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3tan(x+43)=2cos(x+43)

Solución

3 tan (x + 4 3^{\circ}) = 2 cos (x + 4 3^{\circ})

Solución

x = - 1 3^{\circ} + 36 0^{\circ} n, x = 10 7^{\circ} + 36 0^{\circ} n

radianes

x = - \frac{13 π}{180} + 2 πn, x = \frac{107 π}{180} + 2 πn

Pasos de solución

3 tan (x + 4 3^{\circ}) = 2 cos (x + 4 3^{\circ})

Restar

2 cos (x + 4 3^{\circ})

de ambos lados

3 tan (x + 4 3^{\circ}) - 2 cos (x + 4 3^{\circ}) = 0

Simplificar

3 tan (x + 4 3^{\circ}) - 2 cos (x + 4 3^{\circ}) : 3 tan (\frac{180 x + 774 0 ^{\circ}}{180}) - 2 cos (\frac{180 x + 774 0 ^{\circ}}{180})

3 tan (x + 4 3^{\circ}) - 2 cos (x + 4 3^{\circ})

Simplificar

x + 4 3^{\circ}

en una fracción:

\frac{180 x + 774 0 ^{\circ}}{180}

x + 4 3^{\circ}

Convertir a fracción:

x = \frac{x 180}{180}

= \frac{x \cdot 180}{180} + 4 3^{\circ}

Ya que los denominadores son iguales, combinar las fracciones:

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

= \frac{x \cdot 180 + 774 0 ^{\circ}}{180}

= 3 tan (\frac{180 x + 774 0 ^{\circ}}{180}) - 2 cos (x + 4 3^{\circ})

Simplificar

x + 4 3^{\circ}

en una fracción:

\frac{180 x + 774 0 ^{\circ}}{180}

x + 4 3^{\circ}

Convertir a fracción:

x = \frac{x 180}{180}

= \frac{x \cdot 180}{180} + 4 3^{\circ}

Ya que los denominadores son iguales, combinar las fracciones:

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

= \frac{x \cdot 180 + 774 0 ^{\circ}}{180}

= 3 tan (\frac{180 x + 774 0 ^{\circ}}{180}) - 2 cos (\frac{180 x + 774 0 ^{\circ}}{180})

3 tan (\frac{180 x + 774 0 ^{\circ}}{180}) - 2 cos (\frac{180 x + 774 0 ^{\circ}}{180}) = 0

Expresar con seno, coseno

3 \cdot \frac{sin ( \frac{180 x + 774 0 ^{\circ}}{180} )}{cos ( \frac{180 x + 774 0 ^{\circ}}{180} )} - 2 cos (\frac{180 x + 774 0 ^{\circ}}{180}) = 0

Simplificar

3 \cdot \frac{sin ( \frac{180 x + 774 0 ^{\circ}}{180} )}{cos ( \frac{180 x + 774 0 ^{\circ}}{180} )} - 2 cos (\frac{180 x + 774 0 ^{\circ}}{180}) : \frac{3 sin ( \frac{180 x + 774 0 ^{\circ}}{180} ) - 2 cos ^{2} ( \frac{180 x + 774 0 ^{\circ}}{180} )}{cos ( \frac{180 x + 774 0 ^{\circ}}{180} )}

3 \cdot \frac{sin ( \frac{180 x + 774 0 ^{\circ}}{180} )}{cos ( \frac{180 x + 774 0 ^{\circ}}{180} )} - 2 cos (\frac{180 x + 774 0 ^{\circ}}{180})

Multiplicar

3 \cdot \frac{sin ( \frac{180 x + 774 0 ^{\circ}}{180} )}{cos ( \frac{180 x + 774 0 ^{\circ}}{180} )} : \frac{3 sin ( \frac{180 x + 774 0 ^{\circ}}{180} )}{cos ( \frac{180 x + 774 0 ^{\circ}}{180} )}

3 \cdot \frac{sin ( \frac{180 x + 774 0 ^{\circ}}{180} )}{cos ( \frac{180 x + 774 0 ^{\circ}}{180} )}

Multiplicar fracciones:

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

= \frac{sin ( \frac{180 x + 774 0 ^{\circ}}{180} ) \cdot 3}{cos ( \frac{180 x + 774 0 ^{\circ}}{180} )}

= \frac{3 sin ( \frac{180 x + 774 0 ^{\circ}}{180} )}{cos ( \frac{180 x + 774 0 ^{\circ}}{180} )} - 2 cos (\frac{180 x + 774 0 ^{\circ}}{180})

Convertir a fracción:

2 cos (\frac{180 x + 774 0 ^{\circ}}{180}) = \frac{2 cos ( \frac{180 x + 774 0 ^{\circ}}{180} ) cos ( \frac{180 x + 774 0 ^{\circ}}{180} )}{cos ( \frac{180 x + 774 0 ^{\circ}}{180} )}

= \frac{sin ( \frac{180 x + 774 0 ^{\circ}}{180} ) \cdot 3}{cos ( \frac{180 x + 774 0 ^{\circ}}{180} )} - \frac{2 cos ( \frac{180 x + 774 0 ^{\circ}}{180} ) cos ( \frac{180 x + 774 0 ^{\circ}}{180} )}{cos ( \frac{180 x + 774 0 ^{\circ}}{180} )}

Ya que los denominadores son iguales, combinar las fracciones:

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

= \frac{sin ( \frac{180 x + 774 0 ^{\circ}}{180} ) \cdot 3 - 2 cos ( \frac{180 x + 774 0 ^{\circ}}{180} ) cos ( \frac{180 x + 774 0 ^{\circ}}{180} )}{cos ( \frac{180 x + 774 0 ^{\circ}}{180} )}

sin (\frac{180 x + 774 0 ^{\circ}}{180}) \cdot 3 - 2 cos (\frac{180 x + 774 0 ^{\circ}}{180}) cos (\frac{180 x + 774 0 ^{\circ}}{180}) = 3 sin (\frac{180 x + 774 0 ^{\circ}}{180}) - 2 cos^{2} (\frac{180 x + 774 0 ^{\circ}}{180})

sin (\frac{180 x + 774 0 ^{\circ}}{180}) \cdot 3 - 2 cos (\frac{180 x + 774 0 ^{\circ}}{180}) cos (\frac{180 x + 774 0 ^{\circ}}{180})

2 cos (\frac{180 x + 774 0 ^{\circ}}{180}) cos (\frac{180 x + 774 0 ^{\circ}}{180}) = 2 cos^{2} (\frac{180 x + 774 0 ^{\circ}}{180})

2 cos (\frac{180 x + 774 0 ^{\circ}}{180}) cos (\frac{180 x + 774 0 ^{\circ}}{180})

Aplicar las leyes de los exponentes:

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

cos (\frac{180 x + 774 0 ^{\circ}}{180}) cos (\frac{180 x + 774 0 ^{\circ}}{180}) = cos^{1 + 1} (\frac{180 x + 774 0 ^{\circ}}{180})

= 2 cos^{1 + 1} (\frac{180 x + 774 0 ^{\circ}}{180})

Sumar:

1 + 1 = 2

= 2 cos^{2} (\frac{180 x + 774 0 ^{\circ}}{180})

= 3 sin (\frac{180 x + 774 0 ^{\circ}}{180}) - 2 cos^{2} (\frac{180 x + 774 0 ^{\circ}}{180})

= \frac{3 sin ( \frac{180 x + 774 0 ^{\circ}}{180} ) - 2 cos ^{2} ( \frac{180 x + 774 0 ^{\circ}}{180} )}{cos ( \frac{180 x + 774 0 ^{\circ}}{180} )}

\frac{3 sin ( \frac{180 x + 774 0 ^{\circ}}{180} ) - 2 cos ^{2} ( \frac{180 x + 774 0 ^{\circ}}{180} )}{cos ( \frac{180 x + 774 0 ^{\circ}}{180} )} = 0

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

3 sin (\frac{180 x + 774 0 ^{\circ}}{180}) - 2 cos^{2} (\frac{180 x + 774 0 ^{\circ}}{180}) = 0

Sumar

2 cos^{2} (\frac{180 x + 774 0 ^{\circ}}{180})

a ambos lados

3 sin (\frac{180 x + 774 0 ^{\circ}}{180}) = 2 cos^{2} (\frac{180 x + 774 0 ^{\circ}}{180})

Elevar al cuadrado ambos lados

(3 sin (\frac{180 x + 774 0 ^{\circ}}{180}))^{2} = (2 cos^{2} (\frac{180 x + 774 0 ^{\circ}}{180}))^{2}

Restar

(2 cos^{2} (\frac{180 x + 774 0 ^{\circ}}{180}))^{2}

de ambos lados

9 sin^{2} (\frac{180 x + 774 0 ^{\circ}}{180}) - 4 cos^{4} (\frac{180 x + 774 0 ^{\circ}}{180}) = 0

Factorizar

9 sin^{2} (\frac{180 x + 774 0 ^{\circ}}{180}) - 4 cos^{4} (\frac{180 x + 774 0 ^{\circ}}{180}) : (3 sin (\frac{180 x + 774 0 ^{\circ}}{180}) + 2 cos^{2} (\frac{180 x + 774 0 ^{\circ}}{180})) (3 sin (\frac{180 x + 774 0 ^{\circ}}{180}) - 2 cos^{2} (\frac{180 x + 774 0 ^{\circ}}{180}))

9 sin^{2} (\frac{180 x + 774 0 ^{\circ}}{180}) - 4 cos^{4} (\frac{180 x + 774 0 ^{\circ}}{180})

Reescribir

9 sin^{2} (\frac{180 x + 774 0 ^{\circ}}{180}) - 4 cos^{4} (\frac{180 x + 774 0 ^{\circ}}{180})

como

(3 sin (\frac{180 x + 774 0 ^{\circ}}{180}))^{2} - (2 cos^{2} (\frac{180 x + 774 0 ^{\circ}}{180}))^{2}

9 sin^{2} (\frac{180 x + 774 0 ^{\circ}}{180}) - 4 cos^{4} (\frac{180 x + 774 0 ^{\circ}}{180})

Reescribir

9

como

3^{2}

= 3^{2} sin^{2} (\frac{180 x + 774 0 ^{\circ}}{180}) - 4 cos^{4} (\frac{180 x + 774 0 ^{\circ}}{180})

Reescribir

4

como

2^{2}

= 3^{2} sin^{2} (\frac{180 x + 774 0 ^{\circ}}{180}) - 2^{2} cos^{4} (\frac{180 x + 774 0 ^{\circ}}{180})

Aplicar las leyes de los exponentes:

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

cos^{4} (\frac{180 x + 774 0 ^{\circ}}{180}) = (cos^{2} (\frac{180 x + 774 0 ^{\circ}}{180}))^{2}

= 3^{2} sin^{2} (\frac{180 x + 774 0 ^{\circ}}{180}) - 2^{2} (cos^{2} (\frac{180 x + 774 0 ^{\circ}}{180}))^{2}

Aplicar las leyes de los exponentes:

a^{m} b^{m} = (ab)^{m}

3^{2} sin^{2} (\frac{180 x + 774 0 ^{\circ}}{180}) = (3 sin (\frac{180 x + 774 0 ^{\circ}}{180}))^{2}

= (3 sin (\frac{180 x + 774 0 ^{\circ}}{180}))^{2} - 2^{2} (cos^{2} (\frac{180 x + 774 0 ^{\circ}}{180}))^{2}

Aplicar las leyes de los exponentes:

a^{m} b^{m} = (ab)^{m}

2^{2} (cos^{2} (\frac{180 x + 774 0 ^{\circ}}{180}))^{2} = (2 cos^{2} (\frac{180 x + 774 0 ^{\circ}}{180}))^{2}

= (3 sin (\frac{180 x + 774 0 ^{\circ}}{180}))^{2} - (2 cos^{2} (\frac{180 x + 774 0 ^{\circ}}{180}))^{2}

= (3 sin (\frac{180 x + 774 0 ^{\circ}}{180}))^{2} - (2 cos^{2} (\frac{180 x + 774 0 ^{\circ}}{180}))^{2}

Aplicar la siguiente regla para binomios al cuadrado:

x^{2} - y^{2} = (x + y) (x - y)

(3 sin (\frac{180 x + 774 0 ^{\circ}}{180}))^{2} - (2 cos^{2} (\frac{180 x + 774 0 ^{\circ}}{180}))^{2} = (3 sin (\frac{180 x + 774 0 ^{\circ}}{180}) + 2 cos^{2} (\frac{180 x + 774 0 ^{\circ}}{180})) (3 sin (\frac{180 x + 774 0 ^{\circ}}{180}) - 2 cos^{2} (\frac{180 x + 774 0 ^{\circ}}{180}))

= (3 sin (\frac{180 x + 774 0 ^{\circ}}{180}) + 2 cos^{2} (\frac{180 x + 774 0 ^{\circ}}{180})) (3 sin (\frac{180 x + 774 0 ^{\circ}}{180}) - 2 cos^{2} (\frac{180 x + 774 0 ^{\circ}}{180}))

(3 sin (\frac{180 x + 774 0 ^{\circ}}{180}) + 2 cos^{2} (\frac{180 x + 774 0 ^{\circ}}{180})) (3 sin (\frac{180 x + 774 0 ^{\circ}}{180}) - 2 cos^{2} (\frac{180 x + 774 0 ^{\circ}}{180})) = 0

Resolver cada parte por separado

3 sin (\frac{180 x + 774 0 ^{\circ}}{180}) + 2 cos^{2} (\frac{180 x + 774 0 ^{\circ}}{180}) = 0 or 3 sin (\frac{180 x + 774 0 ^{\circ}}{180}) - 2 cos^{2} (\frac{180 x + 774 0 ^{\circ}}{180}) = 0

3 sin (\frac{180 x + 774 0 ^{\circ}}{180}) + 2 cos^{2} (\frac{180 x + 774 0 ^{\circ}}{180}) = 0 : x = 16 7^{\circ} + 36 0^{\circ} n, x = 28 7^{\circ} + 36 0^{\circ} n

3 sin (\frac{180 x + 774 0 ^{\circ}}{180}) + 2 cos^{2} (\frac{180 x + 774 0 ^{\circ}}{180}) = 0

Re-escribir usando identidades trigonométricas

2 cos^{2} (\frac{180 x + 774 0 ^{\circ}}{180}) + 3 sin (\frac{180 x + 774 0 ^{\circ}}{180})

Utilizar la identidad pitagórica:

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

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

= 2 (1 - sin^{2} (\frac{180 x + 774 0 ^{\circ}}{180})) + 3 sin (\frac{180 x + 774 0 ^{\circ}}{180})

(1 - sin^{2} (\frac{180 x + 774 0 ^{\circ}}{180})) \cdot 2 + 3 sin (\frac{180 x + 774 0 ^{\circ}}{180}) = 0

Usando el método de sustitución

(1 - sin^{2} (\frac{180 x + 774 0 ^{\circ}}{180})) \cdot 2 + 3 sin (\frac{180 x + 774 0 ^{\circ}}{180}) = 0

Sea:

sin (\frac{180 x + 774 0 ^{\circ}}{180}) = u

(1 - u^{2}) \cdot 2 + 3 u = 0

(1 - u^{2}) \cdot 2 + 3 u = 0 : u = - \frac{1}{2}, u = 2

(1 - u^{2}) \cdot 2 + 3 u = 0

Desarrollar

(1 - u^{2}) \cdot 2 + 3 u : 2 - 2 u^{2} + 3 u

(1 - u^{2}) \cdot 2 + 3 u

= 2 (1 - u^{2}) + 3 u

Expandir

2 (1 - u^{2}) : 2 - 2 u^{2}

2 (1 - u^{2})

Poner los parentesis utilizando:

a (b - c) = ab - a c

a = 2, b = 1, c = u^{2}

= 2 \cdot 1 - 2 u^{2}

Multiplicar los numeros:

2 \cdot 1 = 2

= 2 - 2 u^{2}

= 2 - 2 u^{2} + 3 u

2 - 2 u^{2} + 3 u = 0

Escribir en la forma binómica

a x^{2} + b x + c = 0

- 2 u^{2} + 3 u + 2 = 0

Resolver con la fórmula general para ecuaciones de segundo grado:

- 2 u^{2} + 3 u + 2 = 0

Formula general para ecuaciones de segundo grado:

Para

a = - 2, b = 3, c = 2

u_{1, 2} = \frac{- 3 \pm 3 ^{2} - 4 ( - 2 ) \cdot 2}{2 ( - 2 )}

u_{1, 2} = \frac{- 3 \pm 3 ^{2} - 4 ( - 2 ) \cdot 2}{2 ( - 2 )}

3^{2} - 4 (- 2) \cdot 2 = 5

3^{2} - 4 (- 2) \cdot 2

Aplicar la regla

- (- a) = a

= 3^{2} + 4 \cdot 2 \cdot 2

Multiplicar los numeros:

4 \cdot 2 \cdot 2 = 16

= 3^{2} + 16

3^{2} = 9

= 9 + 16

Sumar:

9 + 16 = 25

= 25

Descomponer el número en factores primos:

25 = 5^{2}

= 5^{2}

Aplicar las leyes de los exponentes:

5^{2} = 5

= 5

u_{1, 2} = \frac{- 3 \pm 5}{2 ( - 2 )}

Separar las soluciones

u_{1} = \frac{- 3 + 5}{2 ( - 2 )}, u_{2} = \frac{- 3 - 5}{2 ( - 2 )}

u = \frac{- 3 + 5}{2 ( - 2 )} : - \frac{1}{2}

\frac{- 3 + 5}{2 ( - 2 )}

Quitar los parentesis:

(- a) = - a

= \frac{- 3 + 5}{- 2 \cdot 2}

Sumar/restar lo siguiente:

- 3 + 5 = 2

= \frac{2}{- 2 \cdot 2}

Multiplicar los numeros:

2 \cdot 2 = 4

= \frac{2}{- 4}

Aplicar las propiedades de las fracciones:

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

= - \frac{2}{4}

Eliminar los terminos comunes:

2

= - \frac{1}{2}

u = \frac{- 3 - 5}{2 ( - 2 )} : 2

\frac{- 3 - 5}{2 ( - 2 )}

Quitar los parentesis:

(- a) = - a

= \frac{- 3 - 5}{- 2 \cdot 2}

Restar:

- 3 - 5 = - 8

= \frac{- 8}{- 2 \cdot 2}

Multiplicar los numeros:

2 \cdot 2 = 4

= \frac{- 8}{- 4}

Aplicar las propiedades de las fracciones:

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

= \frac{8}{4}

Dividir:

\frac{8}{4} = 2

= 2

Las soluciones a la ecuación de segundo grado son:

u = - \frac{1}{2}, u = 2

Sustituir en la ecuación

u = sin (\frac{180 x + 774 0 ^{\circ}}{180})

sin (\frac{180 x + 774 0 ^{\circ}}{180}) = - \frac{1}{2}, sin (\frac{180 x + 774 0 ^{\circ}}{180}) = 2

sin (\frac{180 x + 774 0 ^{\circ}}{180}) = - \frac{1}{2}, sin (\frac{180 x + 774 0 ^{\circ}}{180}) = 2

sin (\frac{180 x + 774 0 ^{\circ}}{180}) = - \frac{1}{2} : x = 16 7^{\circ} + 36 0^{\circ} n, x = 28 7^{\circ} + 36 0^{\circ} n

sin (\frac{180 x + 774 0 ^{\circ}}{180}) = - \frac{1}{2}

Soluciones generales para

sin (\frac{180 x + 774 0 ^{\circ}}{180}) = - \frac{1}{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}

\frac{180 x + 774 0 ^{\circ}}{180} = 21 0^{\circ} + 36 0^{\circ} n, \frac{180 x + 774 0 ^{\circ}}{180} = 33 0^{\circ} + 36 0^{\circ} n

\frac{180 x + 774 0 ^{\circ}}{180} = 21 0^{\circ} + 36 0^{\circ} n, \frac{180 x + 774 0 ^{\circ}}{180} = 33 0^{\circ} + 36 0^{\circ} n

Resolver

\frac{180 x + 774 0 ^{\circ}}{180} = 21 0^{\circ} + 36 0^{\circ} n : x = 16 7^{\circ} + 36 0^{\circ} n

\frac{180 x + 774 0 ^{\circ}}{180} = 21 0^{\circ} + 36 0^{\circ} n

Multiplicar ambos lados por

180

\frac{180 x + 774 0 ^{\circ}}{180} = 21 0^{\circ} + 36 0^{\circ} n

Multiplicar ambos lados por

180

\frac{180 ( 180 x + 774 0 ^{\circ} )}{180} = 180 \cdot 21 0^{\circ} + 180 \cdot 36 0^{\circ} n

Simplificar

\frac{180 ( 180 x + 774 0 ^{\circ} )}{180} = 180 \cdot 21 0^{\circ} + 180 \cdot 36 0^{\circ} n

Simplificar

\frac{180 ( 180 x + 774 0 ^{\circ} )}{180} : 180 x + 774 0^{\circ}

\frac{180 ( 180 x + 774 0 ^{\circ} )}{180}

Dividir:

\frac{180}{180} = 1

= 180 x + 774 0^{\circ}

Simplificar

180 \cdot 21 0^{\circ} + 180 \cdot 36 0^{\circ} n : 3780 0^{\circ} + 6480 0^{\circ} n

180 \cdot 21 0^{\circ} + 180 \cdot 36 0^{\circ} n

180 \cdot 21 0^{\circ} = 3780 0^{\circ}

180 \cdot 21 0^{\circ}

Multiplicar fracciones:

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

= 3780 0^{\circ}

Multiplicar los numeros:

7 \cdot 180 = 1260

= 3780 0^{\circ}

Dividir:

\frac{1260}{6} = 210

= 3780 0^{\circ}

180 \cdot 36 0^{\circ} n = 6480 0^{\circ} n

180 \cdot 36 0^{\circ} n

Multiplicar los numeros:

180 \cdot 2 = 360

= 6480 0^{\circ} n

= 3780 0^{\circ} + 6480 0^{\circ} n

180 x + 774 0^{\circ} = 3780 0^{\circ} + 6480 0^{\circ} n

180 x + 774 0^{\circ} = 3780 0^{\circ} + 6480 0^{\circ} n

180 x + 774 0^{\circ} = 3780 0^{\circ} + 6480 0^{\circ} n

Mover

774 0^{\circ}

al lado derecho

180 x + 774 0^{\circ} = 3780 0^{\circ} + 6480 0^{\circ} n

Restar

774 0^{\circ}

de ambos lados

180 x + 774 0^{\circ} - 774 0^{\circ} = 3780 0^{\circ} + 6480 0^{\circ} n - 774 0^{\circ}

Simplificar

180 x = 3006 0^{\circ} + 6480 0^{\circ} n

180 x = 3006 0^{\circ} + 6480 0^{\circ} n

Dividir ambos lados entre

180

180 x = 3006 0^{\circ} + 6480 0^{\circ} n

Dividir ambos lados entre

180

\frac{180 x}{180} = 16 7^{\circ} + \frac{6480 0 ^{\circ} n}{180}

Simplificar

x = 16 7^{\circ} + 36 0^{\circ} n

x = 16 7^{\circ} + 36 0^{\circ} n

Resolver

\frac{180 x + 774 0 ^{\circ}}{180} = 33 0^{\circ} + 36 0^{\circ} n : x = 28 7^{\circ} + 36 0^{\circ} n

\frac{180 x + 774 0 ^{\circ}}{180} = 33 0^{\circ} + 36 0^{\circ} n

Multiplicar ambos lados por

180

\frac{180 x + 774 0 ^{\circ}}{180} = 33 0^{\circ} + 36 0^{\circ} n

Multiplicar ambos lados por

180

\frac{180 ( 180 x + 774 0 ^{\circ} )}{180} = 180 \cdot 33 0^{\circ} + 180 \cdot 36 0^{\circ} n

Simplificar

\frac{180 ( 180 x + 774 0 ^{\circ} )}{180} = 180 \cdot 33 0^{\circ} + 180 \cdot 36 0^{\circ} n

Simplificar

\frac{180 ( 180 x + 774 0 ^{\circ} )}{180} : 180 x + 774 0^{\circ}

\frac{180 ( 180 x + 774 0 ^{\circ} )}{180}

Dividir:

\frac{180}{180} = 1

= 180 x + 774 0^{\circ}

Simplificar

180 \cdot 33 0^{\circ} + 180 \cdot 36 0^{\circ} n : 5940 0^{\circ} + 6480 0^{\circ} n

180 \cdot 33 0^{\circ} + 180 \cdot 36 0^{\circ} n

180 \cdot 33 0^{\circ} = 5940 0^{\circ}

180 \cdot 33 0^{\circ}

Multiplicar fracciones:

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

= 5940 0^{\circ}

Multiplicar los numeros:

11 \cdot 180 = 1980

= 5940 0^{\circ}

Dividir:

\frac{1980}{6} = 330

= 5940 0^{\circ}

180 \cdot 36 0^{\circ} n = 6480 0^{\circ} n

180 \cdot 36 0^{\circ} n

Multiplicar los numeros:

180 \cdot 2 = 360

= 6480 0^{\circ} n

= 5940 0^{\circ} + 6480 0^{\circ} n

180 x + 774 0^{\circ} = 5940 0^{\circ} + 6480 0^{\circ} n

180 x + 774 0^{\circ} = 5940 0^{\circ} + 6480 0^{\circ} n

180 x + 774 0^{\circ} = 5940 0^{\circ} + 6480 0^{\circ} n

Mover

774 0^{\circ}

al lado derecho

180 x + 774 0^{\circ} = 5940 0^{\circ} + 6480 0^{\circ} n

Restar

774 0^{\circ}

de ambos lados

180 x + 774 0^{\circ} - 774 0^{\circ} = 5940 0^{\circ} + 6480 0^{\circ} n - 774 0^{\circ}

Simplificar

180 x = 5166 0^{\circ} + 6480 0^{\circ} n

180 x = 5166 0^{\circ} + 6480 0^{\circ} n

Dividir ambos lados entre

180

180 x = 5166 0^{\circ} + 6480 0^{\circ} n

Dividir ambos lados entre

180

\frac{180 x}{180} = 28 7^{\circ} + \frac{6480 0 ^{\circ} n}{180}

Simplificar

x = 28 7^{\circ} + 36 0^{\circ} n

x = 28 7^{\circ} + 36 0^{\circ} n

x = 16 7^{\circ} + 36 0^{\circ} n, x = 28 7^{\circ} + 36 0^{\circ} n

sin (\frac{180 x + 774 0 ^{\circ}}{180}) = 2 :

Sin solución

sin (\frac{180 x + 774 0 ^{\circ}}{180}) = 2

- 1 \leq sin (x) \leq 1

Sin soluci \overset{o}{ˊ} n

Combinar toda las soluciones

x = 16 7^{\circ} + 36 0^{\circ} n, x = 28 7^{\circ} + 36 0^{\circ} n

3 sin (\frac{180 x + 774 0 ^{\circ}}{180}) - 2 cos^{2} (\frac{180 x + 774 0 ^{\circ}}{180}) = 0 : x = - 1 3^{\circ} + 36 0^{\circ} n, x = 10 7^{\circ} + 36 0^{\circ} n

3 sin (\frac{180 x + 774 0 ^{\circ}}{180}) - 2 cos^{2} (\frac{180 x + 774 0 ^{\circ}}{180}) = 0

Re-escribir usando identidades trigonométricas

- 2 cos^{2} (\frac{180 x + 774 0 ^{\circ}}{180}) + 3 sin (\frac{180 x + 774 0 ^{\circ}}{180})

Utilizar la identidad pitagórica:

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

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

= - 2 (1 - sin^{2} (\frac{180 x + 774 0 ^{\circ}}{180})) + 3 sin (\frac{180 x + 774 0 ^{\circ}}{180})

- (1 - sin^{2} (\frac{180 x + 774 0 ^{\circ}}{180})) \cdot 2 + 3 sin (\frac{180 x + 774 0 ^{\circ}}{180}) = 0

Usando el método de sustitución

- (1 - sin^{2} (\frac{180 x + 774 0 ^{\circ}}{180})) \cdot 2 + 3 sin (\frac{180 x + 774 0 ^{\circ}}{180}) = 0

Sea:

sin (\frac{180 x + 774 0 ^{\circ}}{180}) = u

- (1 - u^{2}) \cdot 2 + 3 u = 0

- (1 - u^{2}) \cdot 2 + 3 u = 0 : u = \frac{1}{2}, u = - 2

- (1 - u^{2}) \cdot 2 + 3 u = 0

Desarrollar

- (1 - u^{2}) \cdot 2 + 3 u : - 2 + 2 u^{2} + 3 u

- (1 - u^{2}) \cdot 2 + 3 u

= - 2 (1 - u^{2}) + 3 u

Expandir

- 2 (1 - u^{2}) : - 2 + 2 u^{2}

- 2 (1 - u^{2})

Poner los parentesis utilizando:

a (b - c) = ab - a c

a = - 2, b = 1, c = u^{2}

= - 2 \cdot 1 - (- 2) u^{2}

Aplicar las reglas de los signos

- (- a) = a

= - 2 \cdot 1 + 2 u^{2}

Multiplicar los numeros:

2 \cdot 1 = 2

= - 2 + 2 u^{2}

= - 2 + 2 u^{2} + 3 u

- 2 + 2 u^{2} + 3 u = 0

Escribir en la forma binómica

a x^{2} + b x + c = 0

2 u^{2} + 3 u - 2 = 0

Resolver con la fórmula general para ecuaciones de segundo grado:

2 u^{2} + 3 u - 2 = 0

Formula general para ecuaciones de segundo grado:

Para

a = 2, b = 3, c = - 2

u_{1, 2} = \frac{- 3 \pm 3 ^{2} - 4 \cdot 2 ( - 2 )}{2 \cdot 2}

u_{1, 2} = \frac{- 3 \pm 3 ^{2} - 4 \cdot 2 ( - 2 )}{2 \cdot 2}

3^{2} - 4 \cdot 2 (- 2) = 5

3^{2} - 4 \cdot 2 (- 2)

Aplicar la regla

- (- a) = a

= 3^{2} + 4 \cdot 2 \cdot 2

Multiplicar los numeros:

4 \cdot 2 \cdot 2 = 16

= 3^{2} + 16

3^{2} = 9

= 9 + 16

Sumar:

9 + 16 = 25

= 25

Descomponer el número en factores primos:

25 = 5^{2}

= 5^{2}

Aplicar las leyes de los exponentes:

5^{2} = 5

= 5

u_{1, 2} = \frac{- 3 \pm 5}{2 \cdot 2}

Separar las soluciones

u_{1} = \frac{- 3 + 5}{2 \cdot 2}, u_{2} = \frac{- 3 - 5}{2 \cdot 2}

u = \frac{- 3 + 5}{2 \cdot 2} : \frac{1}{2}

\frac{- 3 + 5}{2 \cdot 2}

Sumar/restar lo siguiente:

- 3 + 5 = 2

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

Multiplicar los numeros:

2 \cdot 2 = 4

= \frac{2}{4}

Eliminar los terminos comunes:

2

= \frac{1}{2}

u = \frac{- 3 - 5}{2 \cdot 2} : - 2

\frac{- 3 - 5}{2 \cdot 2}

Restar:

- 3 - 5 = - 8

= \frac{- 8}{2 \cdot 2}

Multiplicar los numeros:

2 \cdot 2 = 4

= \frac{- 8}{4}

Aplicar las propiedades de las fracciones:

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

= - \frac{8}{4}

Dividir:

\frac{8}{4} = 2

= - 2

Las soluciones a la ecuación de segundo grado son:

u = \frac{1}{2}, u = - 2

Sustituir en la ecuación

u = sin (\frac{180 x + 774 0 ^{\circ}}{180})

sin (\frac{180 x + 774 0 ^{\circ}}{180}) = \frac{1}{2}, sin (\frac{180 x + 774 0 ^{\circ}}{180}) = - 2

sin (\frac{180 x + 774 0 ^{\circ}}{180}) = \frac{1}{2}, sin (\frac{180 x + 774 0 ^{\circ}}{180}) = - 2

sin (\frac{180 x + 774 0 ^{\circ}}{180}) = \frac{1}{2} : x = - 1 3^{\circ} + 36 0^{\circ} n, x = 10 7^{\circ} + 36 0^{\circ} n

sin (\frac{180 x + 774 0 ^{\circ}}{180}) = \frac{1}{2}

Soluciones generales para

sin (\frac{180 x + 774 0 ^{\circ}}{180}) = \frac{1}{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}

\frac{180 x + 774 0 ^{\circ}}{180} = 3 0^{\circ} + 36 0^{\circ} n, \frac{180 x + 774 0 ^{\circ}}{180} = 15 0^{\circ} + 36 0^{\circ} n

\frac{180 x + 774 0 ^{\circ}}{180} = 3 0^{\circ} + 36 0^{\circ} n, \frac{180 x + 774 0 ^{\circ}}{180} = 15 0^{\circ} + 36 0^{\circ} n

Resolver

\frac{180 x + 774 0 ^{\circ}}{180} = 3 0^{\circ} + 36 0^{\circ} n : x = - 1 3^{\circ} + 36 0^{\circ} n

\frac{180 x + 774 0 ^{\circ}}{180} = 3 0^{\circ} + 36 0^{\circ} n

Multiplicar ambos lados por

180

\frac{180 x + 774 0 ^{\circ}}{180} = 3 0^{\circ} + 36 0^{\circ} n

Multiplicar ambos lados por

180

\frac{180 ( 180 x + 774 0 ^{\circ} )}{180} = 180 \cdot 3 0^{\circ} + 180 \cdot 36 0^{\circ} n

Simplificar

\frac{180 ( 180 x + 774 0 ^{\circ} )}{180} = 180 \cdot 3 0^{\circ} + 180 \cdot 36 0^{\circ} n

Simplificar

\frac{180 ( 180 x + 774 0 ^{\circ} )}{180} : 180 x + 774 0^{\circ}

\frac{180 ( 180 x + 774 0 ^{\circ} )}{180}

Dividir:

\frac{180}{180} = 1

= 180 x + 774 0^{\circ}

Simplificar

180 \cdot 3 0^{\circ} + 180 \cdot 36 0^{\circ} n : 540 0^{\circ} + 6480 0^{\circ} n

180 \cdot 3 0^{\circ} + 180 \cdot 36 0^{\circ} n

180 \cdot 3 0^{\circ} = 540 0^{\circ}

180 \cdot 3 0^{\circ}

Multiplicar fracciones:

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

= 540 0^{\circ}

Dividir:

\frac{180}{6} = 30

= 540 0^{\circ}

180 \cdot 36 0^{\circ} n = 6480 0^{\circ} n

180 \cdot 36 0^{\circ} n

Multiplicar los numeros:

180 \cdot 2 = 360

= 6480 0^{\circ} n

= 540 0^{\circ} + 6480 0^{\circ} n

180 x + 774 0^{\circ} = 540 0^{\circ} + 6480 0^{\circ} n

180 x + 774 0^{\circ} = 540 0^{\circ} + 6480 0^{\circ} n

180 x + 774 0^{\circ} = 540 0^{\circ} + 6480 0^{\circ} n

Mover

774 0^{\circ}

al lado derecho

180 x + 774 0^{\circ} = 540 0^{\circ} + 6480 0^{\circ} n

Restar

774 0^{\circ}

de ambos lados

180 x + 774 0^{\circ} - 774 0^{\circ} = 540 0^{\circ} + 6480 0^{\circ} n - 774 0^{\circ}

Simplificar

180 x = - 234 0^{\circ} + 6480 0^{\circ} n

180 x = - 234 0^{\circ} + 6480 0^{\circ} n

Dividir ambos lados entre

180

180 x = - 234 0^{\circ} + 6480 0^{\circ} n

Dividir ambos lados entre

180

\frac{180 x}{180} = - 1 3^{\circ} + \frac{6480 0 ^{\circ} n}{180}

Simplificar

x = - 1 3^{\circ} + 36 0^{\circ} n

x = - 1 3^{\circ} + 36 0^{\circ} n

Resolver

\frac{180 x + 774 0 ^{\circ}}{180} = 15 0^{\circ} + 36 0^{\circ} n : x = 10 7^{\circ} + 36 0^{\circ} n

\frac{180 x + 774 0 ^{\circ}}{180} = 15 0^{\circ} + 36 0^{\circ} n

Multiplicar ambos lados por

180

\frac{180 x + 774 0 ^{\circ}}{180} = 15 0^{\circ} + 36 0^{\circ} n

Multiplicar ambos lados por

180

\frac{180 ( 180 x + 774 0 ^{\circ} )}{180} = 180 \cdot 15 0^{\circ} + 180 \cdot 36 0^{\circ} n

Simplificar

\frac{180 ( 180 x + 774 0 ^{\circ} )}{180} = 180 \cdot 15 0^{\circ} + 180 \cdot 36 0^{\circ} n

Simplificar

\frac{180 ( 180 x + 774 0 ^{\circ} )}{180} : 180 x + 774 0^{\circ}

\frac{180 ( 180 x + 774 0 ^{\circ} )}{180}

Dividir:

\frac{180}{180} = 1

= 180 x + 774 0^{\circ}

Simplificar

180 \cdot 15 0^{\circ} + 180 \cdot 36 0^{\circ} n : 2700 0^{\circ} + 6480 0^{\circ} n

180 \cdot 15 0^{\circ} + 180 \cdot 36 0^{\circ} n

180 \cdot 15 0^{\circ} = 2700 0^{\circ}

180 \cdot 15 0^{\circ}

Multiplicar fracciones:

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

= 2700 0^{\circ}

Multiplicar los numeros:

5 \cdot 180 = 900

= 2700 0^{\circ}

Dividir:

\frac{900}{6} = 150

= 2700 0^{\circ}

180 \cdot 36 0^{\circ} n = 6480 0^{\circ} n

180 \cdot 36 0^{\circ} n

Multiplicar los numeros:

180 \cdot 2 = 360

= 6480 0^{\circ} n

= 2700 0^{\circ} + 6480 0^{\circ} n

180 x + 774 0^{\circ} = 2700 0^{\circ} + 6480 0^{\circ} n

180 x + 774 0^{\circ} = 2700 0^{\circ} + 6480 0^{\circ} n

180 x + 774 0^{\circ} = 2700 0^{\circ} + 6480 0^{\circ} n

Mover

774 0^{\circ}

al lado derecho

180 x + 774 0^{\circ} = 2700 0^{\circ} + 6480 0^{\circ} n

Restar

774 0^{\circ}

de ambos lados

180 x + 774 0^{\circ} - 774 0^{\circ} = 2700 0^{\circ} + 6480 0^{\circ} n - 774 0^{\circ}

Simplificar

180 x = 1926 0^{\circ} + 6480 0^{\circ} n

180 x = 1926 0^{\circ} + 6480 0^{\circ} n

Dividir ambos lados entre

180

180 x = 1926 0^{\circ} + 6480 0^{\circ} n

Dividir ambos lados entre

180

\frac{180 x}{180} = 10 7^{\circ} + \frac{6480 0 ^{\circ} n}{180}

Simplificar

x = 10 7^{\circ} + 36 0^{\circ} n

x = 10 7^{\circ} + 36 0^{\circ} n

x = - 1 3^{\circ} + 36 0^{\circ} n, x = 10 7^{\circ} + 36 0^{\circ} n

sin (\frac{180 x + 774 0 ^{\circ}}{180}) = - 2 :

Sin solución

sin (\frac{180 x + 774 0 ^{\circ}}{180}) = - 2

- 1 \leq sin (x) \leq 1

Sin soluci \overset{o}{ˊ} n

Combinar toda las soluciones

x = - 1 3^{\circ} + 36 0^{\circ} n, x = 10 7^{\circ} + 36 0^{\circ} n

Combinar toda las soluciones

x = 16 7^{\circ} + 36 0^{\circ} n, x = 28 7^{\circ} + 36 0^{\circ} n, x = - 1 3^{\circ} + 36 0^{\circ} n, x = 10 7^{\circ} + 36 0^{\circ} n

Verificar las soluciones sustituyendo en la ecuación original

Verificar las soluciones sustituyéndolas en

3 tan (x + 4 3^{\circ}) = 2 cos (x + 4 3^{\circ})

Quitar las que no concuerden con la ecuación.

Verificar la solución

16 7^{\circ} + 36 0^{\circ} n :

Falso

16 7^{\circ} + 36 0^{\circ} n

Sustituir

n = 1

16 7^{\circ} + 36 0^{\circ} 1

Multiplicar

3 tan (x + 4 3^{\circ}) = 2 cos (x + 4 3^{\circ})

por

x = 16 7^{\circ} + 36 0^{\circ} 1

3 tan (16 7^{\circ} + 36 0^{\circ} 1 + 4 3^{\circ}) = 2 cos (16 7^{\circ} + 36 0^{\circ} 1 + 4 3^{\circ})

Simplificar

1.73205 \dots = - 1.73205 \dots

\Rightarrow Falso

Verificar la solución

28 7^{\circ} + 36 0^{\circ} n :

Falso

28 7^{\circ} + 36 0^{\circ} n

Sustituir

n = 1

28 7^{\circ} + 36 0^{\circ} 1

Multiplicar

3 tan (x + 4 3^{\circ}) = 2 cos (x + 4 3^{\circ})

por

x = 28 7^{\circ} + 36 0^{\circ} 1

3 tan (28 7^{\circ} + 36 0^{\circ} 1 + 4 3^{\circ}) = 2 cos (28 7^{\circ} + 36 0^{\circ} 1 + 4 3^{\circ})

Simplificar

- 1.73205 \dots = 1.73205 \dots

\Rightarrow Falso

Verificar la solución

- 1 3^{\circ} + 36 0^{\circ} n :

Verdadero

- 1 3^{\circ} + 36 0^{\circ} n

Sustituir

n = 1

- 1 3^{\circ} + 36 0^{\circ} 1

Multiplicar

3 tan (x + 4 3^{\circ}) = 2 cos (x + 4 3^{\circ})

por

x = - 1 3^{\circ} + 36 0^{\circ} 1

3 tan (- 1 3^{\circ} + 36 0^{\circ} 1 + 4 3^{\circ}) = 2 cos (- 1 3^{\circ} + 36 0^{\circ} 1 + 4 3^{\circ})

Simplificar

1.73205 \dots = 1.73205 \dots

\Rightarrow Verdadero

Verificar la solución

10 7^{\circ} + 36 0^{\circ} n :

Verdadero

10 7^{\circ} + 36 0^{\circ} n

Sustituir

n = 1

10 7^{\circ} + 36 0^{\circ} 1

Multiplicar

3 tan (x + 4 3^{\circ}) = 2 cos (x + 4 3^{\circ})

por

x = 10 7^{\circ} + 36 0^{\circ} 1

3 tan (10 7^{\circ} + 36 0^{\circ} 1 + 4 3^{\circ}) = 2 cos (10 7^{\circ} + 36 0^{\circ} 1 + 4 3^{\circ})

Simplificar

- 1.73205 \dots = - 1.73205 \dots

\Rightarrow Verdadero

x = - 1 3^{\circ} + 36 0^{\circ} n, x = 10 7^{\circ} + 36 0^{\circ} n

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