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Popolare Trigonometria >

(sin(x)+cos(x))^2=(1+2sin(x))/(sec(x))

Soluzione

(sin (x) + cos (x))^{2} = \frac{1 + 2 sin ( x )}{sec ( x )}

Soluzione

x = 2 πn

Gradi

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

Fasi della soluzione

(sin (x) + cos (x))^{2} = \frac{1 + 2 sin ( x )}{sec ( x )}

Sottrarre

\frac{1 + 2 sin ( x )}{sec ( x )}

da entrambi i lati

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

Semplifica

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

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

Converti l'elemento in frazione:

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

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

Poiché i denominatori sono uguali, combinare le frazioni:

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

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

Espandi

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

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

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

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

Applicare la formula del quadrato perfetto:

(a + b)^{2} = a^{2} + 2 ab + b^{2}

a = sin (x), b = cos (x)

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

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

Espandi

sec (x) (sin^{2} (x) + 2 sin (x) cos (x) + cos^{2} (x)) : sin^{2} (x) sec (x) + 2 sec (x) sin (x) cos (x) + cos^{2} (x) sec (x)

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

Distribuire le parentesi

= sec (x) sin^{2} (x) + sec (x) \cdot 2 sin (x) cos (x) + sec (x) cos^{2} (x)

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

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

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

- (1 + 2 sin (x))

Distribuire le parentesi

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

Applicare le regole di sottrazione-addizione

+ (- a) = - a

= - 1 - 2 sin (x)

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

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

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

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

sin^{2} (x) sec (x) + 2 sec (x) sin (x) cos (x) + cos^{2} (x) sec (x) - 1 - 2 sin (x) = 0

Esprimere con sen e cos

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

Usare l'identità trigonometrica di base:

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

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

Semplifica

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

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

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

cos^{2} (x) \frac{1}{cos ( x )}

Moltiplica le frazioni:

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

= \frac{1 \cdot cos ^{2} ( x )}{cos ( x )}

Moltiplicare:

1 \cdot cos^{2} (x) = cos^{2} (x)

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

Cancella il fattore comune:

cos (x)

= cos (x)

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

\frac{1}{cos ( x )} sin^{2} (x)

Moltiplica le frazioni:

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

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

Moltiplicare:

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

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

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

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

Moltiplica le frazioni:

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

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

Cancella il fattore comune:

cos (x)

= 1 \cdot 2 sin (x)

Moltiplica i numeri:

1 \cdot 2 = 2

= 2 sin (x)

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

Raggruppa termini simili

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

Aggiungi elementi simili:

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

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

Converti l'elemento in frazione:

cos (x) = \frac{cos ( x ) cos ( x )}{cos ( x )}, 1 = \frac{1 cos ( x )}{cos ( x )}

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

Poiché i denominatori sono uguali, combinare le frazioni:

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

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

cos (x) cos (x) + sin^{2} (x) - 1 \cdot cos (x) = cos^{2} (x) + sin^{2} (x) - cos (x)

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

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

cos (x) cos (x)

Applica la regola degli esponenti:

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

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

= cos^{1 + 1} (x)

Aggiungi i numeri:

1 + 1 = 2

= cos^{2} (x)

1 \cdot cos (x) = cos (x)

1 \cdot cos (x)

Moltiplicare:

1 \cdot cos (x) = cos (x)

= cos (x)

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

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

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

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

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

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

Riscrivere utilizzando identità trigonometriche

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

Usa l'identità pitagorica:

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

= - cos (x) + 1

- cos (x) + 1 = 0

Spostare

1

a destra dell'equazione

- cos (x) + 1 = 0

Sottrarre

1

da entrambi i lati

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

Semplificare

- cos (x) = - 1

- cos (x) = - 1

Dividere entrambi i lati per

- 1

- cos (x) = - 1

Dividere entrambi i lati per

- 1

\frac{- cos ( x )}{- 1} = \frac{- 1}{- 1}

Semplificare

cos (x) = 1

cos (x) = 1

Soluzioni generali per

cos (x) = 1

cos (x)

periodicità tabella con

2 πn

cicli:

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 = 0 + 2 πn

x = 0 + 2 πn

Risolvi

x = 0 + 2 πn : x = 2 πn

x = 0 + 2 πn

0 + 2 πn = 2 πn

x = 2 πn

x = 2 πn

Grafico

Esempi popolari

cot(pi/3)=tan(x)

cot (\frac{π}{3}) = tan (x)

3cot(x)+1=2+4cot(x),0<= x<= 360

3 cot (x) + 1 = 2 + 4 cot (x), 0^{\circ} \leq x \leq 36 0^{\circ}

sin(x)=(48.35)/(50.14)

sin (x) = \frac{48.35}{50.14}

1.47=((sin((x+60)/2))/(sin(30)))

1.47 = (\frac{sin ( \frac{x + 60}{2} )}{sin ( 3 0 ^{\circ} )})

tan(θ)= 14/15

tan (θ) = \frac{14}{15}