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cos(x)sin(x)<0.451

Solución

cos (x) sin (x) < 0.451

Solución

\frac{- π - 1.12437 \dots}{2} + πn < x < 0.56218 \dots + πn

+2

Notación de intervalos

(\frac{- π - 1.12437 \dots}{2} + πn, 0.56218 \dots + πn)

Notación decimal

- 2.13298 \dots + πn < x < 0.56218 \dots + πn

Pasos de solución

cos (x) sin (x) < 0.451

Usar la siguiente identidad:

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

Por lo tanto

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

\frac{sin ( 2 x )}{2} < 0.451

Multiplicar ambos lados por

2

\frac{sin ( 2 x )}{2} < 0.451

Multiplicar ambos lados por

2

\frac{2 sin ( 2 x )}{2} < 0.451 \cdot 2

Simplificar

sin (2 x) < 0.902

sin (2 x) < 0.902

Para

sin (x) < a

, si

- 1 < a \leq 1

entonces

- π - arcsin (a) + 2 πn < x < arcsin (a) + 2 πn

- π - arcsin (0.902) + 2 πn < 2 x < arcsin (0.902) + 2 πn

Pi

a < u < b

entonces

a < u and u < b

- π - arcsin (0.902) + 2 πn < 2 x and 2 x < arcsin (0.902) + 2 πn

- π - arcsin (0.902) + 2 πn < 2 x : x > \frac{- π - arcsin ( 0.902 )}{2} + πn

- π - arcsin (0.902) + 2 πn < 2 x

Intercambiar lados

2 x > - π - arcsin (0.902) + 2 πn

Dividir ambos lados entre

2

2 x > - π - arcsin (0.902) + 2 πn

Dividir ambos lados entre

2

\frac{2 x}{2} > - \frac{π}{2} - \frac{arcsin ( 0.902 )}{2} + \frac{2 πn}{2}

Simplificar

x > - \frac{π}{2} - \frac{arcsin ( 0.902 )}{2} + πn

Simplificar

- \frac{π}{2} - \frac{arcsin ( 0.902 )}{2} : \frac{- π - arcsin ( 0.902 )}{2}

- \frac{π}{2} - \frac{arcsin ( 0.902 )}{2}

Aplicar la regla

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

= \frac{- π - arcsin ( 0.902 )}{2}

x > \frac{- π - arcsin ( 0.902 )}{2} + πn

x > \frac{- π - arcsin ( 0.902 )}{2} + πn

2 x < arcsin (0.902) + 2 πn : x < \frac{arcsin ( 0.902 )}{2} + πn

2 x < arcsin (0.902) + 2 πn

Dividir ambos lados entre

2

2 x < arcsin (0.902) + 2 πn

Dividir ambos lados entre

2

\frac{2 x}{2} < \frac{arcsin ( 0.902 )}{2} + \frac{2 πn}{2}

Simplificar

x < \frac{arcsin ( 0.902 )}{2} + πn

x < \frac{arcsin ( 0.902 )}{2} + πn

Combinar los rangos

x > \frac{- π - arcsin ( 0.902 )}{2} + πn and x < \frac{arcsin ( 0.902 )}{2} + πn

Mezclar intervalos sobrepuestos

\frac{- π - 1.12437 \dots}{2} + πn < x < 0.56218 \dots + πn

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