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Popolare Trigonometria >

tan(β+10)=cot(2β-10)

Soluzione

tan (β + 1 0^{\circ}) = cot (2 β - 1 0^{\circ})

Soluzione

β = 3 0^{\circ} + \frac{36 0 ^{\circ} n}{3}, β = 9 0^{\circ} + \frac{36 0 ^{\circ} n}{3}

Radianti

β = \frac{π}{6} + \frac{2 π}{3} n, β = \frac{π}{2} + \frac{2 π}{3} n

Fasi della soluzione

tan (β + 1 0^{\circ}) = cot (2 β - 1 0^{\circ})

Sottrarre

cot (2 β - 1 0^{\circ})

da entrambi i lati

tan (β + 1 0^{\circ}) - cot (2 β - 1 0^{\circ}) = 0

Semplifica

tan (β + 1 0^{\circ}) - cot (2 β - 1 0^{\circ}) : tan (\frac{18 β + 18 0 ^{\circ}}{18}) - cot (\frac{36 β - 18 0 ^{\circ}}{18})

tan (β + 1 0^{\circ}) - cot (2 β - 1 0^{\circ})

Unisci

β + 1 0^{\circ} : \frac{18 β + 18 0 ^{\circ}}{18}

β + 1 0^{\circ}

Converti l'elemento in frazione:

β = \frac{β 18}{18}

= \frac{β \cdot 18}{18} + 1 0^{\circ}

Poiché i denominatori sono uguali, combinare le frazioni:

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

= \frac{β \cdot 18 + 18 0 ^{\circ}}{18}

= tan (\frac{18 β + 18 0 ^{\circ}}{18}) - cot (2 β - 1 0^{\circ})

Unisci

2 β - 1 0^{\circ} : \frac{36 β - 18 0 ^{\circ}}{18}

2 β - 1 0^{\circ}

Converti l'elemento in frazione:

2 β = \frac{2 β 18}{18}

= \frac{2 β \cdot 18}{18} - 1 0^{\circ}

Poiché i denominatori sono uguali, combinare le frazioni:

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

= \frac{2 β \cdot 18 - 18 0 ^{\circ}}{18}

Moltiplica i numeri:

2 \cdot 18 = 36

= \frac{36 β - 18 0 ^{\circ}}{18}

= tan (\frac{18 β + 18 0 ^{\circ}}{18}) - cot (\frac{36 β - 18 0 ^{\circ}}{18})

tan (\frac{18 β + 18 0 ^{\circ}}{18}) - cot (\frac{36 β - 18 0 ^{\circ}}{18}) = 0

Esprimere con sen e cos

- cot (\frac{- 18 0 ^{\circ} + 36 β}{18}) + tan (\frac{18 0 ^{\circ} + 18 β}{18})

Usare l'identità trigonometrica di base:

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

= - \frac{cos ( \frac{- 18 0 ^{\circ} + 36 β}{18} )}{sin ( \frac{- 18 0 ^{\circ} + 36 β}{18} )} + tan (\frac{18 0 ^{\circ} + 18 β}{18})

Usare l'identità trigonometrica di base:

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

= - \frac{cos ( \frac{- 18 0 ^{\circ} + 36 β}{18} )}{sin ( \frac{- 18 0 ^{\circ} + 36 β}{18} )} + \frac{sin ( \frac{18 0 ^{\circ} + 18 β}{18} )}{cos ( \frac{18 0 ^{\circ} + 18 β}{18} )}

Semplifica

- \frac{cos ( \frac{- 18 0 ^{\circ} + 36 β}{18} )}{sin ( \frac{- 18 0 ^{\circ} + 36 β}{18} )} + \frac{sin ( \frac{18 0 ^{\circ} + 18 β}{18} )}{cos ( \frac{18 0 ^{\circ} + 18 β}{18} )} : \frac{- cos ( \frac{- 18 0 ^{\circ} + 36 β}{18} ) cos ( \frac{18 β + 18 0 ^{\circ}}{18} ) + sin ( \frac{18 0 ^{\circ} + 18 β}{18} ) sin ( \frac{36 β - 18 0 ^{\circ}}{18} )}{sin ( \frac{36 β - 18 0 ^{\circ}}{18} ) cos ( \frac{18 β + 18 0 ^{\circ}}{18} )}

- \frac{cos ( \frac{- 18 0 ^{\circ} + 36 β}{18} )}{sin ( \frac{- 18 0 ^{\circ} + 36 β}{18} )} + \frac{sin ( \frac{18 0 ^{\circ} + 18 β}{18} )}{cos ( \frac{18 0 ^{\circ} + 18 β}{18} )}

Minimo Comune Multiplo di

sin (\frac{- 18 0 ^{\circ} + 36 β}{18}), cos (\frac{18 0 ^{\circ} + 18 β}{18}) : sin (\frac{36 β - 18 0 ^{\circ}}{18}) cos (\frac{18 β + 18 0 ^{\circ}}{18})

sin (\frac{- 18 0 ^{\circ} + 36 β}{18}), cos (\frac{18 0 ^{\circ} + 18 β}{18})

Minimo comune multiplo (mcm)

Calcolo di un'espressione composta da fattori che compaiono in

sin (\frac{- 18 0 ^{\circ} + 36 β}{18})

cos (\frac{18 0 ^{\circ} + 18 β}{18})

= sin (\frac{36 β - 18 0 ^{\circ}}{18}) cos (\frac{18 β + 18 0 ^{\circ}}{18})

Adeguare le frazioni in base al minimo comune multiplo (mcm)

Moltiplicare ogni numeratore per la stessa quantità necessaria a moltiplicare il suo

corrispondente denominatore per trasformarlo in mcm

sin (\frac{36 β - 18 0 ^{\circ}}{18}) cos (\frac{18 β + 18 0 ^{\circ}}{18})

Per

\frac{cos ( \frac{- 18 0 ^{\circ} + 36 β}{18} )}{sin ( \frac{- 18 0 ^{\circ} + 36 β}{18} )} :

moltiplica il numeratore e il denominatore per

cos (\frac{18 β + 18 0 ^{\circ}}{18})

\frac{cos ( \frac{- 18 0 ^{\circ} + 36 β}{18} )}{sin ( \frac{- 18 0 ^{\circ} + 36 β}{18} )} = \frac{cos ( \frac{- 18 0 ^{\circ} + 36 β}{18} ) cos ( \frac{18 β + 18 0 ^{\circ}}{18} )}{sin ( \frac{- 18 0 ^{\circ} + 36 β}{18} ) cos ( \frac{18 β + 18 0 ^{\circ}}{18} )}

Per

\frac{sin ( \frac{18 0 ^{\circ} + 18 β}{18} )}{cos ( \frac{18 0 ^{\circ} + 18 β}{18} )} :

moltiplica il numeratore e il denominatore per

sin (\frac{36 β - 18 0 ^{\circ}}{18})

\frac{sin ( \frac{18 0 ^{\circ} + 18 β}{18} )}{cos ( \frac{18 0 ^{\circ} + 18 β}{18} )} = \frac{sin ( \frac{18 0 ^{\circ} + 18 β}{18} ) sin ( \frac{36 β - 18 0 ^{\circ}}{18} )}{cos ( \frac{18 0 ^{\circ} + 18 β}{18} ) sin ( \frac{36 β - 18 0 ^{\circ}}{18} )}

= - \frac{cos ( \frac{- 18 0 ^{\circ} + 36 β}{18} ) cos ( \frac{18 β + 18 0 ^{\circ}}{18} )}{sin ( \frac{- 18 0 ^{\circ} + 36 β}{18} ) cos ( \frac{18 β + 18 0 ^{\circ}}{18} )} + \frac{sin ( \frac{18 0 ^{\circ} + 18 β}{18} ) sin ( \frac{36 β - 18 0 ^{\circ}}{18} )}{cos ( \frac{18 0 ^{\circ} + 18 β}{18} ) sin ( \frac{36 β - 18 0 ^{\circ}}{18} )}

Poiché i denominatori sono uguali, combinare le frazioni:

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

= \frac{- cos ( \frac{- 18 0 ^{\circ} + 36 β}{18} ) cos ( \frac{18 β + 18 0 ^{\circ}}{18} ) + sin ( \frac{18 0 ^{\circ} + 18 β}{18} ) sin ( \frac{36 β - 18 0 ^{\circ}}{18} )}{sin ( \frac{36 β - 18 0 ^{\circ}}{18} ) cos ( \frac{18 β + 18 0 ^{\circ}}{18} )}

= \frac{- cos ( \frac{- 18 0 ^{\circ} + 36 β}{18} ) cos ( \frac{18 β + 18 0 ^{\circ}}{18} ) + sin ( \frac{18 0 ^{\circ} + 18 β}{18} ) sin ( \frac{36 β - 18 0 ^{\circ}}{18} )}{sin ( \frac{36 β - 18 0 ^{\circ}}{18} ) cos ( \frac{18 β + 18 0 ^{\circ}}{18} )}

\frac{- cos ( \frac{- 18 0 ^{\circ} + 36 β}{18} ) cos ( \frac{18 0 ^{\circ} + 18 β}{18} ) + sin ( \frac{- 18 0 ^{\circ} + 36 β}{18} ) sin ( \frac{18 0 ^{\circ} + 18 β}{18} )}{cos ( \frac{18 0 ^{\circ} + 18 β}{18} ) sin ( \frac{- 18 0 ^{\circ} + 36 β}{18} )} = 0

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

- cos (\frac{- 18 0 ^{\circ} + 36 β}{18}) cos (\frac{18 0 ^{\circ} + 18 β}{18}) + sin (\frac{- 18 0 ^{\circ} + 36 β}{18}) sin (\frac{18 0 ^{\circ} + 18 β}{18}) = 0

Riscrivere utilizzando identità trigonometriche

- cos (\frac{- 18 0 ^{\circ} + 36 β}{18}) cos (\frac{18 0 ^{\circ} + 18 β}{18}) + sin (\frac{- 18 0 ^{\circ} + 36 β}{18}) sin (\frac{18 0 ^{\circ} + 18 β}{18})

Usa la formula della somma degli angoli:

cos (s) cos (t) - sin (s) sin (t) = cos (s + t)

- cos (s) cos (t) + sin (s) sin (t) = - cos (s + t)

= - cos (\frac{- 18 0 ^{\circ} + 36 β}{18} + \frac{18 0 ^{\circ} + 18 β}{18})

- cos (\frac{- 18 0 ^{\circ} + 36 β}{18} + \frac{18 0 ^{\circ} + 18 β}{18}) = 0

Dividere entrambi i lati per

- 1

- cos (\frac{- 18 0 ^{\circ} + 36 β}{18} + \frac{18 0 ^{\circ} + 18 β}{18}) = 0

Dividere entrambi i lati per

- 1

\frac{- cos ( \frac{- 18 0 ^{\circ} + 36 β}{18} + \frac{18 0 ^{\circ} + 18 β}{18} )}{- 1} = \frac{0}{- 1}

Semplificare

cos (\frac{- 18 0 ^{\circ} + 36 β}{18} + \frac{18 0 ^{\circ} + 18 β}{18}) = 0

cos (\frac{- 18 0 ^{\circ} + 36 β}{18} + \frac{18 0 ^{\circ} + 18 β}{18}) = 0

Soluzioni generali per

cos (\frac{- 18 0 ^{\circ} + 36 β}{18} + \frac{18 0 ^{\circ} + 18 β}{18}) = 0

cos (x)

periodicità tabella con

36 0^{\circ} n

cicli:

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x 18 0^{\circ} 21 0^{\circ} 22 5^{\circ} 24 0^{\circ} 27 0^{\circ} 30 0^{\circ} 31 5^{\circ} 33 0^{\circ} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

\frac{- 18 0 ^{\circ} + 36 β}{18} + \frac{18 0 ^{\circ} + 18 β}{18} = 9 0^{\circ} + 36 0^{\circ} n, \frac{- 18 0 ^{\circ} + 36 β}{18} + \frac{18 0 ^{\circ} + 18 β}{18} = 27 0^{\circ} + 36 0^{\circ} n

\frac{- 18 0 ^{\circ} + 36 β}{18} + \frac{18 0 ^{\circ} + 18 β}{18} = 9 0^{\circ} + 36 0^{\circ} n, \frac{- 18 0 ^{\circ} + 36 β}{18} + \frac{18 0 ^{\circ} + 18 β}{18} = 27 0^{\circ} + 36 0^{\circ} n

Risolvi

\frac{- 18 0 ^{\circ} + 36 β}{18} + \frac{18 0 ^{\circ} + 18 β}{18} = 9 0^{\circ} + 36 0^{\circ} n : β = 3 0^{\circ} + \frac{36 0 ^{\circ} n}{3}

\frac{- 18 0 ^{\circ} + 36 β}{18} + \frac{18 0 ^{\circ} + 18 β}{18} = 9 0^{\circ} + 36 0^{\circ} n

Moltiplica entrambi i lati per

18

\frac{- 18 0 ^{\circ} + 36 β}{18} + \frac{18 0 ^{\circ} + 18 β}{18} = 9 0^{\circ} + 36 0^{\circ} n

Moltiplica entrambi i lati per

18

\frac{- 18 0 ^{\circ} + 36 β}{18} \cdot 18 + \frac{18 0 ^{\circ} + 18 β}{18} \cdot 18 = 9 0^{\circ} \cdot 18 + 36 0^{\circ} n \cdot 18

Semplificare

\frac{- 18 0 ^{\circ} + 36 β}{18} \cdot 18 + \frac{18 0 ^{\circ} + 18 β}{18} \cdot 18 = 9 0^{\circ} \cdot 18 + 36 0^{\circ} n \cdot 18

Semplificare

\frac{- 18 0 ^{\circ} + 36 β}{18} \cdot 18 : - 18 0^{\circ} + 36 β

\frac{- 18 0 ^{\circ} + 36 β}{18} \cdot 18

Moltiplica le frazioni:

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

= \frac{( - 18 0 ^{\circ} + 36 β ) \cdot 18}{18}

Cancella il fattore comune:

18

= - - 18 0^{\circ} + 36 β

Semplificare

\frac{18 0 ^{\circ} + 18 β}{18} \cdot 18 : 18 0^{\circ} + 18 β

\frac{18 0 ^{\circ} + 18 β}{18} \cdot 18

Moltiplica le frazioni:

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

= \frac{( 18 0 ^{\circ} + 18 β ) \cdot 18}{18}

Cancella il fattore comune:

18

= 18 0^{\circ} + 18 β

Semplificare

9 0^{\circ} \cdot 18 : 162 0^{\circ}

9 0^{\circ} \cdot 18

Moltiplica le frazioni:

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

= 162 0^{\circ}

Dividi i numeri:

\frac{18}{2} = 9

= 162 0^{\circ}

Semplificare

36 0^{\circ} n \cdot 18 : 648 0^{\circ} n

36 0^{\circ} n \cdot 18

Moltiplica i numeri:

2 \cdot 18 = 36

= 648 0^{\circ} n

- 18 0^{\circ} + 36 β + 18 0^{\circ} + 18 β = 162 0^{\circ} + 648 0^{\circ} n

54 β = 162 0^{\circ} + 648 0^{\circ} n

54 β = 162 0^{\circ} + 648 0^{\circ} n

54 β = 162 0^{\circ} + 648 0^{\circ} n

Dividere entrambi i lati per

54

54 β = 162 0^{\circ} + 648 0^{\circ} n

Dividere entrambi i lati per

54

\frac{54 β}{54} = 3 0^{\circ} + \frac{648 0 ^{\circ} n}{54}

Semplificare

\frac{54 β}{54} = 3 0^{\circ} + \frac{648 0 ^{\circ} n}{54}

Semplificare

\frac{54 β}{54} : β

\frac{54 β}{54}

Dividi i numeri:

\frac{54}{54} = 1

= β

Semplificare

3 0^{\circ} + \frac{648 0 ^{\circ} n}{54} : 3 0^{\circ} + \frac{36 0 ^{\circ} n}{3}

3 0^{\circ} + \frac{648 0 ^{\circ} n}{54}

Cancellare

3 0^{\circ} : 3 0^{\circ}

3 0^{\circ}

Cancella il fattore comune:

9

= 3 0^{\circ}

= 3 0^{\circ} + \frac{648 0 ^{\circ} n}{54}

Cancellare

\frac{648 0 ^{\circ} n}{54} : \frac{36 0 ^{\circ} n}{3}

\frac{648 0 ^{\circ} n}{54}

Cancella il fattore comune:

18

= \frac{36 0 ^{\circ} n}{3}

= 3 0^{\circ} + \frac{36 0 ^{\circ} n}{3}

β = 3 0^{\circ} + \frac{36 0 ^{\circ} n}{3}

β = 3 0^{\circ} + \frac{36 0 ^{\circ} n}{3}

β = 3 0^{\circ} + \frac{36 0 ^{\circ} n}{3}

Risolvi

\frac{- 18 0 ^{\circ} + 36 β}{18} + \frac{18 0 ^{\circ} + 18 β}{18} = 27 0^{\circ} + 36 0^{\circ} n : β = 9 0^{\circ} + \frac{36 0 ^{\circ} n}{3}

\frac{- 18 0 ^{\circ} + 36 β}{18} + \frac{18 0 ^{\circ} + 18 β}{18} = 27 0^{\circ} + 36 0^{\circ} n

Moltiplica entrambi i lati per

18

\frac{- 18 0 ^{\circ} + 36 β}{18} + \frac{18 0 ^{\circ} + 18 β}{18} = 27 0^{\circ} + 36 0^{\circ} n

Moltiplica entrambi i lati per

18

\frac{- 18 0 ^{\circ} + 36 β}{18} \cdot 18 + \frac{18 0 ^{\circ} + 18 β}{18} \cdot 18 = 27 0^{\circ} \cdot 18 + 36 0^{\circ} n \cdot 18

Semplificare

\frac{- 18 0 ^{\circ} + 36 β}{18} \cdot 18 + \frac{18 0 ^{\circ} + 18 β}{18} \cdot 18 = 27 0^{\circ} \cdot 18 + 36 0^{\circ} n \cdot 18

Semplificare

\frac{- 18 0 ^{\circ} + 36 β}{18} \cdot 18 : - 18 0^{\circ} + 36 β

\frac{- 18 0 ^{\circ} + 36 β}{18} \cdot 18

Moltiplica le frazioni:

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

= \frac{( - 18 0 ^{\circ} + 36 β ) \cdot 18}{18}

Cancella il fattore comune:

18

= - - 18 0^{\circ} + 36 β

Semplificare

\frac{18 0 ^{\circ} + 18 β}{18} \cdot 18 : 18 0^{\circ} + 18 β

\frac{18 0 ^{\circ} + 18 β}{18} \cdot 18

Moltiplica le frazioni:

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

= \frac{( 18 0 ^{\circ} + 18 β ) \cdot 18}{18}

Cancella il fattore comune:

18

= 18 0^{\circ} + 18 β

Semplificare

27 0^{\circ} \cdot 18 : 486 0^{\circ}

27 0^{\circ} \cdot 18

Moltiplica le frazioni:

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

= 486 0^{\circ}

Moltiplica i numeri:

3 \cdot 18 = 54

= 486 0^{\circ}

Dividi i numeri:

\frac{54}{2} = 27

= 486 0^{\circ}

Semplificare

36 0^{\circ} n \cdot 18 : 648 0^{\circ} n

36 0^{\circ} n \cdot 18

Moltiplica i numeri:

2 \cdot 18 = 36

= 648 0^{\circ} n

- 18 0^{\circ} + 36 β + 18 0^{\circ} + 18 β = 486 0^{\circ} + 648 0^{\circ} n

54 β = 486 0^{\circ} + 648 0^{\circ} n

54 β = 486 0^{\circ} + 648 0^{\circ} n

54 β = 486 0^{\circ} + 648 0^{\circ} n

Dividere entrambi i lati per

54

54 β = 486 0^{\circ} + 648 0^{\circ} n

Dividere entrambi i lati per

54

\frac{54 β}{54} = 9 0^{\circ} + \frac{648 0 ^{\circ} n}{54}

Semplificare

\frac{54 β}{54} = 9 0^{\circ} + \frac{648 0 ^{\circ} n}{54}

Semplificare

\frac{54 β}{54} : β

\frac{54 β}{54}

Dividi i numeri:

\frac{54}{54} = 1

= β

Semplificare

9 0^{\circ} + \frac{648 0 ^{\circ} n}{54} : 9 0^{\circ} + \frac{36 0 ^{\circ} n}{3}

9 0^{\circ} + \frac{648 0 ^{\circ} n}{54}

Cancellare

9 0^{\circ} : 9 0^{\circ}

9 0^{\circ}

Cancella il fattore comune:

27

= 9 0^{\circ}

= 9 0^{\circ} + \frac{648 0 ^{\circ} n}{54}

Cancellare

\frac{648 0 ^{\circ} n}{54} : \frac{36 0 ^{\circ} n}{3}

\frac{648 0 ^{\circ} n}{54}

Cancella il fattore comune:

18

= \frac{36 0 ^{\circ} n}{3}

= 9 0^{\circ} + \frac{36 0 ^{\circ} n}{3}

β = 9 0^{\circ} + \frac{36 0 ^{\circ} n}{3}

β = 9 0^{\circ} + \frac{36 0 ^{\circ} n}{3}

β = 9 0^{\circ} + \frac{36 0 ^{\circ} n}{3}

β = 3 0^{\circ} + \frac{36 0 ^{\circ} n}{3}, β = 9 0^{\circ} + \frac{36 0 ^{\circ} n}{3}

Grafico

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