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Popular Trigonometría >

(sin(x)+cos(x))^2=(1+2sin(x))/(sec(x))

Solución

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

Solución

x = 2 πn

Grados

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

Pasos de solución

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

Restar

\frac{1 + 2 sin ( x )}{sec ( x )}

de ambos lados

(sin (x) + cos (x))^{2} - \frac{1 + 2 sin ( x )}{sec ( x )} = 0

Simplificar

(sin (x) + cos (x))^{2} - \frac{1 + 2 sin ( x )}{sec ( x )} : \frac{sin ^{2} ( x ) sec ( x ) + 2 sec ( x ) sin ( x ) cos ( x ) + cos ^{2} ( x ) sec ( x ) - 1 - 2 sin ( x )}{sec ( x )}

(sin (x) + cos (x))^{2} - \frac{1 + 2 sin ( x )}{sec ( x )}

Convertir a fracción:

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

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

Ya que los denominadores son iguales, combinar las fracciones:

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

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

Expandir

(sin (x) + cos (x))^{2} sec (x) - (1 + 2 sin (x)) : sin^{2} (x) sec (x) + 2 sec (x) sin (x) cos (x) + cos^{2} (x) sec (x) - 1 - 2 sin (x)

(sin (x) + cos (x))^{2} sec (x) - (1 + 2 sin (x))

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

(sin (x) + cos (x))^{2} : sin^{2} (x) + 2 sin (x) cos (x) + cos^{2} (x)

Aplicar la formula del binomio al cuadrado:

(a + b)^{2} = a^{2} + 2 ab + b^{2}

a = sin (x), b = cos (x)

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

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

Expandir

sec (x) (sin^{2} (x) + 2 sin (x) cos (x) + cos^{2} (x)) : sin^{2} (x) sec (x) + 2 sec (x) sin (x) cos (x) + cos^{2} (x) sec (x)

sec (x) (sin^{2} (x) + 2 sin (x) cos (x) + cos^{2} (x))

Aplicar la siguiente regla de productos notables

= sec (x) sin^{2} (x) + sec (x) \cdot 2 sin (x) cos (x) + sec (x) cos^{2} (x)

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

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

- (1 + 2 sin (x)) : - 1 - 2 sin (x)

- (1 + 2 sin (x))

Poner los parentesis

= - (1) - (2 sin (x))

Aplicar las reglas de los signos

+ (- a) = - a

= - 1 - 2 sin (x)

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

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

\frac{sin ^{2} ( x ) sec ( x ) + 2 sec ( x ) sin ( x ) cos ( x ) + cos ^{2} ( x ) sec ( x ) - 1 - 2 sin ( x )}{sec ( x )} = 0

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

sin^{2} (x) sec (x) + 2 sec (x) sin (x) cos (x) + cos^{2} (x) sec (x) - 1 - 2 sin (x) = 0

Expresar con seno, coseno

- 1 - 2 sin (x) + cos^{2} (x) sec (x) + sec (x) sin^{2} (x) + 2 cos (x) sec (x) sin (x)

Utilizar la identidad trigonométrica básica:

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

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

Simplificar

- 1 - 2 sin (x) + cos^{2} (x) \frac{1}{cos ( x )} + \frac{1}{cos ( x )} sin^{2} (x) + 2 cos (x) \frac{1}{cos ( x )} sin (x) : \frac{cos ^{2} ( x ) + sin ^{2} ( x ) - cos ( x )}{cos ( x )}

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

cos^{2} (x) \frac{1}{cos ( x )} = cos (x)

cos^{2} (x) \frac{1}{cos ( x )}

Multiplicar fracciones:

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

= \frac{1 \cdot cos ^{2} ( x )}{cos ( x )}

Multiplicar:

1 \cdot cos^{2} (x) = cos^{2} (x)

= \frac{cos ^{2} ( x )}{cos ( x )}

Eliminar los terminos comunes:

cos (x)

= cos (x)

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

\frac{1}{cos ( x )} sin^{2} (x)

Multiplicar fracciones:

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

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

Multiplicar:

1 \cdot sin^{2} (x) = sin^{2} (x)

= \frac{sin ^{2} ( x )}{cos ( x )}

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

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

Multiplicar fracciones:

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

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

Eliminar los terminos comunes:

cos (x)

= 1 \cdot 2 sin (x)

Multiplicar los numeros:

1 \cdot 2 = 2

= 2 sin (x)

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

Agrupar términos semejantes

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

Sumar elementos similares:

- 2 sin (x) + 2 sin (x) = 0

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

Convertir a fracción:

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

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

Ya que los denominadores son iguales, combinar las fracciones:

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

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

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

cos (x) cos (x) + sin^{2} (x) - 1 \cdot cos (x)

cos (x) cos (x) = cos^{2} (x)

cos (x) cos (x)

Aplicar las leyes de los exponentes:

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

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

= cos^{1 + 1} (x)

Sumar:

1 + 1 = 2

= cos^{2} (x)

1 \cdot cos (x) = cos (x)

1 \cdot cos (x)

Multiplicar:

1 \cdot cos (x) = cos (x)

= cos (x)

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

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

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

\frac{- cos ( x ) + cos ^{2} ( x ) + sin ^{2} ( x )}{cos ( x )} = 0

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

- cos (x) + cos^{2} (x) + sin^{2} (x) = 0

Re-escribir usando identidades trigonométricas

- cos (x) + cos^{2} (x) + sin^{2} (x)

Utilizar la identidad pitagórica:

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

= - cos (x) + 1

- cos (x) + 1 = 0

Desplace

1

a la derecha

- cos (x) + 1 = 0

Restar

1

de ambos lados

- cos (x) + 1 - 1 = 0 - 1

Simplificar

- cos (x) = - 1

- cos (x) = - 1

Dividir ambos lados entre

- 1

- cos (x) = - 1

Dividir ambos lados entre

- 1

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

Simplificar

cos (x) = 1

cos (x) = 1

Soluciones generales para

cos (x) = 1

cos (x)

tabla de valores periódicos con

2 πn

intervalos:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

x = 0 + 2 πn

x = 0 + 2 πn

Resolver

x = 0 + 2 πn : x = 2 πn

x = 0 + 2 πn

0 + 2 πn = 2 πn

x = 2 πn

x = 2 πn

Gráfica

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