Resolva as equa\u00e7\u00f5es trigonom\u00e9tricas seguintes:<\/p>\n
1.<\/strong> 2.<\/strong> 3.<\/strong> 4.<\/strong> 5.<\/strong> 6.<\/strong> 7.<\/strong> 8.<\/strong> 9.<\/strong> 10.<\/strong>
\nOra,
\n\\[\\begin{array}{*{35}{l}}
\nsen\\,\\theta =sen\\,\\frac{\\pi }{4} & \\Leftrightarrow\u00a0 & \\begin{matrix}
\n\\theta =\\frac{\\pi }{4}+2k\\pi\u00a0 & \\vee\u00a0 & \\theta =(\\pi -\\frac{\\pi }{4})+2k\\pi \\,,\\ k\\in \\mathbb{Z}\u00a0 \\\\
\n\\end{matrix}\u00a0 \\\\
\n{} & \\Leftrightarrow\u00a0 & \\begin{matrix}
\n\\theta =\\frac{\\pi }{4}+2k\\pi\u00a0 & \\vee\u00a0 & \\theta =\\frac{3\\pi }{4}+2k\\pi \\,,\\ k\\in \\mathbb{Z}\u00a0 \\\\
\n\\end{matrix}\u00a0 \\\\
\n\\end{array}\\]
\n
\nOra,
\n\\[\\begin{array}{*{35}{l}}
\ntg\\,\\theta =\\sqrt{3} & \\Leftrightarrow\u00a0 & \\theta =\\frac{\\pi }{3}+k\\pi \\,,\\ k\\in \\mathbb{Z}\u00a0 \\\\
\n\\end{array}\\]
\n
\nOra,
\n\\[\\begin{array}{*{35}{l}}
\nsen\\,\\theta =-sen\\,\\frac{3\\pi }{4} & \\Leftrightarrow\u00a0 & sen\\,\\theta =sen\\,(\\pi +\\frac{3\\pi }{4})\u00a0 \\\\
\n{} & \\Leftrightarrow\u00a0 & \\begin{matrix}
\n\\theta =\\frac{7\\pi }{4}+2k\\pi\u00a0 & \\vee\u00a0 & \\theta =(\\pi -(\\pi +\\frac{3\\pi }{4}))+2k\\pi \\,,\\ k\\in \\mathbb{Z}\u00a0 \\\\
\n\\end{matrix}\u00a0 \\\\
\n{} & \\Leftrightarrow\u00a0 & \\begin{matrix}
\n\\theta =\\frac{7\\pi }{4}+2k\\pi\u00a0 & \\vee\u00a0 & \\theta =-\\frac{3\\pi }{4}+2k\\pi \\,,\\ k\\in \\mathbb{Z}\u00a0 \\\\
\n\\end{matrix}\u00a0 \\\\
\n\\end{array}\\]
\n
\nOra,
\n\\[\\begin{array}{*{35}{l}}
\nsen\\,\\theta =-1 & \\Leftrightarrow\u00a0 & \\theta =\\frac{3\\pi }{2}+2k\\pi \\,,\\ k\\in \\mathbb{Z}\u00a0 \\\\
\n\\end{array}\\]
\n
\nOra,
\n\\[\\begin{array}{*{35}{l}}
\nsen\\,\\theta =\\cos \\theta\u00a0 & \\Leftrightarrow\u00a0 & sen\\,\\theta =sen\\,(\\frac{\\pi }{2}-\\theta )\u00a0 \\\\
\n{} & \\Leftrightarrow\u00a0 & \\begin{array}{*{35}{l}}
\n\\theta =(\\frac{\\pi }{2}-\\theta )+2k\\pi\u00a0 & \\vee\u00a0 & \\theta =(\\pi -(\\frac{\\pi }{2}-\\theta ))+2k\\pi \\,,\\ k\\in \\mathbb{Z}\u00a0 \\\\
\n\\end{array}\u00a0 \\\\
\n{} & \\Leftrightarrow\u00a0 & \\begin{array}{*{35}{l}}
\n2\\theta =\\frac{\\pi }{2}+2k\\pi\u00a0 & \\vee\u00a0 & 0\\times \\theta =\\frac{\\pi }{2}+2k\\pi \\,,\\ k\\in \\mathbb{Z}\u00a0 \\\\
\n\\end{array}\u00a0 \\\\
\n{} & \\Leftrightarrow\u00a0 & \\theta =\\frac{\\pi }{4}+k\\pi \\,,\\ k\\in \\mathbb{Z}\u00a0 \\\\
\n\\end{array}\\]
\nAlternativa:
\n\\[\\begin{array}{*{35}{l}}
\nsen\\,\\theta =\\cos \\theta\u00a0 & \\Leftrightarrow\u00a0 & \\cos (\\frac{\\pi }{2}-\\theta )=\\cos \\theta\u00a0\u00a0 \\\\
\n{} & \\Leftrightarrow\u00a0 & \\begin{array}{*{35}{l}}
\n\\frac{\\pi }{2}-\\theta =\\theta +2k\\pi\u00a0 & \\vee\u00a0 & \\frac{\\pi }{2}-\\theta =-\\theta +2k\\pi \\,,\\ k\\in \\mathbb{Z}\u00a0 \\\\
\n\\end{array}\u00a0 \\\\
\n{} & \\Leftrightarrow\u00a0 & \\begin{array}{*{35}{l}}
\n2\\theta =\\frac{\\pi }{2}+2k\\pi\u00a0 & \\vee\u00a0 & 0\\times \\theta =\\frac{\\pi }{2}+2k\\pi \\,,\\ k\\in \\mathbb{Z}\u00a0 \\\\
\n\\end{array}\u00a0 \\\\
\n{} & \\Leftrightarrow\u00a0 & \\theta =\\frac{\\pi }{4}+k\\pi \\,,\\ k\\in \\mathbb{Z}\u00a0 \\\\
\n\\end{array}\\]
\n
\nOra,
\n\\[\\begin{array}{*{35}{l}}
\n\\cos \\frac{\\theta }{3}=sen\\,\\theta\u00a0 & \\Leftrightarrow\u00a0 & \\cos \\frac{\\theta }{3}=\\cos (\\frac{\\pi }{2}-\\theta )\u00a0 \\\\
\n{} & \\Leftrightarrow\u00a0 & \\frac{\\theta }{3}=\\mp (\\frac{\\pi }{2}-\\theta )+2k\\pi \\,,\\ k\\in \\mathbb{Z}\u00a0 \\\\
\n{} & \\Leftrightarrow\u00a0 & \\begin{array}{*{35}{l}}
\n\\frac{\\theta }{3}+\\theta =\\frac{\\pi }{2}+2k\\pi\u00a0 & \\vee\u00a0 & \\frac{\\theta }{3}-\\theta =-\\frac{\\pi }{2}+2k\\pi \\,,\\ k\\in \\mathbb{Z}\u00a0 \\\\
\n\\end{array}\u00a0 \\\\
\n{} & \\Leftrightarrow\u00a0 & \\begin{array}{*{35}{l}}
\n\\theta +3\\theta =\\frac{3\\pi }{2}+6k\\pi\u00a0 & \\vee\u00a0 & \\theta -3\\theta =-\\frac{3\\pi }{2}+6k\\pi \\,,\\ k\\in \\mathbb{Z}\u00a0 \\\\
\n\\end{array}\u00a0 \\\\
\n{} & \\Leftrightarrow\u00a0 & \\begin{array}{*{35}{l}}
\n\\theta =\\frac{3\\pi }{8}+\\frac{3k\\pi }{2} & \\vee\u00a0 & \\theta =\\frac{3\\pi }{4}+3k\\pi \\,,\\ k\\in \\mathbb{Z}\u00a0 \\\\
\n\\end{array}\u00a0 \\\\
\n\\end{array}\\]
\nAlternativa:
\n\\[\\begin{array}{*{35}{l}}
\n\\cos \\frac{\\theta }{3}=sen\\,\\theta\u00a0 & \\Leftrightarrow\u00a0 & sen\\,(\\frac{\\pi }{2}-\\frac{\\theta }{3})=sen\\,\\theta\u00a0\u00a0 \\\\
\n{} & \\Leftrightarrow\u00a0 & \\begin{array}{*{35}{l}}
\n\\frac{\\pi }{2}-\\frac{\\theta }{3}=\\theta +2k\\pi\u00a0 & \\vee\u00a0 & \\frac{\\pi }{2}-\\frac{\\theta }{3}=(\\pi -\\theta )+2k\\pi \\,,\\ k\\in \\mathbb{Z}\u00a0 \\\\
\n\\end{array}\u00a0 \\\\
\n{} & \\Leftrightarrow\u00a0 & \\begin{array}{*{35}{l}}
\n-\\frac{\\theta }{3}-\\theta =-\\frac{\\pi }{2}+2k\\pi\u00a0 & \\vee\u00a0 & -\\frac{\\theta }{3}+\\theta =\\frac{\\pi }{2}+2k\\pi \\,,\\ k\\in \\mathbb{Z}\u00a0 \\\\
\n\\end{array}\u00a0 \\\\
\n{} & \\Leftrightarrow\u00a0 & \\begin{array}{*{35}{l}}
\n\\theta +3\\theta =\\frac{3\\pi }{2}+6k\\pi\u00a0 & \\vee\u00a0 & \\theta -3\\theta =-\\frac{3\\pi }{2}+6k\\pi \\,,\\ k\\in \\mathbb{Z}\u00a0 \\\\
\n\\end{array}\u00a0 \\\\
\n{} & \\Leftrightarrow\u00a0 & \\begin{array}{*{35}{l}}
\n\\theta =\\frac{3\\pi }{8}+\\frac{3k\\pi }{2} & \\vee\u00a0 & \\theta =\\frac{3\\pi }{4}+3k\\pi \\,,\\ k\\in \\mathbb{Z}\u00a0 \\\\
\n\\end{array}\u00a0 \\\\
\n\\end{array}\\]
\n
\nOra,
\n\\[\\begin{array}{*{35}{l}}
\nt{{g}^{2}}\\,\\theta =1 & \\Leftrightarrow\u00a0 & \\begin{matrix}
\ntg\\,\\theta =-1 & \\vee\u00a0 & tg\\,\\theta =1\u00a0 \\\\
\n\\end{matrix}\u00a0 \\\\
\n{} & \\Leftrightarrow\u00a0 & \\begin{matrix}
\n\\theta =-\\frac{\\pi }{4}+k\\pi\u00a0 & \\vee\u00a0 & \\theta =\\frac{\\pi }{4}+k\\pi \\,,\\ k\\in \\mathbb{Z}\u00a0 \\\\
\n\\end{matrix}\u00a0 \\\\
\n{} & \\Leftrightarrow\u00a0 & \\theta =\\mp \\frac{\\pi }{4}+k\\pi \\,,\\ k\\in \\mathbb{Z}\u00a0 \\\\
\n\\end{array}\\]
\n
\nOra,
\n\\[\\begin{array}{*{35}{l}}
\n1+2\\,sen\\,\\theta =0 & \\Leftrightarrow\u00a0 & sen\\,\\theta =-\\frac{1}{2}\u00a0 \\\\
\n{} & \\Leftrightarrow\u00a0 & \\begin{array}{*{35}{l}}
\n\\theta =\\frac{7\\pi }{6}+2k\\pi\u00a0 & \\vee\u00a0 & \\theta =(\\pi -\\frac{7\\pi }{6})+2k\\pi \\,,\\ k\\in \\mathbb{Z}\u00a0 \\\\
\n\\end{array}\u00a0 \\\\
\n{} & \\Leftrightarrow\u00a0 & \\begin{array}{*{35}{l}}
\n\\theta =\\frac{7\\pi }{6}+2k\\pi\u00a0 & \\vee\u00a0 & \\theta =-\\frac{\\pi }{6}+2k\\pi \\,,\\ k\\in \\mathbb{Z}\u00a0 \\\\
\n\\end{array}\u00a0 \\\\
\n\\end{array}\\]
\n
\nOra,
\n\\[\\begin{array}{*{35}{l}}
\n2\\,sen\\,\\theta +\\sqrt{3}=0 & \\Leftrightarrow\u00a0 & sen\\,\\theta =-\\frac{\\sqrt{3}}{2}\u00a0 \\\\
\n{} & \\Leftrightarrow\u00a0 & \\begin{array}{*{35}{l}}
\n\\theta =\\frac{4\\pi }{3}+2k\\pi\u00a0 & \\vee\u00a0 & \\theta =(\\pi -\\frac{4\\pi }{3})+2k\\pi \\,,\\ k\\in \\mathbb{Z}\u00a0 \\\\
\n\\end{array}\u00a0 \\\\
\n{} & \\Leftrightarrow\u00a0 & \\begin{array}{*{35}{l}}
\n\\theta =\\frac{4\\pi }{3}+2k\\pi\u00a0 & \\vee\u00a0 & \\theta =-\\frac{\\pi }{3}+2k\\pi \\,,\\ k\\in \\mathbb{Z}\u00a0 \\\\
\n\\end{array}\u00a0 \\\\
\n\\end{array}\\]
\n
\nOra,
\n\\[\\begin{array}{*{35}{l}}
\n5-5\\cos \\,(2\\theta )=0 & \\Leftrightarrow\u00a0 & \\cos \\,(2\\theta )=1\u00a0 \\\\
\n{} & \\Leftrightarrow\u00a0 & 2\\theta =\\mp 0+2k\\pi \\,,\\ k\\in \\mathbb{Z}\u00a0 \\\\
\n{} & \\Leftrightarrow\u00a0 & \\theta =k\\pi \\,,\\ k\\in \\mathbb{Z}\u00a0 \\\\
\n\\end{array}\\]<\/p>\n