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sin^2(x)-tan(x)=0

Solución

sin^{2} (x) - tan (x) = 0

Solución

x = 2 πn, x = π + 2 πn

Grados

x = 0^{\circ} + 36 0^{\circ} n, x = 18 0^{\circ} + 36 0^{\circ} n

Pasos de solución

sin^{2} (x) - tan (x) = 0

Expresar con seno, coseno

sin^{2} (x) - tan (x)

Utilizar la identidad trigonométrica básica:

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

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

Simplificar

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

sin^{2} (x) - \frac{sin ( x )}{cos ( x )}

Convertir a fracción:

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

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

Ya que los denominadores son iguales, combinar las fracciones:

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

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

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

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

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

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

Factorizar

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

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

Aplicar las leyes de los exponentes:

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

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

= - sin (x) + sin (x) sin (x)

Factorizar el termino común

sin (x)

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

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

Resolver cada parte por separado

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

sin (x) = 0 : x = 2 πn, x = π + 2 πn

sin (x) = 0

Soluciones generales para

sin (x) = 0

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} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{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} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

x = 0 + 2 πn, x = π + 2 πn

x = 0 + 2 πn, x = π + 2 πn

Resolver

x = 0 + 2 πn : x = 2 πn

x = 0 + 2 πn

0 + 2 πn = 2 πn

x = 2 πn

x = 2 πn, x = π + 2 πn

- 1 + sin (x) cos (x) = 0 :

Sin solución

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

Re-escribir usando identidades trigonométricas

- 1 + sin (x) cos (x)

Utilizar la identidad trigonométrica del ángulo doble:

2 sin (x) cos (x) = sin (2 x)

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

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

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

Mover

1

al lado derecho

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

Sumar

1

a ambos lados

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

Simplificar

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

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

Multiplicar ambos lados por

2

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

Multiplicar ambos lados por

2

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

Simplificar

sin (2 x) = 2

sin (2 x) = 2

- 1 \leq sin (x) \leq 1

Sin soluci \overset{o}{ˊ} n

Combinar toda las soluciones

x = 2 πn, x = π + 2 πn

Ejemplos populares

6sin^3(x)+7sin^2(x)-sin(x)-2=0

6 sin^{3} (x) + 7 sin^{2} (x) - sin (x) - 2 = 0

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

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

tan(x)= 1/(sec(x))

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

2*cos(2x)-1=0

2 \cdot cos (2 x) - 1 = 0

sin^2(x)=((7m-2))/8

sin^{2} (x) = \frac{( 7 m - 2 )}{8}