English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popolare Trigonometria >

dimostrare tan(120)=tan(180-60)

Soluzione

dimostrare

tan (12 0^{\circ}) = tan (18 0^{\circ} - 6 0^{\circ})

Soluzione

Vero

Fasi della soluzione

tan (12 0^{\circ}) = tan (18 0^{\circ} - 6 0^{\circ})

Manipolando il lato sinistro

tan (12 0^{\circ})

Semplifica

tan (12 0^{\circ}) : - 3

tan (12 0^{\circ})

Riscrivere utilizzando identità trigonometriche:

\frac{sin ( 12 0 ^{\circ} )}{cos ( 12 0 ^{\circ} )}

tan (12 0^{\circ})

Usare l'identità trigonometrica di base:

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

= \frac{sin ( 12 0 ^{\circ} )}{cos ( 12 0 ^{\circ} )}

= \frac{sin ( 12 0 ^{\circ} )}{cos ( 12 0 ^{\circ} )}

Usare la seguente identità triviale:

sin (12 0^{\circ}) = \frac{3}{2}

sin (12 0^{\circ})

sin (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} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{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} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

= \frac{3}{2}

Usare la seguente identità triviale:

cos (12 0^{\circ}) = - \frac{1}{2}

cos (12 0^{\circ})

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{1}{2}

= \frac{\frac{3}{2}}{- \frac{1}{2}}

Semplificare

\frac{\frac{3}{2}}{- \frac{1}{2}} : - 3

\frac{\frac{3}{2}}{- \frac{1}{2}}

Applica la regola delle frazioni:

\frac{a}{- b} = - \frac{a}{b}

= - \frac{\frac{3}{2}}{\frac{1}{2}}

Dividi le frazioni:

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

= - \frac{3 \cdot 2}{2 \cdot 1}

Affinare

= - \frac{3 \cdot 2}{2}

Cancella il fattore comune:

2

= - 3

= - 3

= - 3

Manipolando il lato destro

tan (18 0^{\circ} - 6 0^{\circ})

Riscrivere utilizzando identità trigonometriche

tan (18 0^{\circ} - 6 0^{\circ})

Usare l'identità trigonometrica di base:

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

= \frac{sin ( 18 0 ^{\circ} - 6 0 ^{\circ} )}{cos ( 18 0 ^{\circ} - 6 0 ^{\circ} )}

Usa la formula della differenza degli angoli:

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

= \frac{sin ( 18 0 ^{\circ} ) cos ( 6 0 ^{\circ} ) - cos ( 18 0 ^{\circ} ) sin ( 6 0 ^{\circ} )}{cos ( 18 0 ^{\circ} - 6 0 ^{\circ} )}

Usa la formula della differenza degli angoli:

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

= \frac{sin ( 18 0 ^{\circ} ) cos ( 6 0 ^{\circ} ) - cos ( 18 0 ^{\circ} ) sin ( 6 0 ^{\circ} )}{cos ( 18 0 ^{\circ} ) cos ( 6 0 ^{\circ} ) + sin ( 18 0 ^{\circ} ) sin ( 6 0 ^{\circ} )}

\frac{sin ( 18 0 ^{\circ} ) cos ( 6 0 ^{\circ} ) - cos ( 18 0 ^{\circ} ) sin ( 6 0 ^{\circ} )}{cos ( 18 0 ^{\circ} ) cos ( 6 0 ^{\circ} ) + sin ( 18 0 ^{\circ} ) sin ( 6 0 ^{\circ} )} = - 3

\frac{sin ( 18 0 ^{\circ} ) cos ( 6 0 ^{\circ} ) - cos ( 18 0 ^{\circ} ) sin ( 6 0 ^{\circ} )}{cos ( 18 0 ^{\circ} ) cos ( 6 0 ^{\circ} ) + sin ( 18 0 ^{\circ} ) sin ( 6 0 ^{\circ} )}

sin (18 0^{\circ}) cos (6 0^{\circ}) - cos (18 0^{\circ}) sin (6 0^{\circ}) = \frac{3}{2}

sin (18 0^{\circ}) cos (6 0^{\circ}) - cos (18 0^{\circ}) sin (6 0^{\circ})

sin (18 0^{\circ}) cos (6 0^{\circ}) = 0

sin (18 0^{\circ}) cos (6 0^{\circ})

Semplifica

sin (18 0^{\circ}) : 0

sin (18 0^{\circ})

Usare la seguente identità triviale:

sin (18 0^{\circ}) = 0

sin (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} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{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} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

= 0

= 0 \cdot cos (6 0^{\circ})

Semplifica

cos (6 0^{\circ}) : \frac{1}{2}

cos (6 0^{\circ})

Usare la seguente identità triviale:

cos (6 0^{\circ}) = \frac{1}{2}

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{1}{2}

= 0 \cdot \frac{1}{2}

Applicare la regola

0 \cdot a = 0

= 0

cos (18 0^{\circ}) sin (6 0^{\circ}) = - \frac{3}{2}

cos (18 0^{\circ}) sin (6 0^{\circ})

Semplifica

cos (18 0^{\circ}) : - 1

cos (18 0^{\circ})

Usare la seguente identità triviale:

cos (18 0^{\circ}) = (- 1)

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}

= - 1

= - 1 \cdot sin (6 0^{\circ})

Semplifica

sin (6 0^{\circ}) : \frac{3}{2}

sin (6 0^{\circ})

Usare la seguente identità triviale:

sin (6 0^{\circ}) = \frac{3}{2}

sin (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} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{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} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

= \frac{3}{2}

= - 1 \cdot \frac{3}{2}

Moltiplicare:

1 \cdot \frac{3}{2} = \frac{3}{2}

= - \frac{3}{2}

= 0 - (- \frac{3}{2})

Applicare la regola

- (- a) = a

= 0 + \frac{3}{2}

0 + \frac{3}{2} = \frac{3}{2}

= \frac{3}{2}

= \frac{\frac{3}{2}}{cos ( 18 0 ^{\circ} ) cos ( 6 0 ^{\circ} ) + sin ( 18 0 ^{\circ} ) sin ( 6 0 ^{\circ} )}

cos (18 0^{\circ}) cos (6 0^{\circ}) + sin (18 0^{\circ}) sin (6 0^{\circ}) = - \frac{1}{2}

cos (18 0^{\circ}) cos (6 0^{\circ}) + sin (18 0^{\circ}) sin (6 0^{\circ})

cos (18 0^{\circ}) cos (6 0^{\circ}) = - \frac{1}{2}

cos (18 0^{\circ}) cos (6 0^{\circ})

Semplifica

cos (18 0^{\circ}) : - 1

cos (18 0^{\circ})

Usare la seguente identità triviale:

cos (18 0^{\circ}) = (- 1)

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}

= - 1

= - 1 \cdot cos (6 0^{\circ})

Semplifica

cos (6 0^{\circ}) : \frac{1}{2}

cos (6 0^{\circ})

Usare la seguente identità triviale:

cos (6 0^{\circ}) = \frac{1}{2}

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{1}{2}

= - 1 \cdot \frac{1}{2}

Moltiplicare:

1 \cdot \frac{1}{2} = \frac{1}{2}

= - \frac{1}{2}

= - \frac{1}{2} + sin (18 0^{\circ}) sin (6 0^{\circ})

sin (18 0^{\circ}) sin (6 0^{\circ}) = 0

sin (18 0^{\circ}) sin (6 0^{\circ})

Semplifica

sin (18 0^{\circ}) : 0

sin (18 0^{\circ})

Usare la seguente identità triviale:

sin (18 0^{\circ}) = 0

sin (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} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{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} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

= 0

= 0 \cdot sin (6 0^{\circ})

Semplifica

sin (6 0^{\circ}) : \frac{3}{2}

sin (6 0^{\circ})

Usare la seguente identità triviale:

sin (6 0^{\circ}) = \frac{3}{2}

sin (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} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{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} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

= \frac{3}{2}

= 0 \cdot \frac{3}{2}

Applicare la regola

0 \cdot a = 0

= 0

= - \frac{1}{2} + 0

- \frac{1}{2} + 0 = - \frac{1}{2}

= - \frac{1}{2}

= \frac{\frac{3}{2}}{- \frac{1}{2}}

Semplificare

\frac{\frac{3}{2}}{- \frac{1}{2}}

Applica la regola delle frazioni:

\frac{a}{- b} = - \frac{a}{b}

= - \frac{\frac{3}{2}}{\frac{1}{2}}

Dividi le frazioni:

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

= - \frac{3 \cdot 2}{2 \cdot 1}

Affinare

= - \frac{3 \cdot 2}{2}

Cancella il fattore comune:

2

= - 3

= - 3

= - 3

= - 3

Abbiamo mostrato che i due lati possono prendere la stessa forma

\Rightarrow Vero

Esempi popolari

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

p ro v e sin (2 x) tan (x) = 2 sin^{2} (x)

dimostrare (1-4sec^2(x))/(1+2sec(x))=1-2sec(x)

p ro v e \frac{1 - 4 sec ^{2} ( x )}{1 + 2 sec ( x )} = 1 - 2 sec (x)

dimostrare tan(x)=cot(pi/2-x)

p ro v e tan (x) = cot (\frac{π}{2} - x)

dimostrare sec(2x)=(csc^2(x))/(csc^2(x)-2)

p ro v e sec (2 x) = \frac{csc ^{2} ( x )}{csc ^{2} ( x ) - 2}

dimostrare (1-cos(a))/(1+cos(a))=tan^2(a/2)

p ro v e \frac{1 - cos ( a )}{1 + cos ( a )} = tan^{2} (\frac{a}{2})