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tan(x+pi)-cos(x+pi/2)=0

Solución

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

Solución

x = 2 πn, x = π + 2 πn

Grados

x = 0^{\circ} + 36 0^{\circ} n, x = 18 0^{\circ} + 36 0^{\circ} n

Pasos de solución

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

Re-escribir usando identidades trigonométricas

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

Re-escribir usando identidades trigonométricas

cos (x + \frac{π}{2})

Utilizar la identidad de suma de ángulos:

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

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

Simplificar

cos (x) cos (\frac{π}{2}) - sin (x) sin (\frac{π}{2}) : - sin (x)

cos (x) cos (\frac{π}{2}) - sin (x) sin (\frac{π}{2})

cos (x) cos (\frac{π}{2}) = 0

cos (x) cos (\frac{π}{2})

Simplificar

cos (\frac{π}{2}) : 0

cos (\frac{π}{2})

Utilizar la siguiente identidad trivial:

cos (\frac{π}{2}) = 0

cos (x)

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

= 0

= 0 \cdot cos (x)

Aplicar la regla

0 \cdot a = 0

= 0

sin (x) sin (\frac{π}{2}) = sin (x)

sin (x) sin (\frac{π}{2})

Simplificar

sin (\frac{π}{2}) : 1

sin (\frac{π}{2})

Utilizar la siguiente identidad trivial:

sin (\frac{π}{2}) = 1

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}

= 1

= 1 \cdot sin (x)

Multiplicar:

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

= sin (x)

= 0 - sin (x)

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

= - sin (x)

= - sin (x)

Utilizar la identidad trigonométrica básica:

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

= \frac{sin ( x + π )}{cos ( x + π )}

Utilizar la identidad de suma de ángulos:

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

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

Utilizar la identidad de suma de ángulos:

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

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

Simplificar

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

\frac{sin ( x ) cos ( π ) + cos ( x ) sin ( π )}{cos ( x ) cos ( π ) - sin ( x ) sin ( π )}

sin (x) cos (π) + cos (x) sin (π) = - sin (x)

sin (x) cos (π) + cos (x) sin (π)

sin (x) cos (π) = - sin (x)

sin (x) cos (π)

Simplificar

cos (π) : - 1

cos (π)

Utilizar la siguiente identidad trivial:

cos (π) = (- 1)

cos (x)

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

= - 1

= (- 1) sin (x)

Simplificar

= - sin (x)

= - sin (x) + sin (π) cos (x)

cos (x) sin (π) = 0

cos (x) sin (π)

Simplificar

sin (π) : 0

sin (π)

Utilizar la siguiente identidad trivial:

sin (π) = 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}

= 0

= 0 \cdot cos (x)

Aplicar la regla

0 \cdot a = 0

= 0

= - sin (x) + 0

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

= - sin (x)

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

cos (x) cos (π) - sin (x) sin (π) = - cos (x)

cos (x) cos (π) - sin (x) sin (π)

cos (x) cos (π) = - cos (x)

cos (x) cos (π)

Simplificar

cos (π) : - 1

cos (π)

Utilizar la siguiente identidad trivial:

cos (π) = (- 1)

cos (x)

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

= - 1

= (- 1) cos (x)

Simplificar

= - cos (x)

= - cos (x) - sin (π) sin (x)

sin (x) sin (π) = 0

sin (x) sin (π)

Simplificar

sin (π) : 0

sin (π)

Utilizar la siguiente identidad trivial:

sin (π) = 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}

= 0

= 0 \cdot sin (x)

Aplicar la regla

0 \cdot a = 0

= 0

= - cos (x) - 0

- cos (x) - 0 = - cos (x)

= - cos (x)

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

Aplicar las propiedades de las fracciones:

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

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

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

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

Simplificar

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

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

Aplicar la regla

- (- a) = a

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

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

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

Simplificar

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

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

Convertir a fracción:

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

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

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

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

sin (x) + sin (x) cos (x) = 0

Factorizar

sin (x) + sin (x) cos (x) : sin (x) (cos (x) + 1)

sin (x) + sin (x) cos (x)

Factorizar el termino común

sin (x)

= sin (x) (1 + cos (x))

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

Resolver cada parte por separado

sin (x) = 0 or cos (x) + 1 = 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

cos (x) + 1 = 0 : x = π + 2 πn

cos (x) + 1 = 0

Mover

1

al lado derecho

cos (x) + 1 = 0

Restar

1

de ambos lados

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

Simplificar

cos (x) = - 1

cos (x) = - 1

Soluciones generales para

cos (x) = - 1

cos (x)

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

x = π + 2 πn

x = π + 2 πn

Combinar toda las soluciones

x = 2 πn, x = π + 2 πn

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solvefor x,tan(x)=1 resolver para

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cot (t) = - 3

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tan (x) = \frac{3}{8}