Equações trigonométricas 1

Trigonometria: Infinito 11 A - Parte 1 Pág. 98 Ex. 63

by AMMA · Published 21 de Outubro de 2010 · Updated 13 de Janeiro de 2022

Enunciado

Recorrendo ao círculo trigonométrico, resolva, se possível, no intervalo $\left[ 0,\pi \right]$, as seguintes equações:

$\cos \theta =-\frac{1}{2}$
$sen\,\theta =\frac{1}{2}$
$\cos \theta =\frac{\sqrt{3}}{2}$
$tg\,\theta =1$
$tg\,\theta =-\sqrt{3}$
$sen\,\theta =-\frac{\sqrt{3}}{2}$
$sen\,\theta =-0,6$
$\cos \theta =-0,6$
$tg\,\theta =-98$
$\begin{matrix}
\cos \theta =-\frac{1}{2} & \wedge & sen\,\theta =\frac{\sqrt{3}}{2} \\
\end{matrix}$
$\begin{matrix}
\cos \theta =-\frac{1}{2} & \wedge & sen\,\theta =-\frac{\sqrt{3}}{2} \\
\end{matrix}$
$\cos \theta =2,3$

Resolução

$\begin{matrix}
\begin{matrix}
\cos \theta =-\frac{1}{2} & \wedge & \theta \in \left[ 0,\pi \right] \\
\end{matrix} & \Leftrightarrow & \theta =\frac{2\pi }{3} \\
\end{matrix}$
$\begin{matrix}
\begin{matrix}
sen\,\theta =\frac{1}{2} & \wedge & \theta \in \left[ 0,\pi \right] \\
\end{matrix} & \Leftrightarrow & \theta =\frac{\pi }{6}\vee \theta =\frac{5\pi }{6} \\
\end{matrix}$
$\begin{matrix}
\begin{matrix}
\cos \theta =\frac{\sqrt{3}}{2} & \wedge & \theta \in \left[ 0,\pi \right] \\
\end{matrix} & \Leftrightarrow & \theta =\frac{\pi }{6} \\
\end{matrix}$
$\begin{matrix}
\begin{matrix}
tg\,\theta =1 & \wedge & \theta \in \left[ 0,\pi \right] \\
\end{matrix} & \Leftrightarrow & \theta =\frac{\pi }{4} \\
\end{matrix}$
$\begin{matrix}
\begin{matrix}
tg\,\theta =-\sqrt{3} & \wedge & \theta \in \left[ 0,\pi \right] \\
\end{matrix} & \Leftrightarrow & \theta =\frac{2\pi }{3} \\
\end{matrix}$
$\begin{matrix}
\begin{matrix}
sen\,\theta =-\frac{\sqrt{3}}{2} & \wedge & \theta \in \left[ 0,\pi \right] \\
\end{matrix} & \Leftrightarrow & \theta \in \left\{ {} \right\} \\
\end{matrix}$
$\begin{matrix}
\begin{matrix}
sen\,\theta =-0,6 & \wedge & \theta \in \left[ 0,\pi \right] \\
\end{matrix} & \Leftrightarrow & \theta \in \left\{ {} \right\} \\
\end{matrix}$
$\begin{matrix}
\begin{matrix}
\cos \theta =-0,6 & \wedge & \theta \in \left[ 0,\pi \right] \\
\end{matrix} & \Leftrightarrow & \theta \simeq 2,2 \\
\end{matrix}$
$\begin{matrix}
\begin{matrix}
tg\,\theta =-98 & \wedge & \theta \in \left[ 0,\pi \right] \\
\end{matrix} & \Leftrightarrow & \theta \simeq 1,6 \\
\end{matrix}$
$\begin{matrix}
\begin{matrix}
\cos \theta =-\frac{1}{2} & \wedge & sen\,\theta =\frac{\sqrt{3}}{2} & \wedge & \theta \in \left[ 0,\pi \right] \\
\end{matrix} & \Leftrightarrow & \theta =\frac{2\pi }{3} \\
\end{matrix}$
$\begin{matrix}
\begin{matrix}
\cos \theta =-\frac{1}{2} & \wedge & sen\,\theta =-\frac{\sqrt{3}}{2} & \wedge & \theta \in \left[ 0,\pi \right] \\
\end{matrix} & \Leftrightarrow & \theta \in \left\{ {} \right\} \\
\end{matrix}$
$\begin{matrix}
\begin{matrix}
\cos \theta =2,3 & \wedge & \theta \in \left[ 0,\pi \right] \\
\end{matrix} & \Leftrightarrow & \theta \in \left\{ {} \right\} \\
\end{matrix}$