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Popular Trigonometría >

tan(3x+26)=cot(5x)

Solución

tan (3 x + 2 6^{\circ}) = cot (5 x)

Solución

x = 8^{\circ} + \frac{18 0 ^{\circ} n}{4}, x = 30. 5^{\circ} + \frac{18 0 ^{\circ} n}{4}

Radianes

x = \frac{2 π}{45} + \frac{π}{4} n, x = \frac{61 π}{360} + \frac{π}{4} n

Pasos de solución

tan (3 x + 2 6^{\circ}) = cot (5 x)

Restar

cot (5 x)

de ambos lados

tan (3 x + 2 6^{\circ}) - cot (5 x) = 0

Expresar con seno, coseno

- cot (5 x) + tan (2 6^{\circ} + 3 x)

Utilizar la identidad trigonométrica básica:

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

= - \frac{cos ( 5 x )}{sin ( 5 x )} + tan (2 6^{\circ} + 3 x)

Utilizar la identidad trigonométrica básica:

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

= - \frac{cos ( 5 x )}{sin ( 5 x )} + \frac{sin ( 2 6 ^{\circ} + 3 x )}{cos ( 2 6 ^{\circ} + 3 x )}

Simplificar

- \frac{cos ( 5 x )}{sin ( 5 x )} + \frac{sin ( 2 6 ^{\circ} + 3 x )}{cos ( 2 6 ^{\circ} + 3 x )} : \frac{- cos ( 5 x ) cos ( \frac{270 x + 234 0 ^{\circ}}{90} ) + sin ( \frac{234 0 ^{\circ} + 270 x}{90} ) sin ( 5 x )}{sin ( 5 x ) cos ( \frac{270 x + 234 0 ^{\circ}}{90} )}

- \frac{cos ( 5 x )}{sin ( 5 x )} + \frac{sin ( 2 6 ^{\circ} + 3 x )}{cos ( 2 6 ^{\circ} + 3 x )}

\frac{sin ( 2 6 ^{\circ} + 3 x )}{cos ( 2 6 ^{\circ} + 3 x )} = \frac{sin ( \frac{234 0 ^{\circ} + 270 x}{90} )}{cos ( \frac{234 0 ^{\circ} + 270 x}{90} )}

\frac{sin ( 2 6 ^{\circ} + 3 x )}{cos ( 2 6 ^{\circ} + 3 x )}

Simplificar

2 6^{\circ} + 3 x

en una fracción:

\frac{234 0 ^{\circ} + 270 x}{90}

2 6^{\circ} + 3 x

Convertir a fracción:

3 x = \frac{3 x 90}{90}

= 2 6^{\circ} + \frac{3 x \cdot 90}{90}

Ya que los denominadores son iguales, combinar las fracciones:

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

= \frac{234 0 ^{\circ} + 3 x \cdot 90}{90}

Multiplicar los numeros:

3 \cdot 90 = 270

= \frac{234 0 ^{\circ} + 270 x}{90}

= \frac{sin ( 2 6 ^{\circ} + 3 x )}{cos ( \frac{234 0 ^{\circ} + 270 x}{90} )}

Simplificar

2 6^{\circ} + 3 x

en una fracción:

\frac{234 0 ^{\circ} + 270 x}{90}

2 6^{\circ} + 3 x

Convertir a fracción:

3 x = \frac{3 x 90}{90}

= 2 6^{\circ} + \frac{3 x \cdot 90}{90}

Ya que los denominadores son iguales, combinar las fracciones:

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

= \frac{234 0 ^{\circ} + 3 x \cdot 90}{90}

Multiplicar los numeros:

3 \cdot 90 = 270

= \frac{234 0 ^{\circ} + 270 x}{90}

= \frac{sin ( \frac{234 0 ^{\circ} + 270 x}{90} )}{cos ( \frac{234 0 ^{\circ} + 270 x}{90} )}

= - \frac{cos ( 5 x )}{sin ( 5 x )} + \frac{sin ( \frac{270 x + 234 0 ^{\circ}}{90} )}{cos ( \frac{270 x + 234 0 ^{\circ}}{90} )}

Mínimo común múltiplo de

sin (5 x), cos (\frac{234 0 ^{\circ} + 270 x}{90}) : sin (5 x) cos (\frac{270 x + 234 0 ^{\circ}}{90})

sin (5 x), cos (\frac{234 0 ^{\circ} + 270 x}{90})

Mínimo común múltiplo (MCM)

Calcular una expresión que este compuesta de factores que aparezcan tanto en

sin (5 x)

cos (\frac{234 0 ^{\circ} + 270 x}{90})

= sin (5 x) cos (\frac{270 x + 234 0 ^{\circ}}{90})

Reescribir las fracciones basandose en el mínimo común denominador

Multiplicar cada numerador por la misma cantidad necesaria para multiplicar el denominador correspondiente y convertirlo en el mínimo común denominador

Para

\frac{cos ( 5 x )}{sin ( 5 x )} :

multiplicar el denominador y el numerador por

cos (\frac{270 x + 234 0 ^{\circ}}{90})

\frac{cos ( 5 x )}{sin ( 5 x )} = \frac{cos ( 5 x ) cos ( \frac{270 x + 234 0 ^{\circ}}{90} )}{sin ( 5 x ) cos ( \frac{270 x + 234 0 ^{\circ}}{90} )}

Para

\frac{sin ( \frac{234 0 ^{\circ} + 270 x}{90} )}{cos ( \frac{234 0 ^{\circ} + 270 x}{90} )} :

multiplicar el denominador y el numerador por

sin (5 x)

\frac{sin ( \frac{234 0 ^{\circ} + 270 x}{90} )}{cos ( \frac{234 0 ^{\circ} + 270 x}{90} )} = \frac{sin ( \frac{234 0 ^{\circ} + 270 x}{90} ) sin ( 5 x )}{cos ( \frac{234 0 ^{\circ} + 270 x}{90} ) sin ( 5 x )}

= - \frac{cos ( 5 x ) cos ( \frac{270 x + 234 0 ^{\circ}}{90} )}{sin ( 5 x ) cos ( \frac{270 x + 234 0 ^{\circ}}{90} )} + \frac{sin ( \frac{234 0 ^{\circ} + 270 x}{90} ) sin ( 5 x )}{cos ( \frac{234 0 ^{\circ} + 270 x}{90} ) sin ( 5 x )}

Ya que los denominadores son iguales, combinar las fracciones:

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

= \frac{- cos ( 5 x ) cos ( \frac{270 x + 234 0 ^{\circ}}{90} ) + sin ( \frac{234 0 ^{\circ} + 270 x}{90} ) sin ( 5 x )}{sin ( 5 x ) cos ( \frac{270 x + 234 0 ^{\circ}}{90} )}

= \frac{- cos ( 5 x ) cos ( \frac{270 x + 234 0 ^{\circ}}{90} ) + sin ( \frac{234 0 ^{\circ} + 270 x}{90} ) sin ( 5 x )}{sin ( 5 x ) cos ( \frac{270 x + 234 0 ^{\circ}}{90} )}

\frac{- cos ( 5 x ) cos ( \frac{234 0 ^{\circ} + 270 x}{90} ) + sin ( 5 x ) sin ( \frac{234 0 ^{\circ} + 270 x}{90} )}{cos ( \frac{234 0 ^{\circ} + 270 x}{90} ) sin ( 5 x )} = 0

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

- cos (5 x) cos (\frac{234 0 ^{\circ} + 270 x}{90}) + sin (5 x) sin (\frac{234 0 ^{\circ} + 270 x}{90}) = 0

Re-escribir usando identidades trigonométricas

- cos (5 x) cos (\frac{234 0 ^{\circ} + 270 x}{90}) + sin (5 x) sin (\frac{234 0 ^{\circ} + 270 x}{90})

Utilizar la identidad de suma de ángulos:

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

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

= - cos (5 x + \frac{234 0 ^{\circ} + 270 x}{90})

- cos (5 x + \frac{234 0 ^{\circ} + 270 x}{90}) = 0

Dividir ambos lados entre

- 1

- cos (5 x + \frac{234 0 ^{\circ} + 270 x}{90}) = 0

Dividir ambos lados entre

- 1

\frac{- cos ( 5 x + \frac{234 0 ^{\circ} + 270 x}{90} )}{- 1} = \frac{0}{- 1}

Simplificar

cos (5 x + \frac{234 0 ^{\circ} + 270 x}{90}) = 0

cos (5 x + \frac{234 0 ^{\circ} + 270 x}{90}) = 0

Soluciones generales para

cos (5 x + \frac{234 0 ^{\circ} + 270 x}{90}) = 0

cos (x)

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

5 x + \frac{234 0 ^{\circ} + 270 x}{90} = 9 0^{\circ} + 36 0^{\circ} n, 5 x + \frac{234 0 ^{\circ} + 270 x}{90} = 27 0^{\circ} + 36 0^{\circ} n

5 x + \frac{234 0 ^{\circ} + 270 x}{90} = 9 0^{\circ} + 36 0^{\circ} n, 5 x + \frac{234 0 ^{\circ} + 270 x}{90} = 27 0^{\circ} + 36 0^{\circ} n

Resolver

5 x + \frac{234 0 ^{\circ} + 270 x}{90} = 9 0^{\circ} + 36 0^{\circ} n : x = 8^{\circ} + \frac{18 0 ^{\circ} n}{4}

5 x + \frac{234 0 ^{\circ} + 270 x}{90} = 9 0^{\circ} + 36 0^{\circ} n

Multiplicar ambos lados por

90

5 x + \frac{234 0 ^{\circ} + 270 x}{90} = 9 0^{\circ} + 36 0^{\circ} n

Multiplicar ambos lados por

90

5 x \cdot 90 + \frac{234 0 ^{\circ} + 270 x}{90} \cdot 90 = 9 0^{\circ} \cdot 90 + 36 0^{\circ} n \cdot 90

Simplificar

5 x \cdot 90 + \frac{234 0 ^{\circ} + 270 x}{90} \cdot 90 = 9 0^{\circ} \cdot 90 + 36 0^{\circ} n \cdot 90

Simplificar

5 x \cdot 90 : 450 x

5 x \cdot 90

Multiplicar los numeros:

5 \cdot 90 = 450

= 450 x

Simplificar

\frac{234 0 ^{\circ} + 270 x}{90} \cdot 90 : 234 0^{\circ} + 270 x

\frac{234 0 ^{\circ} + 270 x}{90} \cdot 90

Multiplicar fracciones:

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

= \frac{( 234 0 ^{\circ} + 270 x ) \cdot 90}{90}

Eliminar los terminos comunes:

90

= 234 0^{\circ} + 270 x

Simplificar

9 0^{\circ} \cdot 90 : 810 0^{\circ}

9 0^{\circ} \cdot 90

Multiplicar fracciones:

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

= 810 0^{\circ}

Dividir:

\frac{90}{2} = 45

= 810 0^{\circ}

Simplificar

36 0^{\circ} n \cdot 90 : 3240 0^{\circ} n

36 0^{\circ} n \cdot 90

Multiplicar los numeros:

2 \cdot 90 = 180

= 3240 0^{\circ} n

450 x + 234 0^{\circ} + 270 x = 810 0^{\circ} + 3240 0^{\circ} n

720 x + 234 0^{\circ} = 810 0^{\circ} + 3240 0^{\circ} n

720 x + 234 0^{\circ} = 810 0^{\circ} + 3240 0^{\circ} n

720 x + 234 0^{\circ} = 810 0^{\circ} + 3240 0^{\circ} n

Desplace

234 0^{\circ}

a la derecha

720 x + 234 0^{\circ} = 810 0^{\circ} + 3240 0^{\circ} n

Restar

234 0^{\circ}

de ambos lados

720 x + 234 0^{\circ} - 234 0^{\circ} = 810 0^{\circ} + 3240 0^{\circ} n - 234 0^{\circ}

Simplificar

720 x = 576 0^{\circ} + 3240 0^{\circ} n

720 x = 576 0^{\circ} + 3240 0^{\circ} n

Dividir ambos lados entre

720

720 x = 576 0^{\circ} + 3240 0^{\circ} n

Dividir ambos lados entre

720

\frac{720 x}{720} = 8^{\circ} + \frac{3240 0 ^{\circ} n}{720}

Simplificar

\frac{720 x}{720} = 8^{\circ} + \frac{3240 0 ^{\circ} n}{720}

Simplificar

\frac{720 x}{720} : x

\frac{720 x}{720}

Dividir:

\frac{720}{720} = 1

= x

Simplificar

8^{\circ} + \frac{3240 0 ^{\circ} n}{720} : 8^{\circ} + \frac{18 0 ^{\circ} n}{4}

8^{\circ} + \frac{3240 0 ^{\circ} n}{720}

Cancelar

8^{\circ} : 8^{\circ}

8^{\circ}

Eliminar los terminos comunes:

16

= 8^{\circ}

= 8^{\circ} + \frac{3240 0 ^{\circ} n}{720}

Cancelar

\frac{3240 0 ^{\circ} n}{720} : \frac{18 0 ^{\circ} n}{4}

\frac{3240 0 ^{\circ} n}{720}

Eliminar los terminos comunes:

180

= \frac{18 0 ^{\circ} n}{4}

= 8^{\circ} + \frac{18 0 ^{\circ} n}{4}

x = 8^{\circ} + \frac{18 0 ^{\circ} n}{4}

x = 8^{\circ} + \frac{18 0 ^{\circ} n}{4}

x = 8^{\circ} + \frac{18 0 ^{\circ} n}{4}

Resolver

5 x + \frac{234 0 ^{\circ} + 270 x}{90} = 27 0^{\circ} + 36 0^{\circ} n : x = 30. 5^{\circ} + \frac{18 0 ^{\circ} n}{4}

5 x + \frac{234 0 ^{\circ} + 270 x}{90} = 27 0^{\circ} + 36 0^{\circ} n

Multiplicar ambos lados por

90

5 x + \frac{234 0 ^{\circ} + 270 x}{90} = 27 0^{\circ} + 36 0^{\circ} n

Multiplicar ambos lados por

90

5 x \cdot 90 + \frac{234 0 ^{\circ} + 270 x}{90} \cdot 90 = 27 0^{\circ} \cdot 90 + 36 0^{\circ} n \cdot 90

Simplificar

5 x \cdot 90 + \frac{234 0 ^{\circ} + 270 x}{90} \cdot 90 = 27 0^{\circ} \cdot 90 + 36 0^{\circ} n \cdot 90

Simplificar

5 x \cdot 90 : 450 x

5 x \cdot 90

Multiplicar los numeros:

5 \cdot 90 = 450

= 450 x

Simplificar

\frac{234 0 ^{\circ} + 270 x}{90} \cdot 90 : 234 0^{\circ} + 270 x

\frac{234 0 ^{\circ} + 270 x}{90} \cdot 90

Multiplicar fracciones:

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

= \frac{( 234 0 ^{\circ} + 270 x ) \cdot 90}{90}

Eliminar los terminos comunes:

90

= 234 0^{\circ} + 270 x

Simplificar

27 0^{\circ} \cdot 90 : 2430 0^{\circ}

27 0^{\circ} \cdot 90

Multiplicar fracciones:

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

= 2430 0^{\circ}

Multiplicar los numeros:

3 \cdot 90 = 270

= 2430 0^{\circ}

Dividir:

\frac{270}{2} = 135

= 2430 0^{\circ}

Simplificar

36 0^{\circ} n \cdot 90 : 3240 0^{\circ} n

36 0^{\circ} n \cdot 90

Multiplicar los numeros:

2 \cdot 90 = 180

= 3240 0^{\circ} n

450 x + 234 0^{\circ} + 270 x = 2430 0^{\circ} + 3240 0^{\circ} n

720 x + 234 0^{\circ} = 2430 0^{\circ} + 3240 0^{\circ} n

720 x + 234 0^{\circ} = 2430 0^{\circ} + 3240 0^{\circ} n

720 x + 234 0^{\circ} = 2430 0^{\circ} + 3240 0^{\circ} n

Desplace

234 0^{\circ}

a la derecha

720 x + 234 0^{\circ} = 2430 0^{\circ} + 3240 0^{\circ} n

Restar

234 0^{\circ}

de ambos lados

720 x + 234 0^{\circ} - 234 0^{\circ} = 2430 0^{\circ} + 3240 0^{\circ} n - 234 0^{\circ}

Simplificar

720 x = 2196 0^{\circ} + 3240 0^{\circ} n

720 x = 2196 0^{\circ} + 3240 0^{\circ} n

Dividir ambos lados entre

720

720 x = 2196 0^{\circ} + 3240 0^{\circ} n

Dividir ambos lados entre

720

\frac{720 x}{720} = 30. 5^{\circ} + \frac{3240 0 ^{\circ} n}{720}

Simplificar

\frac{720 x}{720} = 30. 5^{\circ} + \frac{3240 0 ^{\circ} n}{720}

Simplificar

\frac{720 x}{720} : x

\frac{720 x}{720}

Dividir:

\frac{720}{720} = 1

= x

Simplificar

30. 5^{\circ} + \frac{3240 0 ^{\circ} n}{720} : 30. 5^{\circ} + \frac{18 0 ^{\circ} n}{4}

30. 5^{\circ} + \frac{3240 0 ^{\circ} n}{720}

Cancelar

30. 5^{\circ} : 30. 5^{\circ}

30. 5^{\circ}

Eliminar los terminos comunes:

2

= 30. 5^{\circ}

= 30. 5^{\circ} + \frac{3240 0 ^{\circ} n}{720}

Cancelar

\frac{3240 0 ^{\circ} n}{720} : \frac{18 0 ^{\circ} n}{4}

\frac{3240 0 ^{\circ} n}{720}

Eliminar los terminos comunes:

180

= \frac{18 0 ^{\circ} n}{4}

= 30. 5^{\circ} + \frac{18 0 ^{\circ} n}{4}

x = 30. 5^{\circ} + \frac{18 0 ^{\circ} n}{4}

x = 30. 5^{\circ} + \frac{18 0 ^{\circ} n}{4}

x = 30. 5^{\circ} + \frac{18 0 ^{\circ} n}{4}

x = 8^{\circ} + \frac{18 0 ^{\circ} n}{4}, x = 30. 5^{\circ} + \frac{18 0 ^{\circ} n}{4}

Gráfica

Ejemplos populares

sin(x)= 3/5 , pi/2 <= x<= pi

sin (x) = \frac{3}{5}, \frac{π}{2} \leq x \leq π

cos(2θ)-sin(2θ)=0

cos (2 θ) - sin (2 θ) = 0

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

2 sin (x) \cdot cos (x) = cos (x)

4sin(x)+3csc(x)=7,0<= x<= 360

4 sin (x) + 3 csc (x) = 7, 0^{\circ} \leq x \leq 36 0^{\circ}

1/(cos(x))=3

\frac{1}{cos ( x )} = 3