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Popular >

cos(105)cos(x)=-1/4

Solución

cos (10 5^{\circ}) cos (x^{\circ}) = - \frac{1}{4}

Solución

Sin soluci \overset{o}{ˊ} n para x \in R

Pasos de solución

cos (10 5^{\circ}) cos (x^{\circ}) = - \frac{1}{4}

cos (10 5^{\circ}) = \frac{2 ( 1 - 3 )}{4}

cos (10 5^{\circ})

Re-escribir usando identidades trigonométricas:

cos (6 0^{\circ}) cos (4 5^{\circ}) - sin (6 0^{\circ}) sin (4 5^{\circ})

cos (10 5^{\circ})

Escribir

cos (10 5^{\circ})

como

cos (6 0^{\circ} + 4 5^{\circ})

= cos (6 0^{\circ} + 4 5^{\circ})

Utilizar la identidad de suma de ángulos:

cos (s + t) = cos (s) cos (t) - sin (s) sin (t)

= cos (6 0^{\circ}) cos (4 5^{\circ}) - sin (6 0^{\circ}) sin (4 5^{\circ})

= cos (6 0^{\circ}) cos (4 5^{\circ}) - sin (6 0^{\circ}) sin (4 5^{\circ})

Utilizar la siguiente identidad trivial:

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

cos (6 0^{\circ})

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}

= \frac{1}{2}

Utilizar la siguiente identidad trivial:

cos (4 5^{\circ}) = \frac{2}{2}

cos (4 5^{\circ})

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}

= \frac{2}{2}

Utilizar la siguiente identidad trivial:

sin (6 0^{\circ}) = \frac{3}{2}

sin (6 0^{\circ})

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} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{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} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

= \frac{3}{2}

Utilizar la siguiente identidad trivial:

sin (4 5^{\circ}) = \frac{2}{2}

sin (4 5^{\circ})

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} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{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} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

= \frac{2}{2}

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

Simplificar

\frac{1}{2} \cdot \frac{2}{2} - \frac{3}{2} \cdot \frac{2}{2} : \frac{2 ( 1 - 3 )}{4}

\frac{1}{2} \cdot \frac{2}{2} - \frac{3}{2} \cdot \frac{2}{2}

Factorizar el termino común

\frac{2}{2}

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

\frac{1}{2} - \frac{3}{2} = \frac{1 - 3}{2}

\frac{1}{2} - \frac{3}{2}

Aplicar la regla

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

= \frac{1 - 3}{2}

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

Multiplicar fracciones:

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

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

Multiplicar los numeros:

2 \cdot 2 = 4

= \frac{2 ( 1 - 3 )}{4}

= \frac{2 ( 1 - 3 )}{4}

\frac{2 ( 1 - 3 )}{4} cos (x^{\circ}) = - \frac{1}{4}

Multiplicar ambos lados por

4

\frac{2 ( 1 - 3 )}{4} cos (x^{\circ}) = - \frac{1}{4}

Multiplicar ambos lados por

4

4 \cdot \frac{2 ( 1 - 3 )}{4} cos (x^{\circ}) = 4 (- \frac{1}{4})

Simplificar

4 \cdot \frac{2 ( 1 - 3 )}{4} cos (x^{\circ}) = 4 (- \frac{1}{4})

Simplificar

4 \cdot \frac{2 ( 1 - 3 )}{4} cos (x^{\circ}) : 2 (1 - 3) cos (x^{\circ})

4 \cdot \frac{2 ( 1 - 3 )}{4} cos (x^{\circ})

Multiplicar fracciones:

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

= \frac{2 ( 1 - 3 ) \cdot 4 cos ( x ^{\circ} )}{4}

Eliminar los terminos comunes:

4

= 2 (1 - 3) cos (x^{\circ})

Simplificar

4 (- \frac{1}{4}) : - 1

4 (- \frac{1}{4})

Quitar los parentesis:

(- a) = - a

= - 4 \cdot \frac{1}{4}

Multiplicar fracciones:

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

= - \frac{1 \cdot 4}{4}

Eliminar los terminos comunes:

4

= - 1

2 (1 - 3) cos (x^{\circ}) = - 1

2 (1 - 3) cos (x^{\circ}) = - 1

2 (1 - 3) cos (x^{\circ}) = - 1

Dividir ambos lados entre

2 (1 - 3)

2 (1 - 3) cos (x^{\circ}) = - 1

Dividir ambos lados entre

2 (1 - 3)

\frac{2 ( 1 - 3 ) cos ( x ^{\circ} )}{2 ( 1 - 3 )} = \frac{- 1}{2 ( 1 - 3 )}

Simplificar

\frac{2 ( 1 - 3 ) cos ( x ^{\circ} )}{2 ( 1 - 3 )} = \frac{- 1}{2 ( 1 - 3 )}

Simplificar

\frac{2 ( 1 - 3 ) cos ( x ^{\circ} )}{2 ( 1 - 3 )} : cos (x^{\circ})

\frac{2 ( 1 - 3 ) cos ( x ^{\circ} )}{2 ( 1 - 3 )}

Eliminar los terminos comunes:

2

= \frac{( 1 - 3 ) cos ( x ^{\circ} )}{1 - 3}

Eliminar los terminos comunes:

1 - 3

= cos (x^{\circ})

Simplificar

\frac{- 1}{2 ( 1 - 3 )} : \frac{2 ( 1 + 3 )}{4}

\frac{- 1}{2 ( 1 - 3 )}

Aplicar las propiedades de las fracciones:

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

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

Racionalizar

- \frac{1}{2 ( 1 - 3 )} : \frac{2 ( 1 + 3 )}{4}

- \frac{1}{2 ( 1 - 3 )}

Multiplicar por el conjugado

\frac{2}{2}

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

1 \cdot 2 = 2

22 = 2

22

Aplicar las leyes de los exponentes:

a a = a

22 = 2

= 2

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

Multiplicar por el conjugado

\frac{1 + 3}{1 + 3}

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

(1 - 3) \cdot 2 (1 + 3) = - 4

(1 - 3) \cdot 2 (1 + 3)

= 2 (1 - 3) (1 + 3)

Expandir

(1 - 3) (1 + 3) : - 2

(1 - 3) (1 + 3)

Aplicar la siguiente regla para binomios al cuadrado:

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

a = 1, b = 3

= 1^{2} - (3)^{2}

Simplificar

1^{2} - (3)^{2} : - 2

1^{2} - (3)^{2}

Aplicar la regla

1^{a} = 1

1^{2} = 1

= 1 - (3)^{2}

(3)^{2} = 3

(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

= 1 - 3

Restar:

1 - 3 = - 2

= - 2

= - 2

= 2 (- 2)

Expandir

2 (- 2) : - 4

2 (- 2)

Aplicar la siguiente regla de productos notables

= 2 (- 2)

Aplicar las reglas de los signos

+ (- a) = - a

= - 2 \cdot 2

Multiplicar los numeros:

2 \cdot 2 = 4

= - 4

= - 4

= - \frac{2 ( 1 + 3 )}{- 4}

Aplicar las propiedades de las fracciones:

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

= - (- \frac{2 ( 1 + 3 )}{4})

Aplicar la regla

- (- a) = a

= \frac{2 ( 1 + 3 )}{4}

= \frac{2 ( 1 + 3 )}{4}

cos (x^{\circ}) = \frac{2 ( 1 + 3 )}{4}

cos (x^{\circ}) = \frac{2 ( 1 + 3 )}{4}

cos (x^{\circ}) = \frac{2 ( 1 + 3 )}{4}

Aplicar propiedades trigonométricas inversas

cos (x^{\circ}) = \frac{2 ( 1 + 3 )}{4}

Soluciones generales para

cos (x^{\circ}) = \frac{2 ( 1 + 3 )}{4}

cos (x) = a \Rightarrow x = arccos (a) + 36 0^{\circ} n, x = 36 0^{\circ} - arccos (a) + 36 0^{\circ} n

x^{\circ} = arccos (\frac{2 ( 1 + 3 )}{4}) + 36 0^{\circ} n, x^{\circ} = 36 0^{\circ} - arccos (\frac{2 ( 1 + 3 )}{4}) + 36 0^{\circ} n

x^{\circ} = arccos (\frac{2 ( 1 + 3 )}{4}) + 36 0^{\circ} n, x^{\circ} = 36 0^{\circ} - arccos (\frac{2 ( 1 + 3 )}{4}) + 36 0^{\circ} n

Sin soluci \overset{o}{ˊ} n para x \in R

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sin(2x)-3cos(x)=0

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