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Popular >

cot^2(x)-cos^2(x)=-1

Solución

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

Solución

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

Pasos de solución

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

Restar

- 1

de ambos lados

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

Re-escribir usando identidades trigonométricas

1 - cos^{2} (x) + cot^{2} (x)

Utilizar la identidad pitagórica:

1 + cot^{2} (x) = csc^{2} (x)

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

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

Factorizar

- cos^{2} (x) + csc^{2} (x) : (csc (x) + cos (x)) (csc (x) - cos (x))

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

Aplicar la siguiente regla para binomios al cuadrado:

x^{2} - y^{2} = (x + y) (x - y)

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

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

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

Resolver cada parte por separado

csc (x) + cos (x) = 0 or csc (x) - cos (x) = 0

csc (x) + cos (x) = 0 :

Sin solución

csc (x) + cos (x) = 0

Expresar con seno, coseno

cos (x) + csc (x)

Utilizar la identidad trigonométrica básica:

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

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

Simplificar

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

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

Convertir a fracción:

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

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

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

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

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

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

Re-escribir usando identidades trigonométricas

1 + cos (x) sin (x)

Utilizar la identidad trigonométrica del ángulo doble:

2 sin (x) cos (x) = sin (2 x)

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

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

1 + \frac{sin ( 2 x )}{2} = 0

Mover

1

al lado derecho

1 + \frac{sin ( 2 x )}{2} = 0

Restar

1

de ambos lados

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

Simplificar

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

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

Multiplicar ambos lados por

2

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

Multiplicar ambos lados por

2

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

Simplificar

sin (2 x) = - 2

sin (2 x) = - 2

- 1 \leq sin (x) \leq 1

Sin soluci \overset{o}{ˊ} n

csc (x) - cos (x) = 0 :

Sin solución

csc (x) - cos (x) = 0

Expresar con seno, coseno

- cos (x) + csc (x)

Utilizar la identidad trigonométrica básica:

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

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

Simplificar

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

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

Convertir a fracción:

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

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

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

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

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

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

Re-escribir usando identidades trigonométricas

1 - cos (x) sin (x)

Utilizar la identidad trigonométrica del ángulo doble:

2 sin (x) cos (x) = sin (2 x)

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

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

1 - \frac{sin ( 2 x )}{2} = 0

Mover

1

al lado derecho

1 - \frac{sin ( 2 x )}{2} = 0

Restar

1

de ambos lados

1 - \frac{sin ( 2 x )}{2} - 1 = 0 - 1

Simplificar

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

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

Multiplicar ambos lados por

2

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

Multiplicar ambos lados por

2

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

Simplificar

- sin (2 x) = - 2

- sin (2 x) = - 2

Dividir ambos lados entre

- 1

- sin (2 x) = - 2

Dividir ambos lados entre

- 1

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

Simplificar

sin (2 x) = 2

sin (2 x) = 2

- 1 \leq sin (x) \leq 1

Sin soluci \overset{o}{ˊ} n

Combinar toda las soluciones

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

Ejemplos populares

sin^2(θ)=2*sin(θ)+3

sin^{2} (θ) = 2 \cdot sin (θ) + 3

tan(3x)=1,-2pi<= x<= 2pi

tan (3 x) = 1, - 2 π \leq x \leq 2 π

sin(x)=(-3)/2

sin (x) = \frac{- 3}{2}

cos(x)sin(2x)+3sin^2(x)=-3sin(x)

cos (x) sin (2 x) + 3 sin^{2} (x) = - 3 sin (x)

sin^2(x)=-cos(x)

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