Trigonometria | Lista de Propriedades Trigonométricas

As propriedades trigonométricas são propriedades e identidades que envolvem funções trigonométricas e que são verdadeiras para todos valores das variáveis envolvidas.

Em geral, são úteis em simplificações e transformações em aplicações, como no cálculo diferencial e integral.

Além disso, incluímos na lista as regras de derivação e integração das funções trigonométricas, bem como algumas fórmulas de integração que julgamos interessantes e úteis nos estudos de Equações Diferenciais, Estatística e correlatos.

Razões Trigonométricas de Ângulos Compostos

1) \text{sen} (A+B) =  \text{sen}(A) \text{cos}(B) + \text{cos}(A) \text{sen}(B);

2) \text{sen} (A-B) =  \text{sen}(A) \text{cos}(B) - \text{cos}(A) \text{sen}(B);

3) \text{cos} (A+B) =  \text{cos}(A) \text{cos}(B) - \text{sen}(A) \text{sen}(B);

4) \text{cos} (A-B) =  \text{cos}(A) \text{cos}(B) + \text{sen}(A) \text{sen}(B);

5) \text{tan} (A+B) =  \dfrac{\text{tan} (A) + \text{tan} (B)}{1 - \text{tan} (A) \text{tan} (B)} ;

6) \text{tan} (A-B) =  \dfrac{\text{tan} (A) - \text{tan} (B)}{1 + \text{tan} (A) \text{tan} (B)} ;

7) \text{cot} (A+B) =  \dfrac{\text{cot} (A) \text{cot} (B) -1 }{\text{cot} (B) - \text{cot} (A)} ;

8) \text{cot} (A-B) =  \dfrac{\text{cot} (A) \text{cot} (B) + 1 }{\text{cot} (B) - \text{cot} (A)} ;

9) \text{sen} (A+B) \text{sen} (A-B) = \text{sen}^2 (A) - \text{sen}^2 (B) = \text{cos}^2 (B) - \text{cos}^2 (A);

10) \text{cos} (A+B) \text{cos} (A-B) = \text{cos}^2 (A) - \text{sen}^2 (B) = \text{cos}^2 (B) - \text{sen}^2 (A);

11) \text{sen} (A+B+C) = \text{cos}(A) \text{cos}(B) \text{sen}(C) + \text{cos}(A)\text{sen}(B)\text{cos}(C)+ \text{sen}(A)\text{cos}(B)\text{cos}(C)-\text{sen}(A)\text{sen}(B)\text{cos}(C);

12) \text{cos} (A+B+C) = \text{cos}(A) \text{cos}(B) \text{cos}(C) - \text{sen}(A)\text{sen}(B)\text{cos}(C) - \text{sen}(A)\text{cos}(B)\text{sen}(C)-\text{cos}(A)\text{sen}(B)\text{cos}(C);

13) \text{tan} (A+B+C) = \dfrac{ \text{tan}(A) + \text{tan} (B) + \text{tan}(C)- \text{tan}(A) \text{tan}(B) \text{tan}(C)}{1- \text{tan}(A) \text{tan}(B)- \text{tan}(B) \text{tan}(C)- \text{tan}(C) \text{tan}(A)};

14) Se A+B+C = 0 então \text{tan}(A)+ \text{tan}(B)+ \text{tan}(C) = \text{tan}(A) \text{tan}(B) \text{tan}(C) .

15) \text{sen} (2A) = 2 \text{sen}(A) \text{cos}(A) = \dfrac{2\text{tan}(A) }{1 + \text{tan}^2 (A)}  ;

16) \text{cos} (2A) = \text{cos}^2 (A) - \text{sen}^2 (A)= 2 \text{cos}^2 (A) -1 = 1 - 2 \text{sen}^2 (A) =  \dfrac{1-\text{tan}^2(A) }{1 + \text{tan}^2 (A)}  ;

17) \text{tan} (2A) = \dfrac{2\text{tan}(A) }{1 + \text{tan}^2 (A)}  ;

18) \text{sen} (3A)  = 3 \text{sen} (A) - 4 \text{sen}^3 (A) ;

19) \text{cos} (3A)  = 4 \text{cos}^3 (A) - 3 \text{cos} (A) ;

20) \text{tan} (3A) = \dfrac{3 \text{tan} (A) - \text{tan}^{3} (A) }{1 - 3 \text{tan}^2 (A) }

21) \text{sen} (A) = 2 \text{sen} \left( \dfrac{A}{2} \right) \text{cos} \left( \dfrac{A}{2} \right) = \dfrac{2 \text{tan} \left( \dfrac{A}{2} \right) }{1 +  \text{tan}^2 \left( \dfrac{A}{2} \right) }

22) \text{cos} (A) = \dfrac{1 -  \text{tan}^2 \left( \dfrac{A}{2} \right) }{1 +  \text{tan}^2 \left( \dfrac{A}{2} \right) }

23) \text{tan} (A) = \dfrac{2\text{tan} \left( \dfrac{A}{2} \right) }{1 -  \text{tan}^2 \left( \dfrac{A}{2} \right) }

24) 1 -  \text{cos} (A) = 2 \text{sen}^2 \left( \dfrac{A}{2} \right) ;

25) 1 +  \text{cos} (A) = 2 \text{cos}^2 \left( \dfrac{A}{2} \right) ;

26) \dfrac{1 -  \text{cos} (A)}{1 +  \text{cos} (A) } = \text{tan}^2 \left( \dfrac{A}{2} \right) ;

27) \text{sen} \left( \dfrac{A}{2} \right) + \text{cos} \left( \dfrac{A}{2} \right) = \pm \sqrt{1 +\text{sen}(A) };

28) \text{sen} \left( \dfrac{A}{2} \right) - \text{cos} \left( \dfrac{A}{2} \right) = \pm \sqrt{1 -\text{sen}(A) };

Identidades Trigonométricas

1) \sin^2{x} + \cos^2{x} = 1 ;

2) 1 + \tan^2{x} = \sec^2{x} ;

3) 1+ \cot^2{x} = \csc^2{x} ;

4) 1 \pm \sin{x} = 1 \pm \cos(\dfrac{\pi}{2} - x) ;

Relações Entre Seno, Cosseno e Tangente- Fórmulas de Transformação

1) \sin{(x)}\sin{(y)} = \dfrac{1}{2}\left[ - \cos{(x+y)} + \cos{(x-y)}\right] ;

2) \cos{(x)}\cos{(y)} = \dfrac{1}{2}\left[ \cos{(x+y)} + \cos{(x-y)}\right] ;

3) \cos{(x)}\sin{(y)} = \dfrac{1}{2}\left[ \sin{(x+y)} + \sin{(x-y)}\right] ;

4) \cos{(x)}\sin{(x)} = \dfrac{1}{2}\sin{(2x)} ;

5) \sin{(C)}+\sin{(D)} = 2 \sin{\left( \dfrac{C+D}{2} \right)} \cos{\left( \dfrac{C- D}{2} \right)};

6) \sin{(C)} - \sin{(D)} = 2 \cos{\left( \dfrac{C+D}{2} \right)} \sin{\left( \dfrac{C- D}{2} \right)};

7) \cos{(C)}+\cos{(D)} = 2 \cos{\left( \dfrac{C+D}{2} \right)} \cos{\left( \dfrac{C- D}{2} \right)};

8) \cos{(C)} - \cos{(D)} = 2 \sin{\left( \dfrac{C+D}{2} \right)} \sin{\left( \dfrac{D-C}{2} \right)} = - 2 \sin{\left( \dfrac{C+D}{2} \right)} \sin{\left( \dfrac{C- D}{2} \right)};

Potências Envolvendo Seno e Cosseno

1) \sin ^2 {(x)} = \dfrac{1}{2} - \dfrac{1}{2} \cos{(2x)} ;

2) \cos ^2 {(x)} = \dfrac{1}{2} + \dfrac{1}{2} \cos{(2x)} ;

3) \sin ^3 {(x)} = \dfrac{3}{4} \sin{(x)} - \dfrac{1}{4} \sin{(3x)} ;

4) \cos ^3 {(x)} = \dfrac{3}{4} \cos{(x)} - \dfrac{1}{4} \cos {(3x)} ;

5) \sin ^4 {(x)} = \dfrac{3}{8}  - \dfrac{1}{2} \cos{(2x)} + \dfrac{1}{8} \cos{(4x)} ;

6) \cos ^4 {(x)} = \dfrac{3}{8}  + \dfrac{1}{2} \cos{(2x)} + \dfrac{1}{8} \cos{(4x)} ;

Valores trigonométricos de ângulos canônicos

Valores trigonométricos de ângulos especiais

Resultados Importantes

1) \text{sen}( \theta ) = 0 \Rightarrow \theta = n \pi ;

2) \text{cos}( \theta ) = 0 \Rightarrow \theta = (2n+1) \dfrac{ \pi }{2};

3) \text{tan}( \theta ) = 0 \Rightarrow \theta = n \pi ;

4) \text{sen}( \theta ) = \text{sen}( \alpha ) \Rightarrow \theta = n \pi +(-1)^n \alpha onde \alpha \in \left[ - \dfrac{\pi}{2} ,  \dfrac{\pi}{2} \right] ;

5) \text{cos}( \theta ) = \text{cos}( \alpha ) \Rightarrow \theta = 2n \pi \pm \alpha onde \alpha \in \left[ 0 ,  \pi \right] ;

6) \text{tan}( \theta ) = \text{tan}( \alpha ) \Rightarrow \theta = n \pi +\alpha onde \alpha \in \left( - \dfrac{\pi}{2} ,  \dfrac{\pi}{2} \right) ;

7) \text{sen}^2 ( \theta ) = \text{sen}^2( \alpha ), \text{cos}^2 ( \theta ) = \text{cos}^2( \alpha ), \text{tan}^2 ( \theta ) = \text{tan}^2( \alpha ) \Rightarrow \theta = n \pi \pm \alpha ;

8) \text{sen}( \theta ) = 1 \Rightarrow \theta = (4n+1) \dfrac{ \pi }{2};

9) \text{cos}( \theta ) = 1 \Rightarrow \theta = 2n \pi;

10) \text{cos}( \theta ) = -1 \Rightarrow \theta = (2n+1) \pi ;

Algumas Substituições Canônicas

Fórmulas de Diferenciação

1) y = \sin{u} \Rightarrow y' = u' \cos{u} ;

2) y = \cos{u} \Rightarrow y' = -u' \sin{u} ;

3) y = \tan{u} \Rightarrow y' = u' \sec^2{u} ;

4) y = \cot{u} \Rightarrow y' = -u' \csc^2{u} ;

5) y = \sec{u} \Rightarrow y' = u' \sec{u} \cdot \tan{u} ;

6) y = \csc{u} \Rightarrow y' = -u' \csc{u} \cdot \cot{u} ;

7) y = \arcsin{u} \Rightarrow y' = \dfrac{u'}{\sqrt{1-u^2}} ;

8) y = \arccos{u} \Rightarrow y' = \dfrac{-u'}{\sqrt{1-u^2}} ;

9) y = \arctan{u} \Rightarrow y' = \dfrac{u'}{\sqrt{1+u^2}} ;

10) y = arccot u \Rightarrow y' = \dfrac{-u'}{\sqrt{1+u^2}} ;

11) y = arcsec u, |u| \geq 1 \Rightarrow y' = \dfrac{u'}{|u| \sqrt{u^2-1}}, |u| > 1 ;

12) y = arccsc u, |u| \geq 1 \Rightarrow y' = \dfrac{-u'}{|u| \sqrt{u^2-1}}, |u| > 1 ;

Fórmulas de Integração Indefinida

1) \int \sin{u} \cdot du = -\cos{u} + c ;

2) \int \cos{u} \cdot du = \sin{u} +c ;

3) \int \tan{u} \cdot du = \ln{|\sec{u}|} + c ;

4) \int \cot{u} \cdot du = \ln{|\sin{u}|} + c ;

5) \int \sec{u} \cdot du = \ln{|\sec{u} + \tan{u}|} + c ;

6) \int \csc{u} \cdot du = \ln{|\csc{u} - \cot{u}|} + c ;

7) \int \sec{u} \cdot \tan{u} \cdot du = \sec{u} + c ;

8) \int \csc{u}\cdot \cot{u} \cdot du = -\csc{u} + c ;

9) \int \sec^2u \cdot du = \tan{u} + c ;

10) \int \csc^2{u} \cdot du = -\cot{u} + c ;

11) \int{sen( a t) e^{bt}dt} = \dfrac{e^{bt} [b sen(at) -a cos(at)]}{b^2 +a^2} ;

Fórmulas de Recorrência

1) \int \sin^n au \cdot du = -\dfrac{\sin^{n-1}au \cdot \cos{au}}{an} + (\dfrac{n-1}{n}) \cdot \int \sin^{n-2} au \cdot du ;

2) \int \cos^n au \cdot du = \dfrac{ \sin{au} \cos^{n-1} au}{an} + \dfrac{n-1}{b} \cdot \int \cos^{n-2} au \cdot du ;

3) \int \tan^n au \cdot du = \dfrac{ \tan^{n-1}{au} }{a(n-1)} - \int \tan^{n-2} au \cdot du ;

4) \int \cot^n au \cdot du = -\dfrac{ \cot^{n-1}{au} }{a(n-1)} - \int \cot^{n-2} au \cdot du ;

5) \int \sec^n au \cdot du = \dfrac{ \sec^{n-2}{au} \tan{au} }{a(n-1)} + \dfrac{n-2}{n-1} \int \sec^{n-2} au \cdot du;

6) \int \csc^n au \cdot du = -\dfrac{ \csc^{n-2}{au} \cot{au}}{a(n-1)} + \dfrac{n-2}{n-1} \int \csc^{n-2} au \cdot du ;

Fórmulas de Integração Definida

1) \dfrac{1}{2 \pi}\int\limits_{0}^{2 \pi}{\sin ^2 {x} dx} = \dfrac{1}{2 \pi}\int\limits_{0}^{2 \pi}{\cos ^2 {x} dx} = \dfrac{1}{2} ;

2) \dfrac{1}{2 \pi}\int\limits_{0}^{2 \pi}{\sin ^4 {x} dx} = \dfrac{1}{2 \pi}\int\limits_{0}^{2 \pi}{\cos ^4 {x} dx} = \dfrac{3}{8} ;

3) \dfrac{1}{2 \pi}\int\limits_{0}^{2 \pi}{\sin ^2 {x} \cos ^2 {x} dx}  = \dfrac{1}{8};

4) \dfrac{1}{2 \pi}\int\limits_{0}^{2 \pi}{\sin ^6 {x} dx} = \dfrac{1}{2 \pi}\int\limits_{0}^{2 \pi}{\cos ^6 {x} dx} = \dfrac{5}{16} ;

5) \dfrac{1}{2 \pi}\int\limits_{0}^{2 \pi}{\sin ^4 {x} \cos ^2 {x} dx}  = \dfrac{1}{2 \pi}\int\limits_{0}^{2 \pi}{\sin ^2 {x} \cos ^4 {x} dx} = \dfrac{1}{16} ;