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Popular >

cos(x)=(2tan(x))/(1+tan^2(x))

Solución

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

Solución

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

Grados

x = 3 0^{\circ} + 36 0^{\circ} n, x = 15 0^{\circ} + 36 0^{\circ} n

Pasos de solución

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

Restar

\frac{2 tan ( x )}{1 + tan ^{2} ( x )}

de ambos lados

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

Simplificar

cos (x) - \frac{2 tan ( x )}{1 + tan ^{2} ( x )} : \frac{cos ( x ) ( 1 + tan ^{2} ( x ) ) - 2 tan ( x )}{1 + tan ^{2} ( x )}

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

Convertir a fracción:

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

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

Ya que los denominadores son iguales, combinar las fracciones:

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

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

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

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

cos (x) (1 + tan^{2} (x)) - 2 tan (x) = 0

Expresar con seno, coseno

(1 + tan^{2} (x)) cos (x) - 2 tan (x)

Utilizar la identidad trigonométrica básica:

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

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

Simplificar

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

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

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

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

Aplicar las leyes de los exponentes:

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

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

Simplificar

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

en una fracción:

\frac{cos ^{2} ( x ) + sin ^{2} ( x )}{cos ^{2} ( x )}

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

Convertir a fracción:

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

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

Multiplicar:

1 \cdot cos^{2} (x) = cos^{2} (x)

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

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

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

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

Multiplicar fracciones:

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

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

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

Multiplicar

\frac{cos ^{2} ( x ) + sin ^{2} ( x )}{cos ^{2} ( x )} cos (x) : \frac{cos ^{2} ( x ) + sin ^{2} ( x )}{cos ( x )}

\frac{cos ^{2} ( x ) + sin ^{2} ( x )}{cos ^{2} ( x )} cos (x)

Multiplicar fracciones:

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

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

Eliminar los terminos comunes:

cos (x)

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

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

Ya que los denominadores son iguales, combinar las fracciones:

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

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

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

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

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

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

Re-escribir usando identidades trigonométricas

cos^{2} (x) + sin^{2} (x) - 2 sin (x)

Utilizar la identidad pitagórica:

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

= - 2 sin (x) + 1

- 2 sin (x) + 1 = 0

Mover

1

al lado derecho

- 2 sin (x) + 1 = 0

Restar

1

de ambos lados

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

Simplificar

- 2 sin (x) = - 1

- 2 sin (x) = - 1

Dividir ambos lados entre

- 2

- 2 sin (x) = - 1

Dividir ambos lados entre

- 2

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

Simplificar

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

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

Soluciones generales para

sin (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}

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

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

Ejemplos populares

(1+cos(4x))sin(2x)=cos^2(2x)

(1 + cos (4 x)) sin (2 x) = cos^{2} (2 x)

cos(2x)-cos(6x)-sin(4x)=0

cos (2 x) - cos (6 x) - sin (4 x) = 0

cot(θ)= 15/8

cot (θ) = \frac{15}{8}

sqrt(2)sin(2x)=sin(4x)

2 sin (2 x) = sin (4 x)

solvefor u,sin(u)=x resolver para

u, sin (u) = x