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solvefor x,sin(x)+cot(x)cos(x)= 1/f

Solución

resolver para

x, sin (x) + cot (x) cos (x) = \frac{1}{f}

Solución

x = arcsin (f) + 2 πn, x = π + arcsin (- f) + 2 πn

Pasos de solución

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

Restar

\frac{1}{f}

de ambos lados

sin (x) + cot (x) cos (x) - \frac{1}{f} = 0

Simplificar

sin (x) + cot (x) cos (x) - \frac{1}{f} : \frac{f sin ( x ) + f cot ( x ) cos ( x ) - 1}{f}

sin (x) + cot (x) cos (x) - \frac{1}{f}

Convertir a fracción:

sin (x) = \frac{sin ( x ) f}{f}, cot (x) cos (x) = \frac{cot ( x ) cos ( x ) f}{f}

= \frac{sin ( x ) f}{f} + \frac{cot ( x ) cos ( x ) f}{f} - \frac{1}{f}

Ya que los denominadores son iguales, combinar las fracciones:

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

= \frac{sin ( x ) f + cot ( x ) cos ( x ) f - 1}{f}

\frac{f sin ( x ) + f cot ( x ) cos ( x ) - 1}{f} = 0

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

f sin (x) + f cot (x) cos (x) - 1 = 0

Re-escribir usando identidades trigonométricas

- 1 + sin (x) f + cos (x) cot (x) f

Utilizar la identidad trigonométrica básica:

cot (x) = \frac{cos ( x )}{sin ( x )}

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

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

cos (x) \frac{cos ( x )}{sin ( x )} f

Multiplicar fracciones:

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

= \frac{cos ( x ) cos ( x ) f}{sin ( x )}

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

cos (x) cos (x) f

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) f

Sumar:

1 + 1 = 2

= cos^{2} (x) f

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

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

Utilizar la identidad pitagórica:

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

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

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

Combinar las fracciones usando el mínimo común denominador:

\frac{f}{sin ( x )}

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

Convertir a fracción:

f sin (x) = \frac{sin ( x ) f sin ( x )}{sin ( x )}

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

Ya que los denominadores son iguales, combinar las fracciones:

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

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

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

(1 - sin^{2} (x)) f + sin (x) f sin (x)

sin (x) f sin (x) = f sin^{2} (x)

sin (x) f sin (x)

Aplicar las leyes de los exponentes:

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

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

= f sin^{1 + 1} (x)

Sumar:

1 + 1 = 2

= f sin^{2} (x)

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

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

Expandir

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

(1 - sin^{2} (x)) f + f sin^{2} (x)

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

Expandir

f (1 - sin^{2} (x)) : f - f sin^{2} (x)

f (1 - sin^{2} (x))

Poner los parentesis utilizando:

a (b - c) = ab - a c

a = f, b = 1, c = sin^{2} (x)

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

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

Multiplicar:

1 \cdot f = f

= f - f sin^{2} (x)

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

Sumar elementos similares:

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

= f

= \frac{f}{sin ( x )}

= \frac{f}{sin ( x )} - 1

- 1 + \frac{f}{sin ( x )} = 0

- 1 + \frac{f}{sin ( x )} = 0

Multiplicar ambos lados por

sin (x)

- 1 + \frac{f}{sin ( x )} = 0

Multiplicar ambos lados por

sin (x)

- 1 \cdot sin (x) + \frac{f}{sin ( x )} sin (x) = 0 \cdot sin (x)

Simplificar

- 1 \cdot sin (x) + \frac{f}{sin ( x )} sin (x) = 0 \cdot sin (x)

Simplificar

- 1 \cdot sin (x) : - sin (x)

- 1 \cdot sin (x)

Multiplicar:

1 \cdot sin (x) = sin (x)

= - sin (x)

Simplificar

\frac{f}{sin ( x )} sin (x) : f

\frac{f}{sin ( x )} sin (x)

Multiplicar fracciones:

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

= \frac{f sin ( x )}{sin ( x )}

Eliminar los terminos comunes:

sin (x)

= f

Simplificar

0 \cdot sin (x) : 0

0 \cdot sin (x)

Aplicar la regla

0 \cdot a = 0

= 0

- sin (x) + f = 0

- sin (x) + f = 0

- sin (x) + f = 0

Mover

f

al lado derecho

- sin (x) + f = 0

Restar

f

de ambos lados

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

Simplificar

- sin (x) = - f

- sin (x) = - f

Dividir ambos lados entre

- 1

- sin (x) = - f

Dividir ambos lados entre

- 1

\frac{- sin ( x )}{- 1} = \frac{- f}{- 1}

Simplificar

sin (x) = f

sin (x) = f

Aplicar propiedades trigonométricas inversas

sin (x) = f

Soluciones generales para

sin (x) = f

sin (x) = a \Rightarrow x = arcsin (a) + 2 πn, x = π + arcsin (a) + 2 πn

x = arcsin (f) + 2 πn, x = π + arcsin (- f) + 2 πn

x = arcsin (f) + 2 πn, x = π + arcsin (- f) + 2 πn

Ejemplos populares

1/(sin(x))=2

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

tan(x)csc(x)-2tan(x)=0

tan (x) csc (x) - 2 tan (x) = 0

1-2sin(θ)=0

1 - 2 sin (θ) = 0

cos(2θ)+3cos(θ)=4

cos (2 θ) + 3 cos (θ) = 4

0=2cos(x)-3sin(x)

0 = 2 cos (x) - 3 sin (x)