English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

1+sin(2a)=sin^2(a)

Solución

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

Solución

a = \frac{π}{2} + 2 πn, a = \frac{3 π}{2} + 2 πn, a = - 0.46364 \dots + πn

Grados

a = 9 0^{\circ} + 36 0^{\circ} n, a = 27 0^{\circ} + 36 0^{\circ} n, a = - 26.56505 \dots^{\circ} + 18 0^{\circ} n

Pasos de solución

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

Restar

sin^{2} (a)

de ambos lados

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

Re-escribir usando identidades trigonométricas

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

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

1 + sin (2 a)

Utilizar la identidad pitagórica:

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

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

Utilizar la identidad trigonométrica del ángulo doble:

sin (2 a) = 2 sin (a) cos (a)

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

Aplicar la formula del binomio al cuadrado:

(a + b)^{2} = a^{2} + 2 ab + b^{2}

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

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

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

Simplificar

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

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

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

Aplicar la formula del binomio al cuadrado:

(a + b)^{2} = a^{2} + 2 ab + b^{2}

a = sin (a), b = cos (a)

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

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

Sumar elementos similares:

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

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

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

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

Factorizar

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

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

Aplicar las leyes de los exponentes:

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

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

= cos (a) cos (a) + 2 sin (a) cos (a)

Factorizar el termino común

cos (a)

= cos (a) (cos (a) + 2 sin (a))

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

Resolver cada parte por separado

cos (a) = 0 or cos (a) + 2 sin (a) = 0

cos (a) = 0 : a = \frac{π}{2} + 2 πn, a = \frac{3 π}{2} + 2 πn

cos (a) = 0

Soluciones generales para

cos (a) = 0

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

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

cos (a) + 2 sin (a) = 0 : a = arctan (- \frac{1}{2}) + πn

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

Re-escribir usando identidades trigonométricas

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

Dividir ambos lados entre

cos (a), cos (a) \neq = 0

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

Simplificar

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

Utilizar la identidad trigonométrica básica:

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

1 + 2 tan (a) = 0

1 + 2 tan (a) = 0

Mover

1

al lado derecho

1 + 2 tan (a) = 0

Restar

1

de ambos lados

1 + 2 tan (a) - 1 = 0 - 1

Simplificar

2 tan (a) = - 1

2 tan (a) = - 1

Dividir ambos lados entre

2

2 tan (a) = - 1

Dividir ambos lados entre

2

\frac{2 tan ( a )}{2} = \frac{- 1}{2}

Simplificar

tan (a) = - \frac{1}{2}

tan (a) = - \frac{1}{2}

Aplicar propiedades trigonométricas inversas

tan (a) = - \frac{1}{2}

Soluciones generales para

tan (a) = - \frac{1}{2}

tan (x) = - a \Rightarrow x = arctan (- a) + πn

a = arctan (- \frac{1}{2}) + πn

a = arctan (- \frac{1}{2}) + πn

Combinar toda las soluciones

a = \frac{π}{2} + 2 πn, a = \frac{3 π}{2} + 2 πn, a = arctan (- \frac{1}{2}) + πn

Mostrar soluciones en forma decimal

a = \frac{π}{2} + 2 πn, a = \frac{3 π}{2} + 2 πn, a = - 0.46364 \dots + πn

Ejemplos populares

((cos^3(a)))/((2cos^2(a)-1))=cos(a)

\frac{( cos ^{3} ( a ) )}{( 2 cos ^{2} ( a ) - 1 )} = cos (a)

cos(x-45)=0

cos (x - 4 5^{\circ}) = 0

7tan^2(x)-15=0

7 tan^{2} (x) - 15 = 0

1+cos^2(x)-2cos^2(x/2)=0

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

2cos^2(x)+5sin(x)=5

2 cos^{2} (x) + 5 sin (x) = 5