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Popular >

csc^2(x)=4cot^2(x)

Solución

csc^{2} (x) = 4 cot^{2} (x)

Solución

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

Grados

x = 12 0^{\circ} + 36 0^{\circ} n, x = 24 0^{\circ} + 36 0^{\circ} n, x = 6 0^{\circ} + 36 0^{\circ} n, x = 30 0^{\circ} + 36 0^{\circ} n

Pasos de solución

csc^{2} (x) = 4 cot^{2} (x)

Restar

4 cot^{2} (x)

de ambos lados

csc^{2} (x) - 4 cot^{2} (x) = 0

Factorizar

csc^{2} (x) - 4 cot^{2} (x) : (csc (x) + 2 cot (x)) (csc (x) - 2 cot (x))

csc^{2} (x) - 4 cot^{2} (x)

Reescribir

csc^{2} (x) - 4 cot^{2} (x)

como

csc^{2} (x) - (2 cot (x))^{2}

csc^{2} (x) - 4 cot^{2} (x)

Reescribir

4

como

2^{2}

= csc^{2} (x) - 2^{2} cot^{2} (x)

Aplicar las leyes de los exponentes:

a^{m} b^{m} = (ab)^{m}

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

= csc^{2} (x) - (2 cot (x))^{2}

= csc^{2} (x) - (2 cot (x))^{2}

Aplicar la siguiente regla para binomios al cuadrado:

x^{2} - y^{2} = (x + y) (x - y)

csc^{2} (x) - (2 cot (x))^{2} = (csc (x) + 2 cot (x)) (csc (x) - 2 cot (x))

= (csc (x) + 2 cot (x)) (csc (x) - 2 cot (x))

(csc (x) + 2 cot (x)) (csc (x) - 2 cot (x)) = 0

Resolver cada parte por separado

csc (x) + 2 cot (x) = 0 or csc (x) - 2 cot (x) = 0

csc (x) + 2 cot (x) = 0 : x = \frac{2 π}{3} + 2 πn, x = \frac{4 π}{3} + 2 πn

csc (x) + 2 cot (x) = 0

Expresar con seno, coseno

csc (x) + 2 cot (x)

Utilizar la identidad trigonométrica básica:

csc (x) = \frac{1}{sin ( x )}

= \frac{1}{sin ( x )} + 2 cot (x)

Utilizar la identidad trigonométrica básica:

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

= \frac{1}{sin ( x )} + 2 \cdot \frac{cos ( x )}{sin ( x )}

Simplificar

\frac{1}{sin ( x )} + 2 \cdot \frac{cos ( x )}{sin ( x )} : \frac{1 + 2 cos ( x )}{sin ( x )}

\frac{1}{sin ( x )} + 2 \cdot \frac{cos ( x )}{sin ( x )}

Multiplicar

2 \cdot \frac{cos ( x )}{sin ( x )} : \frac{2 cos ( x )}{sin ( x )}

2 \cdot \frac{cos ( x )}{sin ( x )}

Multiplicar fracciones:

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

= \frac{cos ( x ) \cdot 2}{sin ( x )}

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

Aplicar la regla

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

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

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

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

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

1 + 2 cos (x) = 0

Mover

1

al lado derecho

1 + 2 cos (x) = 0

Restar

1

de ambos lados

1 + 2 cos (x) - 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{2 π}{3} + 2 πn, x = \frac{4 π}{3} + 2 πn

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

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

csc (x) - 2 cot (x) = 0

Expresar con seno, coseno

csc (x) - 2 cot (x)

Utilizar la identidad trigonométrica básica:

csc (x) = \frac{1}{sin ( x )}

= \frac{1}{sin ( x )} - 2 cot (x)

Utilizar la identidad trigonométrica básica:

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

= \frac{1}{sin ( x )} - 2 \cdot \frac{cos ( x )}{sin ( x )}

Simplificar

\frac{1}{sin ( x )} - 2 \cdot \frac{cos ( x )}{sin ( x )} : \frac{1 - 2 cos ( x )}{sin ( x )}

\frac{1}{sin ( x )} - 2 \cdot \frac{cos ( x )}{sin ( x )}

Multiplicar

2 \cdot \frac{cos ( x )}{sin ( x )} : \frac{2 cos ( x )}{sin ( x )}

2 \cdot \frac{cos ( x )}{sin ( x )}

Multiplicar fracciones:

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

= \frac{cos ( x ) \cdot 2}{sin ( x )}

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

Aplicar la regla

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

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

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

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

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

1 - 2 cos (x) = 0

Mover

1

al lado derecho

1 - 2 cos (x) = 0

Restar

1

de ambos lados

1 - 2 cos (x) - 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

Combinar toda las soluciones

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

Ejemplos populares

cot^2(3x)-tan^2(3x)=0

cot^{2} (3 x) - tan^{2} (3 x) = 0

solvefor x,cos(x)=10 resolver para

x, cos (x) = 10

3cos(θ)=2,1

3 cos (θ) = 2, 1

3csc(x)-7=0

3 csc (x) - 7 = 0

tan(2θ)=0.6583

tan (2 θ) = 0.6583