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csc(x)tan(x)=(2sqrt(3))/3

Solución

csc (x) tan (x) = \frac{2 3}{3}

Solución

x = \frac{π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

Grados

x = 3 0^{\circ} + 36 0^{\circ} n, x = 33 0^{\circ} + 36 0^{\circ} n

Pasos de solución

csc (x) tan (x) = \frac{2 3}{3}

Restar

\frac{2 3}{3}

de ambos lados

csc (x) tan (x) - \frac{2}{3} = 0

Simplificar

csc (x) tan (x) - \frac{2}{3} : \frac{3 csc ( x ) tan ( x ) - 2}{3}

csc (x) tan (x) - \frac{2}{3}

Convertir a fracción:

csc (x) tan (x) = \frac{csc ( x ) tan ( x ) 3}{3}

= \frac{csc ( x ) tan ( x ) 3}{3} - \frac{2}{3}

Ya que los denominadores son iguales, combinar las fracciones:

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

= \frac{csc ( x ) tan ( x ) 3 - 2}{3}

\frac{3 csc ( x ) tan ( x ) - 2}{3} = 0

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

3 csc (x) tan (x) - 2 = 0

Expresar con seno, coseno

- 2 + csc (x) 3 tan (x)

Utilizar la identidad trigonométrica básica:

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

= - 2 + \frac{1}{sin ( x )} 3 tan (x)

Utilizar la identidad trigonométrica básica:

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

= - 2 + \frac{1}{sin ( x )} 3 \frac{sin ( x )}{cos ( x )}

Simplificar

- 2 + \frac{1}{sin ( x )} 3 \frac{sin ( x )}{cos ( x )} : \frac{- 2 cos ( x ) + 3}{cos ( x )}

- 2 + \frac{1}{sin ( x )} 3 \frac{sin ( x )}{cos ( x )}

\frac{1}{sin ( x )} 3 \frac{sin ( x )}{cos ( x )} = \frac{3}{cos ( x )}

\frac{1}{sin ( x )} 3 \frac{sin ( x )}{cos ( x )}

Multiplicar fracciones:

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

= \frac{1 \cdot sin ( x ) 3}{sin ( x ) cos ( x )}

Eliminar los terminos comunes:

sin (x)

= \frac{1 \cdot 3}{cos ( x )}

Multiplicar:

1 \cdot 3 = 3

= \frac{3}{cos ( x )}

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

Convertir a fracción:

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

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

Ya que los denominadores son iguales, combinar las fracciones:

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

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

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

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

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

3 - 2 cos (x) = 0

Mover

3

al lado derecho

3 - 2 cos (x) = 0

Restar

3

de ambos lados

3 - 2 cos (x) - 3 = 0 - 3

Simplificar

- 2 cos (x) = - 3

- 2 cos (x) = - 3

Dividir ambos lados entre

- 2

- 2 cos (x) = - 3

Dividir ambos lados entre

- 2

\frac{- 2 cos ( x )}{- 2} = \frac{- 3}{- 2}

Simplificar

cos (x) = \frac{3}{2}

cos (x) = \frac{3}{2}

Soluciones generales para

cos (x) = \frac{3}{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}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

x = \frac{π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

Ejemplos populares

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

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

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

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

cos^2(2x)-1/4 =0

cos^{2} (2 x) - \frac{1}{4} = 0

cosh(x)= 17/15

cosh (x) = \frac{17}{15}

4cos(x)+2sqrt(3)=0

4 cos (x) + 23 = 0