Exercício 17. Complete a tabela abaixo:
\boldsymbol{t} \sin \boldsymbol{t} \cos \boldsymbol{t} \tan \boldsymbol{t} \csc \boldsymbol{t} \operatorname{sec} \boldsymbol{t} \cot \boldsymbol{t}
0 0 1 undefined
\frac{\pi}{2}
\pi 0 undefined
\frac{3 \pi}{2}

Passo 1

O enunciado pede para completar uma tabela de valores das razões trigonométricas \sin t, \cos t, \tan t, \csc t, \operatorname{sec} t e \cot t
nos quatro ângulos quadrantes 0, \tfrac{\pi}{2}, \pi e \tfrac{3\pi}{2}.
Vamos usar as definições
\tan t = \frac{\sin t}{\cos t},\quad \csc t = \frac{1}{\sin t},\quad \operatorname{sec} t = \frac{1}{\cos t},\quad \cot t = \frac{\cos t}{\sin t}
e conhecer os valores de seno e cosseno nos ângulos múltiplos de \tfrac{\pi}{2}.

Passo 2

Para t=0: \sin 0 = 0 e \cos 0 = 1.
Logo
\tan 0 = \frac{0}{1} = 0,\quad \csc 0 = \frac{1}{0}\text{ (undefined)},\quad \operatorname{sec} 0 = \frac{1}{1} = 1,\quad \cot 0 = \frac{1}{0}\text{ (undefined)}.

Passo 3

Para t=\tfrac{\pi}{2}: \sin \tfrac{\pi}{2} = 1 e \cos \tfrac{\pi}{2} = 0.
Logo
\tan \tfrac{\pi}{2} = \frac{1}{0}\text{ (undefined)},\quad \csc \tfrac{\pi}{2} = \frac{1}{1} = 1,\quad \operatorname{sec} \tfrac{\pi}{2} = \frac{1}{0}\text{ (undefined)},\quad \cot \tfrac{\pi}{2} = \frac{0}{1} = 0.

Passo 4

Para t=\pi: \sin \pi = 0 e \cos \pi = -1.
Logo
\tan \pi = \frac{0}{-1} = 0,\quad \csc \pi = \frac{1}{0}\text{ (undefined)},\quad \operatorname{sec} \pi = \frac{1}{-1} = -1,\quad \cot \pi = \frac{-1}{0}\text{ (undefined)}.

Passo 5

Para t=\tfrac{3\pi}{2}: \sin \tfrac{3\pi}{2} = -1 e \cos \tfrac{3\pi}{2} = 0.
Logo
\tan \tfrac{3\pi}{2} = \frac{-1}{0}\text{ (undefined)},\quad \csc \tfrac{3\pi}{2} = \frac{1}{-1} = -1,\quad \operatorname{sec} \tfrac{3\pi}{2} = \frac{1}{0}\text{ (undefined)},\quad \cot \tfrac{3\pi}{2} = \frac{0}{-1} = 0.

Resposta

\begin{array}{|c|c|c|c|c|c|c|} \hline t & \sin t & \cos t & \tan t & \csc t & \operatorname{sec} t & \cot t\\\ \hline 0 & 0 & 1 & 0 & \text{undefined} & 1 & \text{undefined}\\\ \hline \tfrac{\pi}{2} & 1 & 0 & \text{undefined} & 1 & \text{undefined} & 0\\\ \hline \pi & 0 & -1 & 0 & \text{undefined} & -1 & \text{undefined}\\\ \hline \tfrac{3\pi}{2} & -1 & 0 & \text{undefined} & -1 & \text{undefined} & 0\\\ \hline \end{array}

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