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

Solución

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

Solución

A = \frac{π}{2} + 2 πn, A = \frac{3 π}{2} + 2 πn

Grados

A = 9 0^{\circ} + 36 0^{\circ} n, A = 27 0^{\circ} + 36 0^{\circ} n

Pasos de solución

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

Restar

\frac{1}{sin ^{2} ( A )}

de ambos lados

cos^{2} (A) + sin^{2} (A) - \frac{1}{sin ^{2} ( A )} = 0

Simplificar

cos^{2} (A) + sin^{2} (A) - \frac{1}{sin ^{2} ( A )} : \frac{cos ^{2} ( A ) sin ^{2} ( A ) + sin ^{4} ( A ) - 1}{sin ^{2} ( A )}

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

Convertir a fracción:

cos^{2} (A) = \frac{cos ^{2} ( A ) sin ^{2} ( A )}{sin ^{2} ( A )}, sin^{2} (A) = \frac{sin ^{2} ( A ) sin ^{2} ( A )}{sin ^{2} ( A )}

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

Ya que los denominadores son iguales, combinar las fracciones:

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

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

cos^{2} (A) sin^{2} (A) + sin^{2} (A) sin^{2} (A) - 1 = cos^{2} (A) sin^{2} (A) + sin^{4} (A) - 1

cos^{2} (A) sin^{2} (A) + sin^{2} (A) sin^{2} (A) - 1

sin^{2} (A) sin^{2} (A) = sin^{4} (A)

sin^{2} (A) sin^{2} (A)

Aplicar las leyes de los exponentes:

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

sin^{2} (A) sin^{2} (A) = sin^{2 + 2} (A)

= sin^{2 + 2} (A)

Sumar:

2 + 2 = 4

= sin^{4} (A)

= cos^{2} (A) sin^{2} (A) + sin^{4} (A) - 1

= \frac{cos ^{2} ( A ) sin ^{2} ( A ) + sin ^{4} ( A ) - 1}{sin ^{2} ( A )}

\frac{cos ^{2} ( A ) sin ^{2} ( A ) + sin ^{4} ( A ) - 1}{sin ^{2} ( A )} = 0

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

cos^{2} (A) sin^{2} (A) + sin^{4} (A) - 1 = 0

Re-escribir usando identidades trigonométricas

- 1 + sin^{4} (A) + cos^{2} (A) sin^{2} (A)

Utilizar la identidad pitagórica:

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

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

= - 1 + sin^{4} (A) + (1 - sin^{2} (A)) sin^{2} (A)

Simplificar

- 1 + sin^{4} (A) + (1 - sin^{2} (A)) sin^{2} (A) : sin^{2} (A) - 1

- 1 + sin^{4} (A) + (1 - sin^{2} (A)) sin^{2} (A)

= - 1 + sin^{4} (A) + sin^{2} (A) (1 - sin^{2} (A))

Expandir

sin^{2} (A) (1 - sin^{2} (A)) : sin^{2} (A) - sin^{4} (A)

sin^{2} (A) (1 - sin^{2} (A))

Poner los parentesis utilizando:

a (b - c) = ab - a c

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

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

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

Simplificar

1 \cdot sin^{2} (A) - sin^{2} (A) sin^{2} (A) : sin^{2} (A) - sin^{4} (A)

1 \cdot sin^{2} (A) - sin^{2} (A) sin^{2} (A)

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

1 \cdot sin^{2} (A)

Multiplicar:

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

= sin^{2} (A)

sin^{2} (A) sin^{2} (A) = sin^{4} (A)

sin^{2} (A) sin^{2} (A)

Aplicar las leyes de los exponentes:

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

sin^{2} (A) sin^{2} (A) = sin^{2 + 2} (A)

= sin^{2 + 2} (A)

Sumar:

2 + 2 = 4

= sin^{4} (A)

= sin^{2} (A) - sin^{4} (A)

= sin^{2} (A) - sin^{4} (A)

= - 1 + sin^{4} (A) + sin^{2} (A) - sin^{4} (A)

Simplificar

- 1 + sin^{4} (A) + sin^{2} (A) - sin^{4} (A) : sin^{2} (A) - 1

- 1 + sin^{4} (A) + sin^{2} (A) - sin^{4} (A)

Agrupar términos semejantes

= sin^{4} (A) + sin^{2} (A) - sin^{4} (A) - 1

Sumar elementos similares:

sin^{4} (A) - sin^{4} (A) = 0

= sin^{2} (A) - 1

= sin^{2} (A) - 1

= sin^{2} (A) - 1

- 1 + sin^{2} (A) = 0

Usando el método de sustitución

- 1 + sin^{2} (A) = 0

Sea:

sin (A) = u

- 1 + u^{2} = 0

- 1 + u^{2} = 0 : u = 1, u = - 1

- 1 + u^{2} = 0

Mover

1

al lado derecho

- 1 + u^{2} = 0

Sumar

1

a ambos lados

- 1 + u^{2} + 1 = 0 + 1

Simplificar

u^{2} = 1

u^{2} = 1

Para

x^{2} = f (a)

las soluciones son

x = f (a), - f (a)

u = 1, u = - 1

1 = 1

1

Aplicar la regla

1 = 1

= 1

- 1 = - 1

- 1

Aplicar la regla

1 = 1

= - 1

u = 1, u = - 1

Sustituir en la ecuación

u = sin (A)

sin (A) = 1, sin (A) = - 1

sin (A) = 1, sin (A) = - 1

sin (A) = 1 : A = \frac{π}{2} + 2 πn

sin (A) = 1

Soluciones generales para

sin (A) = 1

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 = \frac{π}{2} + 2 πn

A = \frac{π}{2} + 2 πn

sin (A) = - 1 : A = \frac{3 π}{2} + 2 πn

sin (A) = - 1

Soluciones generales para

sin (A) = - 1

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 = \frac{3 π}{2} + 2 πn

A = \frac{3 π}{2} + 2 πn

Combinar toda las soluciones

A = \frac{π}{2} + 2 πn, A = \frac{3 π}{2} + 2 πn

Ejemplos populares

cos(4θ)=1

cos (4 θ) = 1

tan(2x)+sec(2x)=3

tan (2 x) + sec (2 x) = 3

6sin(x)-5cos(x)=7

6 sin (x) - 5 cos (x) = 7

sin(x)tan(x)=1

sin (x) tan (x) = 1

cos(x)= 1/2 ,0<= x<2pi

cos (x) = \frac{1}{2}, 0 \leq x < 2 π