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sin^2(a)=1-cos(2a)

Solución

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

Restar

1 - cos (2 a)

de ambos lados

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

Re-escribir usando identidades trigonométricas

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

Utilizar la identidad trigonométrica del ángulo doble:

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

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

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

Simplificar

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

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

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

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

Poner los parentesis

= - (cos (2 a)) - (- cos^{2} (a))

Aplicar las reglas de los signos

- (- a) = a, - (a) = - a

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

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

Sumar elementos similares:

cos (2 a) - cos (2 a) = 0

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

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

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

Usando el método de sustitución

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

Sea:

cos (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 = cos (a)

cos (a) = 1, cos (a) = - 1

cos (a) = 1, cos (a) = - 1

cos (a) = 1 : a = 2 πn

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

cos (a) = - 1 : a = π + 2 πn

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 = π + 2 πn

a = π + 2 πn

Combinar toda las soluciones

a = 2 πn, a = π + 2 πn

Ejemplos populares

cos^4(x)=cos^2(x)

cos^{4} (x) = cos^{2} (x)

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

tan (x) = 2 + tan^{3} (x)

2*sin(a)=sin(3a)

2 \cdot sin (a) = sin (3 a)

(cos(x)+5sin(x))/(2-3)=0

\frac{cos ( x ) + 5 sin ( x )}{2 - 3} = 0

sin(y)= 2/3

sin (y) = \frac{2}{3}