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Popolare Trigonometria >

cos^2(2x)>(2sqrt(3))/4

Soluzione

cos^{2} (2 x) > \frac{2 3}{4}

Soluzione

- \frac{arccos ( \frac{3}{2} )}{2} + πn < x < \frac{arccos ( \frac{3}{2} )}{2} + πn or \frac{arccos ( - \frac{3}{2} )}{2} + πn < x < \frac{2 π - arccos ( - \frac{3}{2} )}{2} + πn

Notazione dell’intervallo

- \frac{arccos ( \frac{3}{2} )}{2} + πn, \frac{arccos ( \frac{3}{2} )}{2} + πn \cup \frac{arccos ( - \frac{3}{2} )}{2} + πn, \frac{2 π - arccos ( - \frac{3}{2} )}{2} + πn

Decimale

- 0.18736 \dots + πn < x < 0.18736 \dots + πn or 1.38342 \dots + πn < x < 1.75816 \dots + πn

Fasi della soluzione

cos^{2} (2 x) > \frac{2 3}{4}

Per

u^{n} > a

, se

n

è pari allora

cos (2 x) < - \frac{2 3}{4} or cos (2 x) > \frac{2 3}{4}

\frac{2 3}{4} = \frac{3}{2}

\frac{2 3}{4}

\frac{2 3}{4} = \frac{3}{2}

\frac{2 3}{4}

Cancella il fattore comune:

2

= \frac{3}{2}

= \frac{3}{2}

cos (2 x) < - \frac{3}{2} or cos (2 x) > \frac{3}{2}

cos (2 x) < - \frac{3}{2} : \frac{arccos ( - \frac{3}{2} )}{2} + πn < x < \frac{2 π - arccos ( - \frac{3}{2} )}{2} + πn

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

Per

cos (x) < a

, se

- 1 < a \leq 1

allora

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

arccos - \frac{3}{2} + 2 πn < 2 x < 2 π - arccos - \frac{3}{2} + 2 πn

a < u < b

allora

a < u and u < b

arccos - \frac{3}{2} + 2 πn < 2 x and 2 x < 2 π - arccos - \frac{3}{2} + 2 πn

arccos - \frac{3}{2} + 2 πn < 2 x : x > \frac{arccos ( - \frac{3}{2} )}{2} + πn

arccos - \frac{3}{2} + 2 πn < 2 x

Scambia i lati

2 x > arccos - \frac{3}{2} + 2 πn

Dividere entrambi i lati per

2

2 x > arccos - \frac{3}{2} + 2 πn

Dividere entrambi i lati per

2

\frac{2 x}{2} > \frac{arccos ( - \frac{3}{2} )}{2} + \frac{2 πn}{2}

Semplificare

x > \frac{arccos ( - \frac{3}{2} )}{2} + πn

x > \frac{arccos ( - \frac{3}{2} )}{2} + πn

2 x < 2 π - arccos - \frac{3}{2} + 2 πn : x < \frac{2 π - arccos ( - \frac{3}{2} )}{2} + πn

2 x < 2 π - arccos - \frac{3}{2} + 2 πn

Dividere entrambi i lati per

2

2 x < 2 π - arccos - \frac{3}{2} + 2 πn

Dividere entrambi i lati per

2

\frac{2 x}{2} < \frac{2 π}{2} - \frac{arccos ( - \frac{3}{2} )}{2} + \frac{2 πn}{2}

Semplificare

x < π - \frac{arccos ( - \frac{3}{2} )}{2} + πn

Semplificare

π - \frac{arccos ( - \frac{3}{2} )}{2} : \frac{2 π - arccos ( - \frac{3}{2} )}{2}

π - \frac{arccos ( - \frac{3}{2} )}{2}

Converti l'elemento in frazione:

π = \frac{π 2}{2}

= \frac{π 2}{2} - \frac{arccos ( - \frac{3}{2} )}{2}

Poiché i denominatori sono uguali, combinare le frazioni:

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

= \frac{π 2 - arccos ( - \frac{3}{2} )}{2}

x < \frac{2 π - arccos ( - \frac{3}{2} )}{2} + πn

x < \frac{2 π - arccos ( - \frac{3}{2} )}{2} + πn

Combina gli intervalli

x > \frac{arccos ( - \frac{3}{2} )}{2} + πn and x < \frac{2 π - arccos ( - \frac{3}{2} )}{2} + πn

Unire gli intervalli sovrapposti

\frac{arccos ( - \frac{3}{2} )}{2} + πn < x < \frac{2 π - arccos ( - \frac{3}{2} )}{2} + πn

cos (2 x) > \frac{3}{2} : - \frac{arccos ( \frac{3}{2} )}{2} + πn < x < \frac{arccos ( \frac{3}{2} )}{2} + πn

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

Per

cos (x) > a

, se

- 1 \leq a < 1

allora

- arccos (a) + 2 πn < x < arccos (a) + 2 πn

- arccos \frac{3}{2} + 2 πn < 2 x < arccos \frac{3}{2} + 2 πn

a < u < b

allora

a < u and u < b

- arccos \frac{3}{2} + 2 πn < 2 x and 2 x < arccos \frac{3}{2} + 2 πn

- arccos \frac{3}{2} + 2 πn < 2 x : x > - \frac{arccos ( \frac{3}{2} )}{2} + πn

- arccos \frac{3}{2} + 2 πn < 2 x

Scambia i lati

2 x > - arccos \frac{3}{2} + 2 πn

Dividere entrambi i lati per

2

2 x > - arccos \frac{3}{2} + 2 πn

Dividere entrambi i lati per

2

\frac{2 x}{2} > - \frac{arccos ( \frac{3}{2} )}{2} + \frac{2 πn}{2}

Semplificare

x > - \frac{arccos ( \frac{3}{2} )}{2} + πn

x > - \frac{arccos ( \frac{3}{2} )}{2} + πn

2 x < arccos \frac{3}{2} + 2 πn : x < \frac{arccos ( \frac{3}{2} )}{2} + πn

2 x < arccos \frac{3}{2} + 2 πn

Dividere entrambi i lati per

2

2 x < arccos \frac{3}{2} + 2 πn

Dividere entrambi i lati per

2

\frac{2 x}{2} < \frac{arccos ( \frac{3}{2} )}{2} + \frac{2 πn}{2}

Semplificare

x < \frac{arccos ( \frac{3}{2} )}{2} + πn

x < \frac{arccos ( \frac{3}{2} )}{2} + πn

Combina gli intervalli

x > - \frac{arccos ( \frac{3}{2} )}{2} + πn and x < \frac{arccos ( \frac{3}{2} )}{2} + πn

Unire gli intervalli sovrapposti

- \frac{arccos ( \frac{3}{2} )}{2} + πn < x < \frac{arccos ( \frac{3}{2} )}{2} + πn

Combina gli intervalli

\frac{arccos ( - \frac{3}{2} )}{2} + πn < x < \frac{2 π - arccos ( - \frac{3}{2} )}{2} + πn or - \frac{arccos ( \frac{3}{2} )}{2} + πn < x < \frac{arccos ( \frac{3}{2} )}{2} + πn

Unire gli intervalli sovrapposti

- \frac{arccos ( \frac{3}{2} )}{2} + πn < x < \frac{arccos ( \frac{3}{2} )}{2} + πn or \frac{arccos ( - \frac{3}{2} )}{2} + πn < x < \frac{2 π - arccos ( - \frac{3}{2} )}{2} + πn

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