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2cos^2(x)-sqrt(3)cos(x)=0,0<x<180

Solución

2 cos^{2} (x) - 3 cos (x) = 0, 0^{\circ} < x < 18 0^{\circ}

Solución

x = 3 0^{\circ}, x = 9 0^{\circ}

radianes

x = \frac{π}{6}, x = \frac{π}{2}

Pasos de solución

2 cos^{2} (x) - 3 cos (x) = 0, 0^{\circ} < x < 18 0^{\circ}

Usando el método de sustitución

2 cos^{2} (x) - 3 cos (x) = 0

Sea:

cos (x) = u

2 u^{2} - 3 u = 0

2 u^{2} - 3 u = 0 : u = \frac{3}{2}, u = 0

2 u^{2} - 3 u = 0

Resolver con la fórmula general para ecuaciones de segundo grado:

2 u^{2} - 3 u = 0

Formula general para ecuaciones de segundo grado:

Para

a = 2, b = - 3, c = 0

u_{1, 2} = \frac{- ( - 3 ) \pm ( - 3 ) ^{2} - 4 \cdot 2 \cdot 0}{2 \cdot 2}

u_{1, 2} = \frac{- ( - 3 ) \pm ( - 3 ) ^{2} - 4 \cdot 2 \cdot 0}{2 \cdot 2}

(- 3)^{2} - 4 \cdot 2 \cdot 0 = 3

(- 3)^{2} - 4 \cdot 2 \cdot 0

(- 3)^{2} = 3

(- 3)^{2}

Aplicar las leyes de los exponentes:

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

n

es par

(- 3)^{2} = (3)^{2}

= (3)^{2}

Aplicar las leyes de los exponentes:

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

= (3^{\frac{1}{2}})^{2}

Aplicar las leyes de los exponentes:

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

= 3^{\frac{1}{2} \cdot 2}

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

\frac{1}{2} \cdot 2

Multiplicar fracciones:

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

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

Eliminar los terminos comunes:

2

= 1

= 3

4 \cdot 2 \cdot 0 = 0

4 \cdot 2 \cdot 0

Aplicar la regla

0 \cdot a = 0

= 0

= 3 - 0

Restar:

3 - 0 = 3

= 3

u_{1, 2} = \frac{- ( - 3 ) \pm 3}{2 \cdot 2}

Separar las soluciones

u_{1} = \frac{- ( - 3 ) + 3}{2 \cdot 2}, u_{2} = \frac{- ( - 3 ) - 3}{2 \cdot 2}

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

\frac{- ( - 3 ) + 3}{2 \cdot 2}

Aplicar la regla

- (- a) = a

= \frac{3 + 3}{2 \cdot 2}

Sumar elementos similares:

3 + 3 = 23

= \frac{2 3}{2 \cdot 2}

Multiplicar los numeros:

2 \cdot 2 = 4

= \frac{2 3}{4}

Eliminar los terminos comunes:

2

= \frac{3}{2}

u = \frac{- ( - 3 ) - 3}{2 \cdot 2} : 0

\frac{- ( - 3 ) - 3}{2 \cdot 2}

Aplicar la regla

- (- a) = a

= \frac{3 - 3}{2 \cdot 2}

Sumar elementos similares:

3 - 3 = 0

= \frac{0}{2 \cdot 2}

Multiplicar los numeros:

2 \cdot 2 = 4

= \frac{0}{4}

Aplicar la regla

\frac{0}{a} = 0, a \neq = 0

= 0

Las soluciones a la ecuación de segundo grado son:

u = \frac{3}{2}, u = 0

Sustituir en la ecuación

u = cos (x)

cos (x) = \frac{3}{2}, cos (x) = 0

cos (x) = \frac{3}{2}, cos (x) = 0

cos (x) = \frac{3}{2}, 0 < x < 18 0^{\circ} : x = 3 0^{\circ}

cos (x) = \frac{3}{2}, 0 < x < 18 0^{\circ}

Soluciones generales para

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

cos (x)

tabla de valores periódicos con

36 0^{\circ} n

intervalos:

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x 18 0^{\circ} 21 0^{\circ} 22 5^{\circ} 24 0^{\circ} 27 0^{\circ} 30 0^{\circ} 31 5^{\circ} 33 0^{\circ} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

x = 3 0^{\circ} + 36 0^{\circ} n, x = 33 0^{\circ} + 36 0^{\circ} n

x = 3 0^{\circ} + 36 0^{\circ} n, x = 33 0^{\circ} + 36 0^{\circ} n

Soluciones para el rango

0 < x < 18 0^{\circ}

x = 3 0^{\circ}

cos (x) = 0, 0 < x < 18 0^{\circ} : x = 9 0^{\circ}

cos (x) = 0, 0 < x < 18 0^{\circ}

Soluciones generales para

cos (x) = 0

cos (x)

tabla de valores periódicos con

36 0^{\circ} n

intervalos:

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x 18 0^{\circ} 21 0^{\circ} 22 5^{\circ} 24 0^{\circ} 27 0^{\circ} 30 0^{\circ} 31 5^{\circ} 33 0^{\circ} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

x = 9 0^{\circ} + 36 0^{\circ} n, x = 27 0^{\circ} + 36 0^{\circ} n

x = 9 0^{\circ} + 36 0^{\circ} n, x = 27 0^{\circ} + 36 0^{\circ} n

Soluciones para el rango

0 < x < 18 0^{\circ}

x = 9 0^{\circ}

Combinar toda las soluciones

x = 3 0^{\circ}, x = 9 0^{\circ}

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