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4=tan^2(45+x/2)

Solución

4 = tan^{2} (4 5^{\circ} + \frac{x}{2})

Solución

x = 2 \cdot 1.10714 \dots + 36 0^{\circ} n - 9 0^{\circ}, x = - 2 \cdot 1.10714 \dots + 36 0^{\circ} n - 9 0^{\circ}

radianes

x = 2 \cdot 1.10714 \dots - \frac{π}{2} + 2 πn, x = - 2 \cdot 1.10714 \dots - \frac{π}{2} + 2 πn

Pasos de solución

4 = tan^{2} (4 5^{\circ} + \frac{x}{2})

Intercambiar lados

tan^{2} (4 5^{\circ} + \frac{x}{2}) = 4

Usando el método de sustitución

tan^{2} (4 5^{\circ} + \frac{x}{2}) = 4

Sea:

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

u^{2} = 4

u^{2} = 4 : u = 2, u = - 2

u^{2} = 4

Para

x^{2} = f (a)

las soluciones son

x = f (a), - f (a)

u = 4, u = - 4

4 = 2

4

Descomponer el número en factores primos:

4 = 2^{2}

= 2^{2}

Aplicar las leyes de los exponentes:

2^{2} = 2

= 2

- 4 = - 2

- 4

4 = 2

4

Descomponer el número en factores primos:

4 = 2^{2}

= 2^{2}

Aplicar las leyes de los exponentes:

2^{2} = 2

= 2

= - 2

u = 2, u = - 2

Sustituir en la ecuación

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

tan (4 5^{\circ} + \frac{x}{2}) = 2, tan (4 5^{\circ} + \frac{x}{2}) = - 2

tan (4 5^{\circ} + \frac{x}{2}) = 2, tan (4 5^{\circ} + \frac{x}{2}) = - 2

tan (4 5^{\circ} + \frac{x}{2}) = 2 : x = 2 arctan (2) + 36 0^{\circ} n - 9 0^{\circ}

tan (4 5^{\circ} + \frac{x}{2}) = 2

Aplicar propiedades trigonométricas inversas

tan (4 5^{\circ} + \frac{x}{2}) = 2

Soluciones generales para

tan (4 5^{\circ} + \frac{x}{2}) = 2

tan (x) = a \Rightarrow x = arctan (a) + 18 0^{\circ} n

4 5^{\circ} + \frac{x}{2} = arctan (2) + 18 0^{\circ} n

4 5^{\circ} + \frac{x}{2} = arctan (2) + 18 0^{\circ} n

Resolver

4 5^{\circ} + \frac{x}{2} = arctan (2) + 18 0^{\circ} n : x = 2 arctan (2) + 36 0^{\circ} n - 9 0^{\circ}

4 5^{\circ} + \frac{x}{2} = arctan (2) + 18 0^{\circ} n

Mover

4 5^{\circ}

al lado derecho

4 5^{\circ} + \frac{x}{2} = arctan (2) + 18 0^{\circ} n

Restar

4 5^{\circ}

de ambos lados

4 5^{\circ} + \frac{x}{2} - 4 5^{\circ} = arctan (2) + 18 0^{\circ} n - 4 5^{\circ}

Simplificar

\frac{x}{2} = arctan (2) + 18 0^{\circ} n - 4 5^{\circ}

\frac{x}{2} = arctan (2) + 18 0^{\circ} n - 4 5^{\circ}

Multiplicar ambos lados por

2

\frac{x}{2} = arctan (2) + 18 0^{\circ} n - 4 5^{\circ}

Multiplicar ambos lados por

2

\frac{2 x}{2} = 2 arctan (2) + 36 0^{\circ} n - 2 \cdot 4 5^{\circ}

Simplificar

\frac{2 x}{2} = 2 arctan (2) + 36 0^{\circ} n - 2 \cdot 4 5^{\circ}

Simplificar

\frac{2 x}{2} : x

\frac{2 x}{2}

Dividir:

\frac{2}{2} = 1

= x

Simplificar

2 arctan (2) + 36 0^{\circ} n - 2 \cdot 4 5^{\circ} : 2 arctan (2) + 36 0^{\circ} n - 9 0^{\circ}

2 arctan (2) + 36 0^{\circ} n - 2 \cdot 4 5^{\circ}

2 \cdot 4 5^{\circ} = 9 0^{\circ}

2 \cdot 4 5^{\circ}

Multiplicar fracciones:

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

= 9 0^{\circ}

Eliminar los terminos comunes:

2

= 9 0^{\circ}

= 2 arctan (2) + 36 0^{\circ} n - 9 0^{\circ}

x = 2 arctan (2) + 36 0^{\circ} n - 9 0^{\circ}

x = 2 arctan (2) + 36 0^{\circ} n - 9 0^{\circ}

x = 2 arctan (2) + 36 0^{\circ} n - 9 0^{\circ}

x = 2 arctan (2) + 36 0^{\circ} n - 9 0^{\circ}

tan (4 5^{\circ} + \frac{x}{2}) = - 2 : x = - 2 arctan (2) + 36 0^{\circ} n - 9 0^{\circ}

tan (4 5^{\circ} + \frac{x}{2}) = - 2

Aplicar propiedades trigonométricas inversas

tan (4 5^{\circ} + \frac{x}{2}) = - 2

Soluciones generales para

tan (4 5^{\circ} + \frac{x}{2}) = - 2

tan (x) = - a \Rightarrow x = arctan (- a) + 18 0^{\circ} n

4 5^{\circ} + \frac{x}{2} = arctan (- 2) + 18 0^{\circ} n

4 5^{\circ} + \frac{x}{2} = arctan (- 2) + 18 0^{\circ} n

Resolver

4 5^{\circ} + \frac{x}{2} = arctan (- 2) + 18 0^{\circ} n : x = - 2 arctan (2) + 36 0^{\circ} n - 9 0^{\circ}

4 5^{\circ} + \frac{x}{2} = arctan (- 2) + 18 0^{\circ} n

Simplificar

arctan (- 2) + 18 0^{\circ} n : - arctan (2) + 18 0^{\circ} n

arctan (- 2) + 18 0^{\circ} n

Utilizar la siguiente propiedad:

arctan (- x) = - arctan (x)

arctan (- 2) = - arctan (2)

= - arctan (2) + 18 0^{\circ} n

4 5^{\circ} + \frac{x}{2} = - arctan (2) + 18 0^{\circ} n

Mover

4 5^{\circ}

al lado derecho

4 5^{\circ} + \frac{x}{2} = - arctan (2) + 18 0^{\circ} n

Restar

4 5^{\circ}

de ambos lados

4 5^{\circ} + \frac{x}{2} - 4 5^{\circ} = - arctan (2) + 18 0^{\circ} n - 4 5^{\circ}

Simplificar

\frac{x}{2} = - arctan (2) + 18 0^{\circ} n - 4 5^{\circ}

\frac{x}{2} = - arctan (2) + 18 0^{\circ} n - 4 5^{\circ}

Multiplicar ambos lados por

2

\frac{x}{2} = - arctan (2) + 18 0^{\circ} n - 4 5^{\circ}

Multiplicar ambos lados por

2

\frac{2 x}{2} = - 2 arctan (2) + 36 0^{\circ} n - 2 \cdot 4 5^{\circ}

Simplificar

\frac{2 x}{2} = - 2 arctan (2) + 36 0^{\circ} n - 2 \cdot 4 5^{\circ}

Simplificar

\frac{2 x}{2} : x

\frac{2 x}{2}

Dividir:

\frac{2}{2} = 1

= x

Simplificar

- 2 arctan (2) + 36 0^{\circ} n - 2 \cdot 4 5^{\circ} : - 2 arctan (2) + 36 0^{\circ} n - 9 0^{\circ}

- 2 arctan (2) + 36 0^{\circ} n - 2 \cdot 4 5^{\circ}

2 \cdot 4 5^{\circ} = 9 0^{\circ}

2 \cdot 4 5^{\circ}

Multiplicar fracciones:

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

= 9 0^{\circ}

Eliminar los terminos comunes:

2

= 9 0^{\circ}

= - 2 arctan (2) + 36 0^{\circ} n - 9 0^{\circ}

x = - 2 arctan (2) + 36 0^{\circ} n - 9 0^{\circ}

x = - 2 arctan (2) + 36 0^{\circ} n - 9 0^{\circ}

x = - 2 arctan (2) + 36 0^{\circ} n - 9 0^{\circ}

x = - 2 arctan (2) + 36 0^{\circ} n - 9 0^{\circ}

Combinar toda las soluciones

x = 2 arctan (2) + 36 0^{\circ} n - 9 0^{\circ}, x = - 2 arctan (2) + 36 0^{\circ} n - 9 0^{\circ}

Mostrar soluciones en forma decimal

x = 2 \cdot 1.10714 \dots + 36 0^{\circ} n - 9 0^{\circ}, x = - 2 \cdot 1.10714 \dots + 36 0^{\circ} n - 9 0^{\circ}

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