x}{2} )}{2} = 3 (cos (x) + 1)

6 \cdot \frac{1 + cos ( 2 \cdot \frac{x}{2} )}{2}

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

\frac{1 + cos ( 2 \cdot \frac{x}{2} )}{2}

Multiplicar

2 \cdot \frac{x}{2} : x

2 \cdot \frac{x}{2}

Multiplicar fracciones:

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

= \frac{x \cdot 2}{2}

Eliminar los terminos comunes:

2

= x

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

= 6 \cdot \frac{cos ( x ) + 1}{2}

Multiplicar fracciones:

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

= \frac{( 1 + cos ( x ) ) \cdot 6}{2}

Dividir:

\frac{6}{2} = 3

= 3 (cos (x) + 1)

= 1 + cos (x) + 3 (cos (x) + 1)

Expandir

3 (cos (x) + 1) : 3 cos (x) + 3

3 (cos (x) + 1)

Poner los parentesis utilizando:

a (b + c) = ab + a c

a = 3, b = cos (x), c = 1

= 3 cos (x) + 3 \cdot 1

Multiplicar los numeros:

3 \cdot 1 = 3

= 3 cos (x) + 3

= 1 + cos (x) + 3 cos (x) + 3

Simplificar

1 + cos (x) + 3 cos (x) + 3 : 4 cos (x) + 4

1 + cos (x) + 3 cos (x) + 3

Sumar elementos similares:

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

= 1 + 4 cos (x) + 3

Agrupar términos semejantes

= 4 cos (x) + 1 + 3

Sumar:

1 + 3 = 4

= 4 cos (x) + 4

= 4 cos (x) + 4

= 4 cos (x) + 4

4 + 4 cos (x) = 0

Desplace

4

a la derecha

4 + 4 cos (x) = 0

Restar

4

de ambos lados

4 + 4 cos (x) - 4 = 0 - 4

Simplificar

4 cos (x) = - 4

4 cos (x) = - 4

Dividir ambos lados entre

4

4 cos (x) = - 4

Dividir ambos lados entre

4

\frac{4 cos ( x )}{4} = \frac{- 4}{4}

Simplificar

cos (x) = - 1

cos (x) = - 1

Soluciones generales para

cos (x) = - 1

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 = π + 2 πn

x = π + 2 πn

Gráfica

Ejemplos populares