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Popular >

4sin(x)-sec(x)=0

Solución

4 sin (x) - sec (x) = 0

Solución

x = \frac{π}{12} + πn, x = \frac{5 π}{12} + πn

Grados

x = 1 5^{\circ} + 18 0^{\circ} n, x = 7 5^{\circ} + 18 0^{\circ} n

Pasos de solución

4 sin (x) - sec (x) = 0

Expresar con seno, coseno

- sec (x) + 4 sin (x)

Utilizar la identidad trigonométrica básica:

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

= - \frac{1}{cos ( x )} + 4 sin (x)

Simplificar

- \frac{1}{cos ( x )} + 4 sin (x) : \frac{- 1 + 4 sin ( x ) cos ( x )}{cos ( x )}

- \frac{1}{cos ( x )} + 4 sin (x)

Convertir a fracción:

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

= - \frac{1}{cos ( x )} + \frac{4 sin ( x ) cos ( x )}{cos ( x )}

Ya que los denominadores son iguales, combinar las fracciones:

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

= \frac{- 1 + 4 sin ( x ) cos ( x )}{cos ( x )}

= \frac{- 1 + 4 sin ( x ) cos ( x )}{cos ( x )}

\frac{- 1 + 4 cos ( x ) sin ( x )}{cos ( x )} = 0

\frac{f ( x )}{g ( x )} = 0 \Rightarrow f (x) = 0

- 1 + 4 cos (x) sin (x) = 0

Re-escribir usando identidades trigonométricas

- 1 + 4 cos (x) sin (x)

Utilizar la identidad trigonométrica del ángulo doble:

2 sin (x) cos (x) = sin (2 x)

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

= - 1 + 4 \cdot \frac{sin ( 2 x )}{2}

- 1 + 4 \cdot \frac{sin ( 2 x )}{2} = 0

4 \cdot \frac{sin ( 2 x )}{2} = 2 sin (2 x)

4 \cdot \frac{sin ( 2 x )}{2}

Multiplicar fracciones:

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

= \frac{sin ( 2 x ) \cdot 4}{2}

Dividir:

\frac{4}{2} = 2

= 2 sin (2 x)

- 1 + 2 sin (2 x) = 0

Mover

1

al lado derecho

- 1 + 2 sin (2 x) = 0

Sumar

1

a ambos lados

- 1 + 2 sin (2 x) + 1 = 0 + 1

Simplificar

2 sin (2 x) = 1

2 sin (2 x) = 1

Dividir ambos lados entre

2

2 sin (2 x) = 1

Dividir ambos lados entre

2

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

Simplificar

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

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

Soluciones generales para

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

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

2 x = \frac{π}{6} + 2 πn, 2 x = \frac{5 π}{6} + 2 πn

2 x = \frac{π}{6} + 2 πn, 2 x = \frac{5 π}{6} + 2 πn

Resolver

2 x = \frac{π}{6} + 2 πn : x = \frac{π}{12} + πn

2 x = \frac{π}{6} + 2 πn

Dividir ambos lados entre

2

2 x = \frac{π}{6} + 2 πn

Dividir ambos lados entre

2

\frac{2 x}{2} = \frac{\frac{π}{6}}{2} + \frac{2 πn}{2}

Simplificar

\frac{2 x}{2} = \frac{\frac{π}{6}}{2} + \frac{2 πn}{2}

Simplificar

\frac{2 x}{2} : x

\frac{2 x}{2}

Dividir:

\frac{2}{2} = 1

= x

Simplificar

\frac{\frac{π}{6}}{2} + \frac{2 πn}{2} : \frac{π}{12} + πn

\frac{\frac{π}{6}}{2} + \frac{2 πn}{2}

\frac{\frac{π}{6}}{2} = \frac{π}{12}

\frac{\frac{π}{6}}{2}

Aplicar las propiedades de las fracciones:

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

= \frac{π}{6 \cdot 2}

Multiplicar los numeros:

6 \cdot 2 = 12

= \frac{π}{12}

\frac{2 πn}{2} = πn

\frac{2 πn}{2}

Dividir:

\frac{2}{2} = 1

= πn

= \frac{π}{12} + πn

x = \frac{π}{12} + πn

x = \frac{π}{12} + πn

x = \frac{π}{12} + πn

Resolver

2 x = \frac{5 π}{6} + 2 πn : x = \frac{5 π}{12} + πn

2 x = \frac{5 π}{6} + 2 πn

Dividir ambos lados entre

2

2 x = \frac{5 π}{6} + 2 πn

Dividir ambos lados entre

2

\frac{2 x}{2} = \frac{\frac{5 π}{6}}{2} + \frac{2 πn}{2}

Simplificar

\frac{2 x}{2} = \frac{\frac{5 π}{6}}{2} + \frac{2 πn}{2}

Simplificar

\frac{2 x}{2} : x

\frac{2 x}{2}

Dividir:

\frac{2}{2} = 1

= x

Simplificar

\frac{\frac{5 π}{6}}{2} + \frac{2 πn}{2} : \frac{5 π}{12} + πn

\frac{\frac{5 π}{6}}{2} + \frac{2 πn}{2}

\frac{\frac{5 π}{6}}{2} = \frac{5 π}{12}

\frac{\frac{5 π}{6}}{2}

Aplicar las propiedades de las fracciones:

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

= \frac{5 π}{6 \cdot 2}

Multiplicar los numeros:

6 \cdot 2 = 12

= \frac{5 π}{12}

\frac{2 πn}{2} = πn

\frac{2 πn}{2}

Dividir:

\frac{2}{2} = 1

= πn

= \frac{5 π}{12} + πn

x = \frac{5 π}{12} + πn

x = \frac{5 π}{12} + πn

x = \frac{5 π}{12} + πn

x = \frac{π}{12} + πn, x = \frac{5 π}{12} + πn

Ejemplos populares

sin^2(x)=(1-cos^2(x))/2

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

cos(x)=-1/2 ,0<x<2pi

cos (x) = - \frac{1}{2}, 0 < x < 2 π

cot(θ)=cos(θ)

cot (θ) = cos (θ)

3sin^2(θ)-cos(θ)+1=0

3 sin^{2} (θ) - cos (θ) + 1 = 0

2-2cos^2(x)=0

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