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Popular >

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

Solución

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

Solución

x = \frac{π}{4} + πn

Grados

x = 4 5^{\circ} + 18 0^{\circ} n

Pasos de solución

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

Restar

sec (x)

de ambos lados

\frac{sin ( x )}{1 - cos ^{2} ( x )} - sec (x) = 0

Simplificar

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

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

Convertir a fracción:

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

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

Ya que los denominadores son iguales, combinar las fracciones:

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

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

\frac{sin ( x ) - sec ( x ) ( 1 - cos ^{2} ( x ) )}{1 - cos ^{2} ( x )} = 0

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

sin (x) - sec (x) (1 - cos^{2} (x)) = 0

Re-escribir usando identidades trigonométricas

sin (x) - (1 - cos^{2} (x)) sec (x)

Utilizar la identidad pitagórica:

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

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

= sin (x) - sec (x) sin^{2} (x)

sin (x) - sec (x) sin^{2} (x) = 0

Factorizar

sin (x) - sec (x) sin^{2} (x) : sin (x) (1 - sec (x) sin (x))

sin (x) - sec (x) sin^{2} (x)

Aplicar las leyes de los exponentes:

a^{b + c} = a^{b} a^{c}

sec (x) sin^{2} (x) = sin (x) sin (x)

= sin (x) - sin (x) sin (x)

Factorizar el termino común

sin (x)

= sin (x) (1 - sec (x) sin (x))

sin (x) (1 - sec (x) sin (x)) = 0

Resolver cada parte por separado

sin (x) = 0 or 1 - sec (x) sin (x) = 0

sin (x) = 0 : x = 2 πn, x = π + 2 πn

sin (x) = 0

Soluciones generales para

sin (x) = 0

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}

x = 0 + 2 πn, x = π + 2 πn

x = 0 + 2 πn, x = π + 2 πn

Resolver

x = 0 + 2 πn : x = 2 πn

x = 0 + 2 πn

0 + 2 πn = 2 πn

x = 2 πn

x = 2 πn, x = π + 2 πn

1 - sec (x) sin (x) = 0 : x = \frac{π}{4} + πn

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

Re-escribir usando identidades trigonométricas

1 - sec (x) sin (x)

sec (x) sin (x) = tan (x)

sec (x) sin (x)

Expresar con seno, coseno

sec (x) sin (x)

Utilizar la identidad trigonométrica básica:

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

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

Simplificar

\frac{1}{cos ( x )} sin (x) : \frac{sin ( x )}{cos ( x )}

\frac{1}{cos ( x )} sin (x)

Multiplicar fracciones:

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

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

Multiplicar:

1 \cdot sin (x) = sin (x)

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

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

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

Utilizar la identidad trigonométrica básica:

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

= tan (x)

= 1 - tan (x)

1 - tan (x) = 0

Mover

1

al lado derecho

1 - tan (x) = 0

Restar

1

de ambos lados

1 - tan (x) - 1 = 0 - 1

Simplificar

- tan (x) = - 1

- tan (x) = - 1

Dividir ambos lados entre

- 1

- tan (x) = - 1

Dividir ambos lados entre

- 1

\frac{- tan ( x )}{- 1} = \frac{- 1}{- 1}

Simplificar

tan (x) = 1

tan (x) = 1

Soluciones generales para

tan (x) = 1

tan (x)

tabla de valores periódicos con

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} tan (x) 0 \frac{3}{3} 13 \pm \infty - 3 - 1 - \frac{3}{3}

x = \frac{π}{4} + πn

x = \frac{π}{4} + πn

Combinar toda las soluciones

x = 2 πn, x = π + 2 πn, x = \frac{π}{4} + πn

Siendo que la ecuación esta indefinida para:

2 πn, π + 2 πn

x = \frac{π}{4} + πn

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cos(2x)-2sin(x)=1,0<x<= 360

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cos(api)-1=0

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3tan(2θ)-sqrt(3)=0

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cos (θ) = \frac{4 k - 1}{3}

5sin(x)-4=0

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