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Popular >

((sec(x))/(cos(x)))^2=tan^2(x)

Solución

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

Solución

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

Pasos de solución

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

Restar

tan^{2} (x)

de ambos lados

\frac{sec ^{2} ( x )}{cos ^{2} ( x )} - tan^{2} (x) = 0

Simplificar

\frac{sec ^{2} ( x )}{cos ^{2} ( x )} - tan^{2} (x) : \frac{sec ^{2} ( x ) - tan ^{2} ( x ) cos ^{2} ( x )}{cos ^{2} ( x )}

\frac{sec ^{2} ( x )}{cos ^{2} ( x )} - tan^{2} (x)

Convertir a fracción:

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

= \frac{sec ^{2} ( x )}{cos ^{2} ( x )} - \frac{tan ^{2} ( x ) cos ^{2} ( x )}{cos ^{2} ( x )}

Ya que los denominadores son iguales, combinar las fracciones:

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

= \frac{sec ^{2} ( x ) - tan ^{2} ( x ) cos ^{2} ( x )}{cos ^{2} ( x )}

\frac{sec ^{2} ( x ) - tan ^{2} ( x ) cos ^{2} ( x )}{cos ^{2} ( x )} = 0

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

sec^{2} (x) - tan^{2} (x) cos^{2} (x) = 0

Factorizar

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

sec^{2} (x) - tan^{2} (x) cos^{2} (x)

Reescribir

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

como

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

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

Aplicar las leyes de los exponentes:

a^{m} b^{m} = (ab)^{m}

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

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

= sec^{2} (x) - (tan (x) cos (x))^{2}

Aplicar la siguiente regla para binomios al cuadrado:

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

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

= (sec (x) + tan (x) cos (x)) (sec (x) - tan (x) cos (x))

(sec (x) + tan (x) cos (x)) (sec (x) - tan (x) cos (x)) = 0

Resolver cada parte por separado

sec (x) + tan (x) cos (x) = 0 or sec (x) - tan (x) cos (x) = 0

sec (x) + tan (x) cos (x) = 0 :

Sin solución

sec (x) + tan (x) cos (x) = 0

Expresar con seno, coseno

sec (x) + cos (x) tan (x)

Utilizar la identidad trigonométrica básica:

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

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

Utilizar la identidad trigonométrica básica:

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

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

Simplificar

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

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

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

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

Multiplicar fracciones:

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

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

Eliminar los terminos comunes:

cos (x)

= sin (x)

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

Convertir a fracción:

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

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

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

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

sec (x) - tan (x) cos (x) = 0 :

Sin solución

sec (x) - tan (x) cos (x) = 0

Expresar con seno, coseno

sec (x) - cos (x) tan (x)

Utilizar la identidad trigonométrica básica:

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

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

Utilizar la identidad trigonométrica básica:

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

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

Simplificar

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

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

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

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

Multiplicar fracciones:

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

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

Eliminar los terminos comunes:

cos (x)

= sin (x)

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

Convertir a fracción:

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

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

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

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

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

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

sin(A)=0.53

sin (A) = 0.53

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tan (a) = 3, cos (a)