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Popular Trigonometría >

(2cos(x)-1)*(2sin(x)+1)=3-4sin^2(x)

Solución

(2 cos (x) - 1) \cdot (2 sin (x) + 1) = 3 - 4 sin^{2} (x)

Solución

x = \frac{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn, x = \frac{π}{4} + πn

Grados

x = 6 0^{\circ} + 36 0^{\circ} n, x = 30 0^{\circ} + 36 0^{\circ} n, x = 4 5^{\circ} + 18 0^{\circ} n

Pasos de solución

(2 cos (x) - 1) (2 sin (x) + 1) = 3 - 4 sin^{2} (x)

Restar

3 - 4 sin^{2} (x)

de ambos lados

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

Re-escribir usando identidades trigonométricas

- 4 + 2 cos (x) - 2 sin (x) + 4 sin^{2} (x) + 4 cos (x) sin (x)

Utilizar la identidad pitagórica:

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

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

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

Simplificar

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

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

Expandir

4 (1 - cos^{2} (x)) : 4 - 4 cos^{2} (x)

4 (1 - cos^{2} (x))

Poner los parentesis utilizando:

a (b - c) = ab - a c

a = 4, b = 1, c = cos^{2} (x)

= 4 \cdot 1 - 4 cos^{2} (x)

Multiplicar los numeros:

4 \cdot 1 = 4

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

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

Simplificar

- 4 + 2 cos (x) - 2 sin (x) + 4 - 4 cos^{2} (x) + 4 cos (x) sin (x) : 2 cos (x) + 4 cos (x) sin (x) - 4 cos^{2} (x) - 2 sin (x)

- 4 + 2 cos (x) - 2 sin (x) + 4 - 4 cos^{2} (x) + 4 cos (x) sin (x)

Agrupar términos semejantes

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

- 4 + 4 = 0

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

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

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

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

Factorizar

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

2 cos (x) - 2 sin (x) - 4 cos^{2} (x) + 4 cos (x) sin (x)

Reescribir

4

como

2 \cdot 2

Reescribir

- 4

como

2 \cdot 2

= 2 cos (x) - 2 sin (x) - 2 \cdot 2 cos^{2} (x) + 2 \cdot 2 sin (x) cos (x)

Factorizar el termino común

2

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

Factorizar

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

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

Factorizar

cos (x) - 2 cos^{2} (x) : cos (x) (1 - 2 cos (x))

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

Aplicar las leyes de los exponentes:

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

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

= cos (x) - 2 cos (x) cos (x)

Factorizar el termino común

cos (x)

= cos (x) (1 - 2 cos (x))

Factorizar

- sin (x) + 2 sin (x) cos (x) : sin (x) (- 1 + 2 cos (x))

- sin (x) + 2 sin (x) cos (x)

Factorizar el termino común

sin (x)

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

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

Reescribir como

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

Factorizar el termino común

(1 - 2 cos (x))

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

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

Factorizar

1 - 2 cos (x) : - (2 cos (x) - 1)

1 - 2 cos (x)

Factorizar el termino común

- 1

= - (2 cos (x) - 1)

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

- 2 (2 cos (x) - 1) (cos (x) - sin (x)) = 0

Resolver cada parte por separado

2 cos (x) - 1 = 0 or cos (x) - sin (x) = 0

2 cos (x) - 1 = 0 : x = \frac{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

2 cos (x) - 1 = 0

Desplace

1

a la derecha

2 cos (x) - 1 = 0

Sumar

1

a ambos lados

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

Simplificar

2 cos (x) = 1

2 cos (x) = 1

Dividir ambos lados entre

2

2 cos (x) = 1

Dividir ambos lados entre

2

\frac{2 cos ( x )}{2} = \frac{1}{2}

Simplificar

cos (x) = \frac{1}{2}

cos (x) = \frac{1}{2}

Soluciones generales para

cos (x) = \frac{1}{2}

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}

x = \frac{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

x = \frac{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

cos (x) - sin (x) = 0 : x = \frac{π}{4} + πn

cos (x) - sin (x) = 0

Re-escribir usando identidades trigonométricas

cos (x) - sin (x) = 0

Dividir ambos lados entre

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

\frac{cos ( x ) - sin ( x )}{cos ( x )} = \frac{0}{cos ( x )}

Simplificar

1 - \frac{sin ( x )}{cos ( x )} = 0

Utilizar la identidad trigonométrica básica:

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

1 - tan (x) = 0

1 - tan (x) = 0

Desplace

1

a la derecha

1 - tan (x) = 0

Restar

1

de ambos lados

1 - tan (x) - 1 = 0 - 1

Simplificar

- tan (x) = - 1

- tan (x) = - 1

Dividir ambos lados entre

- 1

- tan (x) = - 1

Dividir ambos lados entre

- 1

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

Simplificar

tan (x) = 1

tan (x) = 1

Soluciones generales para

tan (x) = 1

tan (x)

tabla de valores periódicos con

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} tan (x) 0 \frac{3}{3} 13 \pm \infty - 3 - 1 - \frac{3}{3}

x = \frac{π}{4} + πn

x = \frac{π}{4} + πn

Combinar toda las soluciones

x = \frac{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn, x = \frac{π}{4} + πn

Gráfica

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