Usando as fórmulas:
\cos (\alpha \pm \beta)=\cos \alpha \cos \beta \mp \operatorname{sen} \alpha \operatorname{sen} \beta
\operatorname{sen}(\alpha \pm \beta)=\operatorname{sen} \alpha \cos \beta \pm \cos \alpha \operatorname{sen} \beta
encontrar:
\operatorname{sen} 2 \pi, \cos 2 \pi, \tan 2 \pi

Passo 1

O enunciado pede para usarmos as fórmulas de soma de ângulos
\cos(\alpha\pm\beta)=\cos\alpha\cos\beta\mp\operatorname{sen}\alpha\,\operatorname{sen}\beta \quad\text{e}\quad \operatorname{sen}(\alpha\pm\beta)=\operatorname{sen}\alpha\cos\beta\pm\cos\alpha\,\operatorname{sen}\beta
para calcular \operatorname{sen}2\pi, \cos2\pi e \tan2\pi. Note que 2\pi=\pi+\pi.

Passo 2

Aplicando a fórmula de seno para \operatorname{sen}2\pi:
\operatorname{sen}2\pi =\operatorname{sen}(\pi+\pi) =\operatorname{sen}\pi\cos\pi+\cos\pi\,\operatorname{sen}\pi.
Como \operatorname{sen}\pi=0 e \cos\pi=-1, temos
\operatorname{sen}2\pi =0\cdot(-1)+(-1)\cdot0 =0.

Passo 3

Aplicando a fórmula de cosseno para \cos2\pi:
\cos2\pi =\cos(\pi+\pi) =\cos\pi\cos\pi-\operatorname{sen}\pi\,\operatorname{sen}\pi.
Usando \cos\pi=-1 e \operatorname{sen}\pi=0, obtemos
\cos2\pi =(-1)\cdot(-1)-0\cdot0 =1.

Passo 4

Finalmente, \tan2\pi é a razão entre seno e cosseno:
\tan2\pi=\frac{\operatorname{sen}2\pi}{\cos2\pi} =\frac{0}{1} =0.

Resposta

\operatorname{sen}2\pi=0 \cos2\pi=1 \tan2\pi=0

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