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Popular Trigonometría >

sin^2(x)=(tan(x))/2

Solución

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

Solución

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

Grados

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

Pasos de solución

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

Restar

\frac{tan ( x )}{2}

de ambos lados

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

Simplificar

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

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

Convertir a fracción:

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

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

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 ) \cdot 2 - tan ( x )}{2}

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

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

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

Expresar con seno, coseno

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

Utilizar la identidad trigonométrica básica:

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

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

Simplificar

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

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

Convertir a fracción:

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

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

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

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

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

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

Factorizar

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

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

Factorizar el termino común

sin (x)

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

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

Resolver cada parte por separado

sin (x) = 0 or - 1 + 2 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 + 2 sin (x) cos (x) = 0 : x = \frac{π}{4} + πn

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

Re-escribir usando identidades trigonométricas

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

Utilizar la identidad trigonométrica del ángulo doble:

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

= - 1 + sin (2 x)

- 1 + sin (2 x) = 0

Desplace

1

a la derecha

- 1 + sin (2 x) = 0

Sumar

1

a ambos lados

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

Simplificar

sin (2 x) = 1

sin (2 x) = 1

Soluciones generales para

sin (2 x) = 1

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}

2 x = \frac{π}{2} + 2 πn

2 x = \frac{π}{2} + 2 πn

Resolver

2 x = \frac{π}{2} + 2 πn : x = \frac{π}{4} + πn

2 x = \frac{π}{2} + 2 πn

Dividir ambos lados entre

2

2 x = \frac{π}{2} + 2 πn

Dividir ambos lados entre

2

\frac{2 x}{2} = \frac{\frac{π}{2}}{2} + \frac{2 πn}{2}

Simplificar

\frac{2 x}{2} = \frac{\frac{π}{2}}{2} + \frac{2 πn}{2}

Simplificar

\frac{2 x}{2} : x

\frac{2 x}{2}

Dividir:

\frac{2}{2} = 1

= x

Simplificar

\frac{\frac{π}{2}}{2} + \frac{2 πn}{2} : \frac{π}{4} + πn

\frac{\frac{π}{2}}{2} + \frac{2 πn}{2}

\frac{\frac{π}{2}}{2} = \frac{π}{4}

\frac{\frac{π}{2}}{2}

Aplicar las propiedades de las fracciones:

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

= \frac{π}{2 \cdot 2}

Multiplicar los numeros:

2 \cdot 2 = 4

= \frac{π}{4}

\frac{2 πn}{2} = πn

\frac{2 πn}{2}

Dividir:

\frac{2}{2} = 1

= πn

= \frac{π}{4} + πn

x = \frac{π}{4} + πn

x = \frac{π}{4} + πn

x = \frac{π}{4} + πn

x = \frac{π}{4} + πn

Combinar toda las soluciones

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

Gráfica

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