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Popular Trigonometría >

solvefor x,sin(x)+cos(y)=sin(x)cos(y)

Solución

resolver para

x, sin (x) + cos (y) = sin (x) cos (y)

Solución

x = arccsc (\frac{cos ( y ) - 1}{cos ( y )}) + 2 πn, x = π + arccsc (- \frac{cos ( y ) - 1}{cos ( y )}) + 2 πn

Pasos de solución

sin (x) + cos (y) = sin (x) cos (y)

Restar

sin (x) cos (y)

de ambos lados

sin (x) + cos (y) - cos (y) sin (x) = 0

Re-escribir usando identidades trigonométricas

cos (y) + sin (x) - cos (y) sin (x)

Utilizar la identidad trigonométrica básica:

sin (x) = \frac{1}{csc ( x )}

= cos (y) + \frac{1}{csc ( x )} - cos (y) \frac{1}{csc ( x )}

Simplificar

cos (y) + \frac{1}{csc ( x )} - cos (y) \frac{1}{csc ( x )} : cos (y) + \frac{1 - cos ( y )}{csc ( x )}

cos (y) + \frac{1}{csc ( x )} - cos (y) \frac{1}{csc ( x )}

cos (y) \frac{1}{csc ( x )} = \frac{cos ( y )}{csc ( x )}

cos (y) \frac{1}{csc ( x )}

Multiplicar fracciones:

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

= \frac{1 \cdot cos ( y )}{csc ( x )}

Multiplicar:

1 \cdot cos (y) = cos (y)

= \frac{cos ( y )}{csc ( x )}

= cos (y) + \frac{1}{csc ( x )} - \frac{cos ( y )}{csc ( x )}

Combinar las fracciones usando el mínimo común denominador:

\frac{1 - cos ( y )}{csc ( x )}

Aplicar la regla

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

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

= cos (y) + \frac{- cos ( y ) + 1}{csc ( x )}

= cos (y) + \frac{1 - cos ( y )}{csc ( x )}

cos (y) + \frac{1 - cos ( y )}{csc ( x )} = 0

Multiplicar ambos lados por

csc (x)

cos (y) + \frac{1 - cos ( y )}{csc ( x )} = 0

Multiplicar ambos lados por

csc (x)

cos (y) csc (x) + \frac{1 - cos ( y )}{csc ( x )} csc (x) = 0 \cdot csc (x)

Simplificar

cos (y) csc (x) + \frac{1 - cos ( y )}{csc ( x )} csc (x) = 0 \cdot csc (x)

Simplificar

\frac{1 - cos ( y )}{csc ( x )} csc (x) : 1 - cos (y)

\frac{1 - cos ( y )}{csc ( x )} csc (x)

Multiplicar fracciones:

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

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

Eliminar los terminos comunes:

csc (x)

= 1 - cos (y)

Simplificar

0 \cdot csc (x) : 0

0 \cdot csc (x)

Aplicar la regla

0 \cdot a = 0

= 0

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

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

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

Desplace

cos (y)

a la derecha

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

Sumar

cos (y)

a ambos lados

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

Simplificar

cos (y) csc (x) + 1 = cos (y)

cos (y) csc (x) + 1 = cos (y)

Desplace

1

a la derecha

cos (y) csc (x) + 1 = cos (y)

Restar

1

de ambos lados

cos (y) csc (x) + 1 - 1 = cos (y) - 1

Simplificar

cos (y) csc (x) = cos (y) - 1

cos (y) csc (x) = cos (y) - 1

Dividir ambos lados entre

cos (y); y \neq = \frac{π}{2} + 2 πn, y \neq = \frac{3 π}{2} + 2 πn

cos (y) csc (x) = cos (y) - 1

Dividir ambos lados entre

cos (y); y \neq = \frac{π}{2} + 2 πn, y \neq = \frac{3 π}{2} + 2 πn

\frac{cos ( y ) csc ( x )}{cos ( y )} = \frac{cos ( y )}{cos ( y )} - \frac{1}{cos ( y )}; y \neq = \frac{π}{2} + 2 πn, y \neq = \frac{3 π}{2} + 2 πn

Simplificar

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

Simplificar

\frac{cos ( y ) csc ( x )}{cos ( y )} : csc (x)

\frac{cos ( y ) csc ( x )}{cos ( y )}

Eliminar los terminos comunes:

cos (y)

= csc (x)

Simplificar

\frac{cos ( y )}{cos ( y )} - \frac{1}{cos ( y )} : \frac{cos ( y ) - 1}{cos ( y )}

\frac{cos ( y )}{cos ( y )} - \frac{1}{cos ( y )}

Aplicar la regla

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

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

csc (x) = \frac{cos ( y ) - 1}{cos ( y )}; y \neq = \frac{π}{2} + 2 πn, y \neq = \frac{3 π}{2} + 2 πn

csc (x) = \frac{cos ( y ) - 1}{cos ( y )}; y \neq = \frac{π}{2} + 2 πn, y \neq = \frac{3 π}{2} + 2 πn

csc (x) = \frac{cos ( y ) - 1}{cos ( y )}; y \neq = \frac{π}{2} + 2 πn, y \neq = \frac{3 π}{2} + 2 πn

Aplicar propiedades trigonométricas inversas

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

Soluciones generales para

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

csc (x) = a \Rightarrow x = arccsc (a) + 2 πn, x = π + arccsc (a) + 2 πn

x = arccsc (\frac{cos ( y ) - 1}{cos ( y )}) + 2 πn, x = π + arccsc (- \frac{cos ( y ) - 1}{cos ( y )}) + 2 πn

x = arccsc (\frac{cos ( y ) - 1}{cos ( y )}) + 2 πn, x = π + arccsc (- \frac{cos ( y ) - 1}{cos ( y )}) + 2 πn

Gráfica

Ejemplos populares

solvefor x,z=arctan(x/y)

so l v e f or x, z = arctan (\frac{x}{y})

-1/2 =cos(θ)

- \frac{1}{2} = cos (θ)

4cos^2(x)-3cos(x)-2=0

4 cos^{2} (x) - 3 cos (x) - 2 = 0

sin(x)=cos(3x)

sin (x) = cos (3 x)

solvefor x,y=5sin(x/4)

so l v e f or x, y = 5 sin (\frac{x}{4})