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sin(x)+cos(x)= 5/7 ,90>x>0

Solución

sin (x) + cos (x) = \frac{5}{7}, 9 0^{\circ} > x > 0^{\circ}

Solución

Sin soluci \overset{o}{ˊ} n para x \in R

Pasos de solución

sin (x) + cos (x) = \frac{5}{7}, 9 0^{\circ} > x > 0^{\circ}

Re-escribir usando identidades trigonométricas

sin (x) + cos (x)

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

sin (x) + cos (x)

Reescribir como

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

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 (x) + sin (4 5^{\circ}) cos (x))

Utilizar la identidad de suma de ángulos:

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

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

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

2 sin (x + 4 5^{\circ}) = \frac{5}{7}

Dividir ambos lados entre

2

2 sin (x + 4 5^{\circ}) = \frac{5}{7}

Dividir ambos lados entre

2

\frac{2 sin ( x + 4 5 ^{\circ} )}{2} = \frac{\frac{5}{7}}{2}

Simplificar

\frac{2 sin ( x + 4 5 ^{\circ} )}{2} = \frac{\frac{5}{7}}{2}

Simplificar

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

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

Eliminar los terminos comunes:

2

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

Simplificar

\frac{\frac{5}{7}}{2} : \frac{5 2}{14}

\frac{\frac{5}{7}}{2}

Aplicar las propiedades de las fracciones:

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

= \frac{5}{7 2}

Racionalizar

\frac{5}{7 2} : \frac{5 2}{14}

\frac{5}{7 2}

Multiplicar por el conjugado

\frac{2}{2}

= \frac{5 2}{7 2 2}

722 = 14

722

Aplicar las leyes de los exponentes:

a a = a

22 = 2

= 7 \cdot 2

Multiplicar los numeros:

7 \cdot 2 = 14

= 14

= \frac{5 2}{14}

= \frac{5 2}{14}

sin (x + 4 5^{\circ}) = \frac{5 2}{14}

sin (x + 4 5^{\circ}) = \frac{5 2}{14}

sin (x + 4 5^{\circ}) = \frac{5 2}{14}

Aplicar propiedades trigonométricas inversas

sin (x + 4 5^{\circ}) = \frac{5 2}{14}

Soluciones generales para

sin (x + 4 5^{\circ}) = \frac{5 2}{14}

sin (x) = a \Rightarrow x = arcsin (a) + 36 0^{\circ} n, x = 18 0^{\circ} - arcsin (a) + 36 0^{\circ} n

x + 4 5^{\circ} = arcsin (\frac{5 2}{14}) + 36 0^{\circ} n, x + 4 5^{\circ} = 18 0^{\circ} - arcsin (\frac{5 2}{14}) + 36 0^{\circ} n

x + 4 5^{\circ} = arcsin (\frac{5 2}{14}) + 36 0^{\circ} n, x + 4 5^{\circ} = 18 0^{\circ} - arcsin (\frac{5 2}{14}) + 36 0^{\circ} n

Resolver

x + 4 5^{\circ} = arcsin (\frac{5 2}{14}) + 36 0^{\circ} n : x = arcsin (\frac{5}{7 2}) + 36 0^{\circ} n - 4 5^{\circ}

x + 4 5^{\circ} = arcsin (\frac{5 2}{14}) + 36 0^{\circ} n

Simplificar

arcsin (\frac{5 2}{14}) + 36 0^{\circ} n : arcsin (\frac{5}{7 2}) + 36 0^{\circ} n

arcsin (\frac{5 2}{14}) + 36 0^{\circ} n

\frac{5 2}{14} = \frac{5}{7 2}

\frac{5 2}{14}

Factorizar

14 : 2 \cdot 7

Factorizar

14 = 2 \cdot 7

= \frac{5 2}{2 \cdot 7}

Cancelar

\frac{5 2}{2 \cdot 7} : \frac{5}{2 \cdot 7}

\frac{5 2}{2 \cdot 7}

Aplicar las leyes de los exponentes:

2 = 2^{\frac{1}{2}}

= \frac{5 \cdot 2 ^{\frac{1}{2}}}{2 \cdot 7}

Aplicar las leyes de los exponentes:

\frac{x ^{a}}{x ^{b}} = \frac{1}{x ^{b - a}}

\frac{2 ^{\frac{1}{2}}}{2 ^{1}} = \frac{1}{2 ^{1 - \frac{1}{2}}}

= \frac{5}{7 \cdot 2 ^{- \frac{1}{2} + 1}}

Restar:

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

= \frac{5}{7 \cdot 2 ^{\frac{1}{2}}}

Aplicar las leyes de los exponentes:

2^{\frac{1}{2}} = 2

= \frac{5}{7 2}

= \frac{5}{2 \cdot 7}

= arcsin (\frac{5}{7 2}) + 36 0^{\circ} n

x + 4 5^{\circ} = arcsin (\frac{5}{7 2}) + 36 0^{\circ} n

Mover

4 5^{\circ}

al lado derecho

x + 4 5^{\circ} = arcsin (\frac{5}{7 2}) + 36 0^{\circ} n

Restar

4 5^{\circ}

de ambos lados

x + 4 5^{\circ} - 4 5^{\circ} = arcsin (\frac{5}{7 2}) + 36 0^{\circ} n - 4 5^{\circ}

Simplificar

x + 4 5^{\circ} - 4 5^{\circ} = arcsin (\frac{5}{7 2}) + 36 0^{\circ} n - 4 5^{\circ}

Simplificar

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

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

Sumar elementos similares:

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

= x

Simplificar

arcsin (\frac{5}{7 2}) + 36 0^{\circ} n - 4 5^{\circ} : arcsin (\frac{5}{7 2}) + 36 0^{\circ} n - 4 5^{\circ}

arcsin (\frac{5}{7 2}) + 36 0^{\circ} n - 4 5^{\circ}

= arcsin (\frac{5 2}{14}) + 36 0^{\circ} n - 4 5^{\circ}

\frac{5 2}{14} = \frac{5}{7 2}

\frac{5 2}{14}

Factorizar

14 : 2 \cdot 7

Factorizar

14 = 2 \cdot 7

= \frac{5 2}{2 \cdot 7}

Cancelar

\frac{5 2}{2 \cdot 7} : \frac{5}{2 \cdot 7}

\frac{5 2}{2 \cdot 7}

Aplicar las leyes de los exponentes:

2 = 2^{\frac{1}{2}}

= \frac{5 \cdot 2 ^{\frac{1}{2}}}{2 \cdot 7}

Aplicar las leyes de los exponentes:

\frac{x ^{a}}{x ^{b}} = \frac{1}{x ^{b - a}}

\frac{2 ^{\frac{1}{2}}}{2 ^{1}} = \frac{1}{2 ^{1 - \frac{1}{2}}}

= \frac{5}{7 \cdot 2 ^{- \frac{1}{2} + 1}}

Restar:

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

= \frac{5}{7 \cdot 2 ^{\frac{1}{2}}}

Aplicar las leyes de los exponentes:

2^{\frac{1}{2}} = 2

= \frac{5}{7 2}

= \frac{5}{2 \cdot 7}

= arcsin (\frac{5}{7 2}) + 36 0^{\circ} n - 4 5^{\circ}

No se puede simplificar mas

= arcsin (\frac{5}{7 2}) + 36 0^{\circ} n - 4 5^{\circ}

x = arcsin (\frac{5}{7 2}) + 36 0^{\circ} n - 4 5^{\circ}

x = arcsin (\frac{5}{7 2}) + 36 0^{\circ} n - 4 5^{\circ}

x = arcsin (\frac{5}{7 2}) + 36 0^{\circ} n - 4 5^{\circ}

Resolver

x + 4 5^{\circ} = 18 0^{\circ} - arcsin (\frac{5 2}{14}) + 36 0^{\circ} n : x = 18 0^{\circ} - arcsin (\frac{5}{7 2}) + 36 0^{\circ} n - 4 5^{\circ}

x + 4 5^{\circ} = 18 0^{\circ} - arcsin (\frac{5 2}{14}) + 36 0^{\circ} n

Simplificar

18 0^{\circ} - arcsin (\frac{5 2}{14}) + 36 0^{\circ} n : 18 0^{\circ} - arcsin (\frac{5}{7 2}) + 36 0^{\circ} n

18 0^{\circ} - arcsin (\frac{5 2}{14}) + 36 0^{\circ} n

\frac{5 2}{14} = \frac{5}{7 2}

\frac{5 2}{14}

Factorizar

14 : 2 \cdot 7

Factorizar

14 = 2 \cdot 7

= \frac{5 2}{2 \cdot 7}

Cancelar

\frac{5 2}{2 \cdot 7} : \frac{5}{2 \cdot 7}

\frac{5 2}{2 \cdot 7}

Aplicar las leyes de los exponentes:

2 = 2^{\frac{1}{2}}

= \frac{5 \cdot 2 ^{\frac{1}{2}}}{2 \cdot 7}

Aplicar las leyes de los exponentes:

\frac{x ^{a}}{x ^{b}} = \frac{1}{x ^{b - a}}

\frac{2 ^{\frac{1}{2}}}{2 ^{1}} = \frac{1}{2 ^{1 - \frac{1}{2}}}

= \frac{5}{7 \cdot 2 ^{- \frac{1}{2} + 1}}

Restar:

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

= \frac{5}{7 \cdot 2 ^{\frac{1}{2}}}

Aplicar las leyes de los exponentes:

2^{\frac{1}{2}} = 2

= \frac{5}{7 2}

= \frac{5}{2 \cdot 7}

= 18 0^{\circ} - arcsin (\frac{5}{7 2}) + 36 0^{\circ} n

x + 4 5^{\circ} = 18 0^{\circ} - arcsin (\frac{5}{7 2}) + 36 0^{\circ} n

Mover

4 5^{\circ}

al lado derecho

x + 4 5^{\circ} = 18 0^{\circ} - arcsin (\frac{5}{7 2}) + 36 0^{\circ} n

Restar

4 5^{\circ}

de ambos lados

x + 4 5^{\circ} - 4 5^{\circ} = 18 0^{\circ} - arcsin (\frac{5}{7 2}) + 36 0^{\circ} n - 4 5^{\circ}

Simplificar

x + 4 5^{\circ} - 4 5^{\circ} = 18 0^{\circ} - arcsin (\frac{5}{7 2}) + 36 0^{\circ} n - 4 5^{\circ}

Simplificar

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

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

Sumar elementos similares:

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

= x

Simplificar

18 0^{\circ} - arcsin (\frac{5}{7 2}) + 36 0^{\circ} n - 4 5^{\circ} : 18 0^{\circ} - arcsin (\frac{5}{7 2}) + 36 0^{\circ} n - 4 5^{\circ}

18 0^{\circ} - arcsin (\frac{5}{7 2}) + 36 0^{\circ} n - 4 5^{\circ}

= 18 0^{\circ} - arcsin (\frac{5 2}{14}) + 36 0^{\circ} n - 4 5^{\circ}

\frac{5 2}{14} = \frac{5}{7 2}

\frac{5 2}{14}

Factorizar

14 : 2 \cdot 7

Factorizar

14 = 2 \cdot 7

= \frac{5 2}{2 \cdot 7}

Cancelar

\frac{5 2}{2 \cdot 7} : \frac{5}{2 \cdot 7}

\frac{5 2}{2 \cdot 7}

Aplicar las leyes de los exponentes:

2 = 2^{\frac{1}{2}}

= \frac{5 \cdot 2 ^{\frac{1}{2}}}{2 \cdot 7}

Aplicar las leyes de los exponentes:

\frac{x ^{a}}{x ^{b}} = \frac{1}{x ^{b - a}}

\frac{2 ^{\frac{1}{2}}}{2 ^{1}} = \frac{1}{2 ^{1 - \frac{1}{2}}}

= \frac{5}{7 \cdot 2 ^{- \frac{1}{2} + 1}}

Restar:

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

= \frac{5}{7 \cdot 2 ^{\frac{1}{2}}}

Aplicar las leyes de los exponentes:

2^{\frac{1}{2}} = 2

= \frac{5}{7 2}

= \frac{5}{2 \cdot 7}

= 18 0^{\circ} - arcsin (\frac{5}{7 2}) + 36 0^{\circ} n - 4 5^{\circ}

No se puede simplificar mas

= 18 0^{\circ} - arcsin (\frac{5}{7 2}) + 36 0^{\circ} n - 4 5^{\circ}

x = 18 0^{\circ} - arcsin (\frac{5}{7 2}) + 36 0^{\circ} n - 4 5^{\circ}

x = 18 0^{\circ} - arcsin (\frac{5}{7 2}) + 36 0^{\circ} n - 4 5^{\circ}

x = 18 0^{\circ} - arcsin (\frac{5}{7 2}) + 36 0^{\circ} n - 4 5^{\circ}

x = arcsin (\frac{5}{7 2}) + 36 0^{\circ} n - 4 5^{\circ}, x = 18 0^{\circ} - arcsin (\frac{5}{7 2}) + 36 0^{\circ} n - 4 5^{\circ}

Soluciones para el rango

9 0^{\circ} > x > 0

Sin soluci \overset{o}{ˊ} n para x \in R

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