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csc^2(x)=(1/(cos(x)))^2

Solución

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

Solución

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

Grados

x = 13 5^{\circ} + 18 0^{\circ} n, x = 4 5^{\circ} + 18 0^{\circ} n

Pasos de solución

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

Restar

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

de ambos lados

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

Simplificar

csc^{2} (x) - \frac{1}{cos ^{2} ( x )} : \frac{csc ^{2} ( x ) cos ^{2} ( x ) - 1}{cos ^{2} ( x )}

csc^{2} (x) - \frac{1}{cos ^{2} ( x )}

Convertir a fracción:

csc^{2} (x) = \frac{csc ^{2} ( x ) cos ^{2} ( x )}{cos ^{2} ( x )}

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

Ya que los denominadores son iguales, combinar las fracciones:

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

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

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

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

csc^{2} (x) cos^{2} (x) - 1 = 0

Factorizar

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

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

Aplicar las leyes de los exponentes:

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

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

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

Aplicar la siguiente regla para binomios al cuadrado:

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

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

= (cos (x) csc (x) + 1) (cos (x) csc (x) - 1)

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

Resolver cada parte por separado

cos (x) csc (x) + 1 = 0 or cos (x) csc (x) - 1 = 0

cos (x) csc (x) + 1 = 0 : x = \frac{3 π}{4} + πn

cos (x) csc (x) + 1 = 0

Re-escribir usando identidades trigonométricas

cos (x) csc (x) + 1

csc (x) cos (x) = cot (x)

csc (x) cos (x)

Expresar con seno, coseno

csc (x) cos (x)

Utilizar la identidad trigonométrica básica:

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

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

Simplificar

\frac{1}{sin ( x )} cos (x) : \frac{cos ( x )}{sin ( x )}

\frac{1}{sin ( x )} cos (x)

Multiplicar fracciones:

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

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

Multiplicar:

1 \cdot cos (x) = cos (x)

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

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

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

Utilizar la identidad trigonométrica básica:

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

= cot (x)

= 1 + cot (x)

1 + cot (x) = 0

Mover

1

al lado derecho

1 + cot (x) = 0

Restar

1

de ambos lados

1 + cot (x) - 1 = 0 - 1

Simplificar

cot (x) = - 1

cot (x) = - 1

Soluciones generales para

cot (x) = - 1

cot (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} cot (x) \mp \infty 31 \frac{3}{3} 0 - \frac{3}{3} - 1 - 3

x = \frac{3 π}{4} + πn

x = \frac{3 π}{4} + πn

cos (x) csc (x) - 1 = 0 : x = \frac{π}{4} + πn

cos (x) csc (x) - 1 = 0

Re-escribir usando identidades trigonométricas

cos (x) csc (x) - 1

csc (x) cos (x) = cot (x)

csc (x) cos (x)

Expresar con seno, coseno

csc (x) cos (x)

Utilizar la identidad trigonométrica básica:

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

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

Simplificar

\frac{1}{sin ( x )} cos (x) : \frac{cos ( x )}{sin ( x )}

\frac{1}{sin ( x )} cos (x)

Multiplicar fracciones:

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

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

Multiplicar:

1 \cdot cos (x) = cos (x)

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

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

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

Utilizar la identidad trigonométrica básica:

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

= cot (x)

= - 1 + cot (x)

- 1 + cot (x) = 0

Mover

1

al lado derecho

- 1 + cot (x) = 0

Sumar

1

a ambos lados

- 1 + cot (x) + 1 = 0 + 1

Simplificar

cot (x) = 1

cot (x) = 1

Soluciones generales para

cot (x) = 1

cot (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} cot (x) \mp \infty 31 \frac{3}{3} 0 - \frac{3}{3} - 1 - 3

x = \frac{π}{4} + πn

x = \frac{π}{4} + πn

Combinar toda las soluciones

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

Ejemplos populares

-2cos^2(x)+3cos(x)=1

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2-10sec(x)=4-9sec(x)

2 - 10 sec (x) = 4 - 9 sec (x)

7=e^{2sin(x)}+3

7 = e^{2 s i n (x)} + 3

sin(u)=-0.9

sin (u) = - 0.9

-3/2 =-1/2 sec(x)

- \frac{3}{2} = - \frac{1}{2} sec (x)