Equações trigonométricas 3

1.
Ora,
\[\begin{array}{*{35}{l}}
sen\,\theta =sen\,\frac{\pi }{4} & \Leftrightarrow  & \begin{matrix}
\theta =\frac{\pi }{4}+2k\pi  & \vee  & \theta =(\pi -\frac{\pi }{4})+2k\pi \,,\ k\in \mathbb{Z}  \\
\end{matrix}  \\
{} & \Leftrightarrow  & \begin{matrix}
\theta =\frac{\pi }{4}+2k\pi  & \vee  & \theta =\frac{3\pi }{4}+2k\pi \,,\ k\in \mathbb{Z}  \\
\end{matrix}  \\
\end{array}\]

2.
Ora,
\[\begin{array}{*{35}{l}}
tg\,\theta =\sqrt{3} & \Leftrightarrow  & \theta =\frac{\pi }{3}+k\pi \,,\ k\in \mathbb{Z}  \\
\end{array}\]

3.
Ora,
\[\begin{array}{*{35}{l}}
sen\,\theta =-sen\,\frac{3\pi }{4} & \Leftrightarrow  & sen\,\theta =sen\,(\pi +\frac{3\pi }{4})  \\
{} & \Leftrightarrow  & \begin{matrix}
\theta =\frac{7\pi }{4}+2k\pi  & \vee  & \theta =(\pi -(\pi +\frac{3\pi }{4}))+2k\pi \,,\ k\in \mathbb{Z}  \\
\end{matrix}  \\
{} & \Leftrightarrow  & \begin{matrix}
\theta =\frac{7\pi }{4}+2k\pi  & \vee  & \theta =-\frac{3\pi }{4}+2k\pi \,,\ k\in \mathbb{Z}  \\
\end{matrix}  \\
\end{array}\]

4.
Ora,
\[\begin{array}{*{35}{l}}
sen\,\theta =-1 & \Leftrightarrow  & \theta =\frac{3\pi }{2}+2k\pi \,,\ k\in \mathbb{Z}  \\
\end{array}\]

5.
Ora,
\[\begin{array}{*{35}{l}}
sen\,\theta =\cos \theta  & \Leftrightarrow  & sen\,\theta =sen\,(\frac{\pi }{2}-\theta )  \\
{} & \Leftrightarrow  & \begin{array}{*{35}{l}}
\theta =(\frac{\pi }{2}-\theta )+2k\pi  & \vee  & \theta =(\pi -(\frac{\pi }{2}-\theta ))+2k\pi \,,\ k\in \mathbb{Z}  \\
\end{array}  \\
{} & \Leftrightarrow  & \begin{array}{*{35}{l}}
2\theta =\frac{\pi }{2}+2k\pi  & \vee  & 0\times \theta =\frac{\pi }{2}+2k\pi \,,\ k\in \mathbb{Z}  \\
\end{array}  \\
{} & \Leftrightarrow  & \theta =\frac{\pi }{4}+k\pi \,,\ k\in \mathbb{Z}  \\
\end{array}\]
Alternativa:
\[\begin{array}{*{35}{l}}
sen\,\theta =\cos \theta  & \Leftrightarrow  & \cos (\frac{\pi }{2}-\theta )=\cos \theta   \\
{} & \Leftrightarrow  & \begin{array}{*{35}{l}}
\frac{\pi }{2}-\theta =\theta +2k\pi  & \vee  & \frac{\pi }{2}-\theta =-\theta +2k\pi \,,\ k\in \mathbb{Z}  \\
\end{array}  \\
{} & \Leftrightarrow  & \begin{array}{*{35}{l}}
2\theta =\frac{\pi }{2}+2k\pi  & \vee  & 0\times \theta =\frac{\pi }{2}+2k\pi \,,\ k\in \mathbb{Z}  \\
\end{array}  \\
{} & \Leftrightarrow  & \theta =\frac{\pi }{4}+k\pi \,,\ k\in \mathbb{Z}  \\
\end{array}\]

6.
Ora,
\[\begin{array}{*{35}{l}}
\cos \frac{\theta }{3}=sen\,\theta  & \Leftrightarrow  & \cos \frac{\theta }{3}=\cos (\frac{\pi }{2}-\theta )  \\
{} & \Leftrightarrow  & \frac{\theta }{3}=\mp (\frac{\pi }{2}-\theta )+2k\pi \,,\ k\in \mathbb{Z}  \\
{} & \Leftrightarrow  & \begin{array}{*{35}{l}}
\frac{\theta }{3}+\theta =\frac{\pi }{2}+2k\pi  & \vee  & \frac{\theta }{3}-\theta =-\frac{\pi }{2}+2k\pi \,,\ k\in \mathbb{Z}  \\
\end{array}  \\
{} & \Leftrightarrow  & \begin{array}{*{35}{l}}
\theta +3\theta =\frac{3\pi }{2}+6k\pi  & \vee  & \theta -3\theta =-\frac{3\pi }{2}+6k\pi \,,\ k\in \mathbb{Z}  \\
\end{array}  \\
{} & \Leftrightarrow  & \begin{array}{*{35}{l}}
\theta =\frac{3\pi }{8}+\frac{3k\pi }{2} & \vee  & \theta =\frac{3\pi }{4}+3k\pi \,,\ k\in \mathbb{Z}  \\
\end{array}  \\
\end{array}\]
Alternativa:
\[\begin{array}{*{35}{l}}
\cos \frac{\theta }{3}=sen\,\theta  & \Leftrightarrow  & sen\,(\frac{\pi }{2}-\frac{\theta }{3})=sen\,\theta   \\
{} & \Leftrightarrow  & \begin{array}{*{35}{l}}
\frac{\pi }{2}-\frac{\theta }{3}=\theta +2k\pi  & \vee  & \frac{\pi }{2}-\frac{\theta }{3}=(\pi -\theta )+2k\pi \,,\ k\in \mathbb{Z}  \\
\end{array}  \\
{} & \Leftrightarrow  & \begin{array}{*{35}{l}}
-\frac{\theta }{3}-\theta =-\frac{\pi }{2}+2k\pi  & \vee  & -\frac{\theta }{3}+\theta =\frac{\pi }{2}+2k\pi \,,\ k\in \mathbb{Z}  \\
\end{array}  \\
{} & \Leftrightarrow  & \begin{array}{*{35}{l}}
\theta +3\theta =\frac{3\pi }{2}+6k\pi  & \vee  & \theta -3\theta =-\frac{3\pi }{2}+6k\pi \,,\ k\in \mathbb{Z}  \\
\end{array}  \\
{} & \Leftrightarrow  & \begin{array}{*{35}{l}}
\theta =\frac{3\pi }{8}+\frac{3k\pi }{2} & \vee  & \theta =\frac{3\pi }{4}+3k\pi \,,\ k\in \mathbb{Z}  \\
\end{array}  \\
\end{array}\]

7.
Ora,
\[\begin{array}{*{35}{l}}
t{{g}^{2}}\,\theta =1 & \Leftrightarrow  & \begin{matrix}
tg\,\theta =-1 & \vee  & tg\,\theta =1  \\
\end{matrix}  \\
{} & \Leftrightarrow  & \begin{matrix}
\theta =-\frac{\pi }{4}+k\pi  & \vee  & \theta =\frac{\pi }{4}+k\pi \,,\ k\in \mathbb{Z}  \\
\end{matrix}  \\
{} & \Leftrightarrow  & \theta =\mp \frac{\pi }{4}+k\pi \,,\ k\in \mathbb{Z}  \\
\end{array}\]

8.
Ora,
\[\begin{array}{*{35}{l}}
1+2\,sen\,\theta =0 & \Leftrightarrow  & sen\,\theta =-\frac{1}{2}  \\
{} & \Leftrightarrow  & \begin{array}{*{35}{l}}
\theta =\frac{7\pi }{6}+2k\pi  & \vee  & \theta =(\pi -\frac{7\pi }{6})+2k\pi \,,\ k\in \mathbb{Z}  \\
\end{array}  \\
{} & \Leftrightarrow  & \begin{array}{*{35}{l}}
\theta =\frac{7\pi }{6}+2k\pi  & \vee  & \theta =-\frac{\pi }{6}+2k\pi \,,\ k\in \mathbb{Z}  \\
\end{array}  \\
\end{array}\]

9.
Ora,
\[\begin{array}{*{35}{l}}
2\,sen\,\theta +\sqrt{3}=0 & \Leftrightarrow  & sen\,\theta =-\frac{\sqrt{3}}{2}  \\
{} & \Leftrightarrow  & \begin{array}{*{35}{l}}
\theta =\frac{4\pi }{3}+2k\pi  & \vee  & \theta =(\pi -\frac{4\pi }{3})+2k\pi \,,\ k\in \mathbb{Z}  \\
\end{array}  \\
{} & \Leftrightarrow  & \begin{array}{*{35}{l}}
\theta =\frac{4\pi }{3}+2k\pi  & \vee  & \theta =-\frac{\pi }{3}+2k\pi \,,\ k\in \mathbb{Z}  \\
\end{array}  \\
\end{array}\]

10.
Ora,
\[\begin{array}{*{35}{l}}
5-5\cos \,(2\theta )=0 & \Leftrightarrow  & \cos \,(2\theta )=1  \\
{} & \Leftrightarrow  & 2\theta =\mp 0+2k\pi \,,\ k\in \mathbb{Z}  \\
{} & \Leftrightarrow  & \theta =k\pi \,,\ k\in \mathbb{Z}  \\
\end{array}\]