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Popular Trigonometría >

tan(45)=(1-cos(u))/(sin(u))

Solución

tan (4 5^{\circ}) = \frac{1 - cos ( u )}{sin ( u )}

Solución

u = 36 0^{\circ} n + 9 0^{\circ}

Radianes

u = \frac{π}{2} + 2 πn

Pasos de solución

tan (4 5^{\circ}) = \frac{1 - cos ( u )}{sin ( u )}

tan (4 5^{\circ}) = 1

tan (4 5^{\circ})

Utilizar la siguiente identidad trivial:

tan (4 5^{\circ}) = 1

tan (4 5^{\circ})

tan (x)

tabla de valores periódicos con

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

= 1

= 1

1 = \frac{1 - cos ( u )}{sin ( u )}

Restar

\frac{1 - cos ( u )}{sin ( u )}

de ambos lados

1 - \frac{1 - cos ( u )}{sin ( u )} = 0

Simplificar

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

1 - \frac{1 - cos ( u )}{sin ( u )}

Convertir a fracción:

1 = \frac{1 sin ( u )}{sin ( u )}

= \frac{1 \cdot sin ( u )}{sin ( u )} - \frac{1 - cos ( u )}{sin ( u )}

Ya que los denominadores son iguales, combinar las fracciones:

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

= \frac{1 \cdot sin ( u ) - ( 1 - cos ( u ) )}{sin ( u )}

Multiplicar:

1 \cdot sin (u) = sin (u)

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

- (1 - cos (u)) : - 1 + cos (u)

- (1 - cos (u))

Poner los parentesis

= - (1) - (- cos (u))

Aplicar las reglas de los signos

- (- a) = a, - (a) = - a

= - 1 + cos (u)

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

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

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

sin (u) - 1 + cos (u) = 0

Re-escribir usando identidades trigonométricas

sin (u) - 1 + cos (u)

sin (u) + cos (u) = 2 sin (u + 4 5^{\circ})

sin (u) + cos (u)

Reescribir como

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

Utilizar la siguiente identidad trivial:

cos (\frac{π}{4}) = \frac{1}{2}

Utilizar la siguiente identidad trivial:

sin (\frac{π}{4}) = \frac{1}{2}

= 2 (cos (4 5^{\circ}) sin (u) + sin (4 5^{\circ}) cos (u))

Utilizar la identidad de suma de ángulos:

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

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

= - 1 + 2 sin (u + 4 5^{\circ})

- 1 + 2 sin (u + 4 5^{\circ}) = 0

Desplace

1

a la derecha

- 1 + 2 sin (u + 4 5^{\circ}) = 0

Sumar

1

a ambos lados

- 1 + 2 sin (u + 4 5^{\circ}) + 1 = 0 + 1

Simplificar

2 sin (u + 4 5^{\circ}) = 1

2 sin (u + 4 5^{\circ}) = 1

Dividir ambos lados entre

2

2 sin (u + 4 5^{\circ}) = 1

Dividir ambos lados entre

2

\frac{2 sin ( u + 4 5 ^{\circ} )}{2} = \frac{1}{2}

Simplificar

\frac{2 sin ( u + 4 5 ^{\circ} )}{2} = \frac{1}{2}

Simplificar

\frac{2 sin ( u + 4 5 ^{\circ} )}{2} : sin (u + 4 5^{\circ})

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

Eliminar los terminos comunes:

2

= sin (u + 4 5^{\circ})

Simplificar

\frac{1}{2} : \frac{2}{2}

\frac{1}{2}

Multiplicar por el conjugado

\frac{2}{2}

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

1 \cdot 2 = 2

22 = 2

22

Aplicar las leyes de los exponentes:

a a = a

22 = 2

= 2

= \frac{2}{2}

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

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

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

Soluciones generales para

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

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}

u + 4 5^{\circ} = 4 5^{\circ} + 36 0^{\circ} n, u + 4 5^{\circ} = 13 5^{\circ} + 36 0^{\circ} n

u + 4 5^{\circ} = 4 5^{\circ} + 36 0^{\circ} n, u + 4 5^{\circ} = 13 5^{\circ} + 36 0^{\circ} n

Resolver

u + 4 5^{\circ} = 4 5^{\circ} + 36 0^{\circ} n : u = 36 0^{\circ} n

u + 4 5^{\circ} = 4 5^{\circ} + 36 0^{\circ} n

Restar

4 5^{\circ}

de ambos lados

u + 4 5^{\circ} - 4 5^{\circ} = 4 5^{\circ} + 36 0^{\circ} n - 4 5^{\circ}

Simplificar

u = 36 0^{\circ} n

Resolver

u + 4 5^{\circ} = 13 5^{\circ} + 36 0^{\circ} n : u = 36 0^{\circ} n + 9 0^{\circ}

u + 4 5^{\circ} = 13 5^{\circ} + 36 0^{\circ} n

Desplace

4 5^{\circ}

a la derecha

u + 4 5^{\circ} = 13 5^{\circ} + 36 0^{\circ} n

Restar

4 5^{\circ}

de ambos lados

u + 4 5^{\circ} - 4 5^{\circ} = 13 5^{\circ} + 36 0^{\circ} n - 4 5^{\circ}

Simplificar

u + 4 5^{\circ} - 4 5^{\circ} = 13 5^{\circ} + 36 0^{\circ} n - 4 5^{\circ}

Simplificar

u + 4 5^{\circ} - 4 5^{\circ} : u

u + 4 5^{\circ} - 4 5^{\circ}

Sumar elementos similares:

4 5^{\circ} - 4 5^{\circ} = 0

= u

Simplificar

13 5^{\circ} + 36 0^{\circ} n - 4 5^{\circ} : 36 0^{\circ} n + 9 0^{\circ}

13 5^{\circ} + 36 0^{\circ} n - 4 5^{\circ}

Agrupar términos semejantes

= 36 0^{\circ} n - 4 5^{\circ} + 13 5^{\circ}

Combinar las fracciones usando el mínimo común denominador:

9 0^{\circ}

Aplicar la regla

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

= \frac{- 18 0 ^{\circ} + 54 0 ^{\circ}}{4}

Sumar elementos similares:

- 18 0^{\circ} + 54 0^{\circ} = 36 0^{\circ}

= 9 0^{\circ}

Eliminar los terminos comunes:

2

= 9 0^{\circ}

= 36 0^{\circ} n + 9 0^{\circ}

u = 36 0^{\circ} n + 9 0^{\circ}

u = 36 0^{\circ} n + 9 0^{\circ}

u = 36 0^{\circ} n + 9 0^{\circ}

u = 36 0^{\circ} n, u = 36 0^{\circ} n + 9 0^{\circ}

Siendo que la ecuación esta indefinida para:

36 0^{\circ} n

u = 36 0^{\circ} n + 9 0^{\circ}

Gráfica

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