Equações trigonométricas 4

1.
Ora,
\[\begin{array}{*{35}{l}}
sen\,\theta =-\cos \frac{\pi }{3} & \Leftrightarrow  & \cos (\frac{\pi }{2}-\theta )=\cos (\pi +\frac{\pi }{3})  \\
{} & \Leftrightarrow  & \frac{\pi }{2}-\theta =\mp \frac{4\pi }{3}+2k\pi \,,\,\,k\in \mathbb{Z}  \\
{} & \Leftrightarrow  & \begin{array}{*{35}{l}}
-\theta =-\frac{\pi }{2}-\frac{4\pi }{3}+2k\pi  & \vee  & -\theta =-\frac{\pi }{2}+\frac{4\pi }{3}+2k\pi \,,\,\,k\in \mathbb{Z}  \\
\end{array}  \\
{} & \Leftrightarrow  & \begin{array}{*{35}{l}}
\theta =\frac{11\pi }{6}+2k\pi  & \vee  & \theta =-\frac{5\pi }{6}+2k\pi \,,\,\,k\in \mathbb{Z}  \\
\end{array}  \\
\end{array}\]
Alternativa 1:
\[\begin{array}{*{35}{l}}
sen\,\theta =-\cos \frac{\pi }{3} & \Leftrightarrow  & sen\,\theta =\cos (\pi +\frac{\pi }{3})  \\
{} & \Leftrightarrow  & sen\,\theta =sen\,(\frac{\pi }{2}-(\pi +\frac{\pi }{3}))  \\
{} & \Leftrightarrow  & \begin{array}{*{35}{l}}
\theta =-\frac{5\pi }{6}+2k\pi  & \vee  & \theta =(\pi -(-\frac{5\pi }{6}))+2k\pi \,,\,\,k\in \mathbb{Z}  \\
\end{array}  \\
{} & \Leftrightarrow  & \begin{array}{*{35}{l}}
\theta =-\frac{5\pi }{6}+2k\pi  & \vee  & \theta =\frac{11\pi }{6}+2k\pi \,,\,\,k\in \mathbb{Z}  \\
\end{array}  \\
\end{array}\]
Alternativa 2:
\[\begin{array}{*{35}{l}}
sen\,\theta =-\cos \frac{\pi }{3} & \Leftrightarrow  & -sen\,(-\theta )=-\cos \frac{\pi }{3}  \\
{} & \Leftrightarrow  & sen\,(-\theta )=sen\,(\frac{\pi }{2}-\frac{\pi }{3})  \\
{} & \Leftrightarrow  & \begin{array}{*{35}{l}}
-\theta =\frac{\pi }{6}+2k\pi  & \vee  & -\theta =(\pi -\frac{\pi }{6})+2k\pi \,,\,\,k\in \mathbb{Z}  \\
\end{array}  \\
{} & \Leftrightarrow  & \begin{array}{*{35}{l}}
\theta =-\frac{\pi }{6}+2k\pi  & \vee  & \theta =-\frac{5\pi }{6}+2k\pi \,,\,\,k\in \mathbb{Z}  \\
\end{array}  \\
\end{array}\]

2.
Ora,
\[\begin{array}{*{35}{l}}
sen\,\theta =\cos \frac{\pi }{5} & \Leftrightarrow  & sen\,\theta =sen\,(\frac{\pi }{2}-\frac{\pi }{5})  \\
{} & \Leftrightarrow  & \begin{array}{*{35}{l}}
\theta =\frac{3\pi }{10}+2k\pi  & \vee  & \theta =(\pi -\frac{3\pi }{10})+2k\pi \,,\,\,k\in \mathbb{Z}  \\
\end{array}  \\
{} & \Leftrightarrow  & \begin{array}{*{35}{l}}
\theta =\frac{3\pi }{10}+2k\pi  & \vee  & \theta =\frac{7\pi }{10}+2k\pi \,,\,\,k\in \mathbb{Z}  \\
\end{array}  \\
\end{array}\]

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

4.
Ora,
\[\begin{array}{*{35}{l}}
tg\,\theta \times \cos \theta =0 & \Leftrightarrow  & \begin{array}{*{35}{l}}
tg\,\theta =0 & \vee  & \cos \theta =0  \\
\end{array}  \\
{} & \Leftrightarrow  & \begin{array}{*{35}{l}}
\theta =k\pi  & \vee  & \theta =\frac{\pi }{2}+k\pi \,,\,\,k\in \mathbb{Z}  \\
\end{array}  \\
\end{array}\]

5.
Ora,
\[\begin{array}{*{35}{l}}
(sen\,\theta )\times (2\cos \theta -1)=0 & \Leftrightarrow  & \begin{array}{*{35}{l}}
sen\,\theta =0 & \vee  & 2\cos \theta -1=0  \\
\end{array}  \\
{} & \Leftrightarrow  & \begin{array}{*{35}{l}}
sen\,\theta =0 & \vee  & \cos \theta =\frac{1}{2}  \\
\end{array}  \\
{} & \Leftrightarrow  & \begin{array}{*{35}{l}}
\theta =k\pi  & \vee  & \theta =\mp \frac{\pi }{3}+2k\pi \,,\,\,k\in \mathbb{Z}  \\
\end{array}  \\
\end{array}\]

6.
Ora,
\[\begin{array}{*{35}{l}}
sen\,(\theta -\frac{\pi }{6})=1 & \Leftrightarrow  & \theta -\frac{\pi }{6}=\frac{\pi }{2}+2k\pi \,,\,\,k\in \mathbb{Z}  \\
{} & \Leftrightarrow  & \theta =\frac{2\pi }{3}+2k\pi \,,\,\,k\in \mathbb{Z}  \\
\end{array}\]

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

8.
Ora,
\[\begin{array}{*{35}{l}}
\cos \,\theta -sen\,\theta \times \cos \theta =0 & \Leftrightarrow  & \cos \,\theta \times (1-sen\,\theta )=0  \\
{} & \Leftrightarrow  & \begin{array}{*{35}{l}}
\cos \,\theta =0 & \vee  & sen\,\theta =1  \\
\end{array}  \\
{} & \Leftrightarrow  & \begin{array}{*{35}{l}}
\theta =\frac{\pi }{2}+k\pi  & \vee  & \theta =\frac{\pi }{2}+2k\pi \,,\,\,k\in \mathbb{Z}  \\
\end{array}  \\
{} & \Leftrightarrow  & \theta =\frac{\pi }{2}+k\pi \,,\,\,k\in \mathbb{Z}\,\,\,\,\,\text{(Porqu }\!\!\hat{\mathrm{e}}\!\!\text{ ?)}  \\
\end{array}\]

9.
Ora,
\[\begin{array}{*{35}{l}}
\cos \,3\theta =\cos \theta  & \Leftrightarrow  & 3\theta =\mp \theta +2k\pi \,,\,\,k\in \mathbb{Z}  \\
{} & \Leftrightarrow  & \begin{array}{*{35}{l}}
2\theta =2k\pi  & \vee  & 4\theta =2k\pi \,,\,\,k\in \mathbb{Z}  \\
\end{array}  \\
{} & \Leftrightarrow  & \begin{array}{*{35}{l}}
\theta =k\pi  & \vee  & \theta =\frac{k\pi }{2}\,,\,\,k\in \mathbb{Z}  \\
\end{array}  \\
{} & \Leftrightarrow  & \theta =\frac{k\pi }{2}\,,\,\,k\in \mathbb{Z}  \\
\end{array}\]

10.
Ora,
\[\begin{array}{*{35}{l}}
\cos \,(2\theta +\frac{\pi }{6})=\frac{\sqrt{3}}{2} & \Leftrightarrow  & 2\theta +\frac{\pi }{6}=\mp \frac{\pi }{6}+2k\pi \,,\,\,k\in \mathbb{Z}  \\
{} & \Leftrightarrow  & \begin{array}{*{35}{l}}
2\theta =2k\pi  & \vee  & 2\theta =-\frac{\pi }{3}+2k\pi \,,\,\,k\in \mathbb{Z}  \\
\end{array}  \\
{} & \Leftrightarrow  & \begin{array}{*{35}{l}}
\theta =k\pi  & \vee  & \theta =-\frac{\pi }{6}+k\pi \,,\,\,k\in \mathbb{Z}  \\
\end{array}  \\
\end{array}\]

11.
Ora,
\[\begin{array}{*{35}{l}}
{{\cos }^{2}}\theta =1 & \Leftrightarrow  & \begin{array}{*{35}{l}}
\cos \,\theta =-1 & \vee  & \cos \,\theta =1  \\
\end{array}  \\
{} & \Leftrightarrow  & \begin{array}{*{35}{l}}
\theta =\pi +2k\pi  & \vee  & \theta =0+2k\pi \,,\,\,k\in \mathbb{Z}  \\
\end{array}  \\
{} & \Leftrightarrow  & \theta =k\pi \,,\,\,k\in \mathbb{Z}\,\,\,\,\,\text{(Porqu }\!\!\hat{\mathrm{e}}\!\!\text{ ?)}  \\
\end{array}\]

12.
Ora,
\[\begin{array}{*{35}{l}}
-1+\sqrt{2}\,sen\,\theta =2 & \Leftrightarrow  & \sqrt{2}\,sen\,\theta =3  \\
{} & \Leftrightarrow  & sen\,\theta =\frac{3}{\sqrt{2}}  \\
{} & \Leftrightarrow  & sen\,\theta =\frac{3\sqrt{2}}{2}  \\
{} & \Leftrightarrow  & \theta \in \left\{ {} \right\}  \\
\end{array}\]