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Popular >

sin^2(a)sec(a)=tan^2(a)

Solución

sin^{2} (a) sec (a) = tan^{2} (a)

Solución

a = 2 πn, a = π + 2 πn

Grados

a = 0^{\circ} + 36 0^{\circ} n, a = 18 0^{\circ} + 36 0^{\circ} n

Pasos de solución

sin^{2} (a) sec (a) = tan^{2} (a)

Restar

tan^{2} (a)

de ambos lados

sin^{2} (a) sec (a) - tan^{2} (a) = 0

Expresar con seno, coseno

- tan^{2} (a) + sec (a) sin^{2} (a)

Utilizar la identidad trigonométrica básica:

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

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

Utilizar la identidad trigonométrica básica:

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

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

Simplificar

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

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

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

(\frac{sin ( a )}{cos ( a )})^{2}

Aplicar las leyes de los exponentes:

(\frac{a}{b})^{c} = \frac{a ^{c}}{b ^{c}}

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

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

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

Multiplicar fracciones:

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

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

Multiplicar:

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

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

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

Mínimo común múltiplo de

cos^{2} (a), cos (a) : cos^{2} (a)

cos^{2} (a), cos (a)

Mínimo común múltiplo (MCM)

Calcular una expresión que este compuesta de factores que aparezcan tanto en

cos^{2} (a)

cos (a)

= cos^{2} (a)

Reescribir las fracciones basandose en el mínimo común denominador

Multiplicar cada numerador por la misma cantidad necesaria para multiplicar el denominador correspondiente y convertirlo en el mínimo común denominador

Para

\frac{sin ^{2} ( a )}{cos ( a )} :

multiplicar el denominador y el numerador por

cos (a)

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

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

Ya que los denominadores son iguales, combinar las fracciones:

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

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

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

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

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

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

Factorizar

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

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

Factorizar el termino común

sin^{2} (a)

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

sin^{2} (a) (cos (a) - 1) = 0

Resolver cada parte por separado

sin^{2} (a) = 0 or cos (a) - 1 = 0

sin^{2} (a) = 0 : a = 2 πn, a = π + 2 πn

sin^{2} (a) = 0

Aplicar la regla

x^{n} = 0 \Rightarrow x = 0

sin (a) = 0

Soluciones generales para

sin (a) = 0

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} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{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} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

a = 0 + 2 πn, a = π + 2 πn

a = 0 + 2 πn, a = π + 2 πn

Resolver

a = 0 + 2 πn : a = 2 πn

a = 0 + 2 πn

0 + 2 πn = 2 πn

a = 2 πn

a = 2 πn, a = π + 2 πn

cos (a) - 1 = 0 : a = 2 πn

cos (a) - 1 = 0

Mover

1

al lado derecho

cos (a) - 1 = 0

Sumar

1

a ambos lados

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

Simplificar

cos (a) = 1

cos (a) = 1

Soluciones generales para

cos (a) = 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}

a = 0 + 2 πn

a = 0 + 2 πn

Resolver

a = 0 + 2 πn : a = 2 πn

a = 0 + 2 πn

0 + 2 πn = 2 πn

a = 2 πn

a = 2 πn

Combinar toda las soluciones

a = 2 πn, a = π + 2 πn

Ejemplos populares

tan^2(θ)-sec(θ)=1

tan^{2} (θ) - sec (θ) = 1

sin(x)= 8/10

sin (x) = \frac{8}{10}

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

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

7cos(5x)+9=12

7 cos (5 x) + 9 = 12

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

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