Resolva as equa\u00e7\u00f5es trigonom\u00e9tricas que se seguem.<\/p>\n
1.<\/strong> 2.<\/strong> 3.<\/strong> 4.<\/strong> 5.<\/strong> 6.<\/strong> 7.<\/strong> 8.<\/strong> 9.<\/strong> 10.<\/strong> 11.<\/strong> 12.<\/strong>
\nOra,
\n\\[\\begin{array}{*{35}{l}}
\nsen\\,\\theta =-\\cos \\frac{\\pi }{3} & \\Leftrightarrow\u00a0 & \\cos (\\frac{\\pi }{2}-\\theta )=\\cos (\\pi +\\frac{\\pi }{3})\u00a0 \\\\
\n{} & \\Leftrightarrow\u00a0 & \\frac{\\pi }{2}-\\theta =\\mp \\frac{4\\pi }{3}+2k\\pi \\,,\\,\\,k\\in \\mathbb{Z}\u00a0 \\\\
\n{} & \\Leftrightarrow\u00a0 & \\begin{array}{*{35}{l}}
\n-\\theta =-\\frac{\\pi }{2}-\\frac{4\\pi }{3}+2k\\pi\u00a0 & \\vee\u00a0 & -\\theta =-\\frac{\\pi }{2}+\\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{11\\pi }{6}+2k\\pi\u00a0 & \\vee\u00a0 & \\theta =-\\frac{5\\pi }{6}+2k\\pi \\,,\\,\\,k\\in \\mathbb{Z}\u00a0 \\\\
\n\\end{array}\u00a0 \\\\
\n\\end{array}\\]
\nAlternativa 1:
\n\\[\\begin{array}{*{35}{l}}
\nsen\\,\\theta =-\\cos \\frac{\\pi }{3} & \\Leftrightarrow\u00a0 & sen\\,\\theta =\\cos (\\pi +\\frac{\\pi }{3})\u00a0 \\\\
\n{} & \\Leftrightarrow\u00a0 & sen\\,\\theta =sen\\,(\\frac{\\pi }{2}-(\\pi +\\frac{\\pi }{3}))\u00a0 \\\\
\n{} & \\Leftrightarrow\u00a0 & \\begin{array}{*{35}{l}}
\n\\theta =-\\frac{5\\pi }{6}+2k\\pi\u00a0 & \\vee\u00a0 & \\theta =(\\pi -(-\\frac{5\\pi }{6}))+2k\\pi \\,,\\,\\,k\\in \\mathbb{Z}\u00a0 \\\\
\n\\end{array}\u00a0 \\\\
\n{} & \\Leftrightarrow\u00a0 & \\begin{array}{*{35}{l}}
\n\\theta =-\\frac{5\\pi }{6}+2k\\pi\u00a0 & \\vee\u00a0 & \\theta =\\frac{11\\pi }{6}+2k\\pi \\,,\\,\\,k\\in \\mathbb{Z}\u00a0 \\\\
\n\\end{array}\u00a0 \\\\
\n\\end{array}\\]
\nAlternativa 2:
\n\\[\\begin{array}{*{35}{l}}
\nsen\\,\\theta =-\\cos \\frac{\\pi }{3} & \\Leftrightarrow\u00a0 & -sen\\,(-\\theta )=-\\cos \\frac{\\pi }{3}\u00a0 \\\\
\n{} & \\Leftrightarrow\u00a0 & sen\\,(-\\theta )=sen\\,(\\frac{\\pi }{2}-\\frac{\\pi }{3})\u00a0 \\\\
\n{} & \\Leftrightarrow\u00a0 & \\begin{array}{*{35}{l}}
\n-\\theta =\\frac{\\pi }{6}+2k\\pi\u00a0 & \\vee\u00a0 & -\\theta =(\\pi -\\frac{\\pi }{6})+2k\\pi \\,,\\,\\,k\\in \\mathbb{Z}\u00a0 \\\\
\n\\end{array}\u00a0 \\\\
\n{} & \\Leftrightarrow\u00a0 & \\begin{array}{*{35}{l}}
\n\\theta =-\\frac{\\pi }{6}+2k\\pi\u00a0 & \\vee\u00a0 & \\theta =-\\frac{5\\pi }{6}+2k\\pi \\,,\\,\\,k\\in \\mathbb{Z}\u00a0 \\\\
\n\\end{array}\u00a0 \\\\
\n\\end{array}\\]
\n
\nOra,
\n\\[\\begin{array}{*{35}{l}}
\nsen\\,\\theta =\\cos \\frac{\\pi }{5} & \\Leftrightarrow\u00a0 & sen\\,\\theta =sen\\,(\\frac{\\pi }{2}-\\frac{\\pi }{5})\u00a0 \\\\
\n{} & \\Leftrightarrow\u00a0 & \\begin{array}{*{35}{l}}
\n\\theta =\\frac{3\\pi }{10}+2k\\pi\u00a0 & \\vee\u00a0 & \\theta =(\\pi -\\frac{3\\pi }{10})+2k\\pi \\,,\\,\\,k\\in \\mathbb{Z}\u00a0 \\\\
\n\\end{array}\u00a0 \\\\
\n{} & \\Leftrightarrow\u00a0 & \\begin{array}{*{35}{l}}
\n\\theta =\\frac{3\\pi }{10}+2k\\pi\u00a0 & \\vee\u00a0 & \\theta =\\frac{7\\pi }{10}+2k\\pi \\,,\\,\\,k\\in \\mathbb{Z}\u00a0 \\\\
\n\\end{array}\u00a0 \\\\
\n\\end{array}\\]
\n
\nOra,
\n\\[\\begin{array}{*{35}{l}}
\n\\cos \\,\\theta =\\cos (\\frac{3\\pi }{2}-\\theta ) & \\Leftrightarrow\u00a0 & \\theta =\\mp (\\frac{3\\pi }{2}-\\theta )+2k\\pi \\,,\\,\\,k\\in \\mathbb{Z}\u00a0 \\\\
\n{} & \\Leftrightarrow\u00a0 & \\begin{array}{*{35}{l}}
\n2\\theta =\\frac{3\\pi }{2}+2k\\pi\u00a0 & \\vee\u00a0 & 0\\times \\theta =-\\frac{3\\pi }{2}+2k\\pi \\,,\\,\\,k\\in \\mathbb{Z}\u00a0 \\\\
\n\\end{array}\u00a0 \\\\
\n{} & \\Leftrightarrow\u00a0 & \\theta =\\frac{3\\pi }{4}+k\\pi \\,,\\,\\,k\\in \\mathbb{Z}\u00a0 \\\\
\n\\end{array}\\]
\n
\nOra,
\n\\[\\begin{array}{*{35}{l}}
\ntg\\,\\theta \\times \\cos \\theta =0 & \\Leftrightarrow\u00a0 & \\begin{array}{*{35}{l}}
\ntg\\,\\theta =0 & \\vee\u00a0 & \\cos \\theta =0\u00a0 \\\\
\n\\end{array}\u00a0 \\\\
\n{} & \\Leftrightarrow\u00a0 & \\begin{array}{*{35}{l}}
\n\\theta =k\\pi\u00a0 & \\vee\u00a0 & \\theta =\\frac{\\pi }{2}+k\\pi \\,,\\,\\,k\\in \\mathbb{Z}\u00a0 \\\\
\n\\end{array}\u00a0 \\\\
\n\\end{array}\\]
\n
\nOra,
\n\\[\\begin{array}{*{35}{l}}
\n(sen\\,\\theta )\\times (2\\cos \\theta -1)=0 & \\Leftrightarrow\u00a0 & \\begin{array}{*{35}{l}}
\nsen\\,\\theta =0 & \\vee\u00a0 & 2\\cos \\theta -1=0\u00a0 \\\\
\n\\end{array}\u00a0 \\\\
\n{} & \\Leftrightarrow\u00a0 & \\begin{array}{*{35}{l}}
\nsen\\,\\theta =0 & \\vee\u00a0 & \\cos \\theta =\\frac{1}{2}\u00a0 \\\\
\n\\end{array}\u00a0 \\\\
\n{} & \\Leftrightarrow\u00a0 & \\begin{array}{*{35}{l}}
\n\\theta =k\\pi\u00a0 & \\vee\u00a0 & \\theta =\\mp \\frac{\\pi }{3}+2k\\pi \\,,\\,\\,k\\in \\mathbb{Z}\u00a0 \\\\
\n\\end{array}\u00a0 \\\\
\n\\end{array}\\]
\n
\nOra,
\n\\[\\begin{array}{*{35}{l}}
\nsen\\,(\\theta -\\frac{\\pi }{6})=1 & \\Leftrightarrow\u00a0 & \\theta -\\frac{\\pi }{6}=\\frac{\\pi }{2}+2k\\pi \\,,\\,\\,k\\in \\mathbb{Z}\u00a0 \\\\
\n{} & \\Leftrightarrow\u00a0 & \\theta =\\frac{2\\pi }{3}+2k\\pi \\,,\\,\\,k\\in \\mathbb{Z}\u00a0 \\\\
\n\\end{array}\\]
\n
\nOra,
\n\\[\\begin{array}{*{35}{l}}
\nse{{n}^{2}}\\,\\theta +sen\\,\\theta =0 & \\Leftrightarrow\u00a0 & sen\\,\\theta \\times (sen\\,\\theta +1)=0\u00a0 \\\\
\n{} & \\Leftrightarrow\u00a0 & \\begin{array}{*{35}{l}}
\nsen\\,\\theta =0 & \\vee\u00a0 & sen\\,\\theta =-1\u00a0 \\\\
\n\\end{array}\u00a0 \\\\
\n{} & \\Leftrightarrow\u00a0 & \\begin{array}{*{35}{l}}
\n\\theta =k\\pi\u00a0 & \\vee\u00a0 & \\theta =-\\frac{\\pi }{2}+2k\\pi \\,,\\,\\,k\\in \\mathbb{Z}\u00a0 \\\\
\n\\end{array}\u00a0 \\\\
\n\\end{array}\\]
\n
\nOra,
\n\\[\\begin{array}{*{35}{l}}
\n\\cos \\,\\theta -sen\\,\\theta \\times \\cos \\theta =0 & \\Leftrightarrow\u00a0 & \\cos \\,\\theta \\times (1-sen\\,\\theta )=0\u00a0 \\\\
\n{} & \\Leftrightarrow\u00a0 & \\begin{array}{*{35}{l}}
\n\\cos \\,\\theta =0 & \\vee\u00a0 & sen\\,\\theta =1\u00a0 \\\\
\n\\end{array}\u00a0 \\\\
\n{} & \\Leftrightarrow\u00a0 & \\begin{array}{*{35}{l}}
\n\\theta =\\frac{\\pi }{2}+k\\pi\u00a0 & \\vee\u00a0 & \\theta =\\frac{\\pi }{2}+2k\\pi \\,,\\,\\,k\\in \\mathbb{Z}\u00a0 \\\\
\n\\end{array}\u00a0 \\\\
\n{} & \\Leftrightarrow\u00a0 & \\theta =\\frac{\\pi }{2}+k\\pi \\,,\\,\\,k\\in \\mathbb{Z}\\,\\,\\,\\,\\,\\text{(Porqu }\\!\\!\\hat{\\mathrm{e}}\\!\\!\\text{ ?)}\u00a0 \\\\
\n\\end{array}\\]
\n
\nOra,
\n\\[\\begin{array}{*{35}{l}}
\n\\cos \\,3\\theta =\\cos \\theta\u00a0 & \\Leftrightarrow\u00a0 & 3\\theta =\\mp \\theta +2k\\pi \\,,\\,\\,k\\in \\mathbb{Z}\u00a0 \\\\
\n{} & \\Leftrightarrow\u00a0 & \\begin{array}{*{35}{l}}
\n2\\theta =2k\\pi\u00a0 & \\vee\u00a0 & 4\\theta =2k\\pi \\,,\\,\\,k\\in \\mathbb{Z}\u00a0 \\\\
\n\\end{array}\u00a0 \\\\
\n{} & \\Leftrightarrow\u00a0 & \\begin{array}{*{35}{l}}
\n\\theta =k\\pi\u00a0 & \\vee\u00a0 & \\theta =\\frac{k\\pi }{2}\\,,\\,\\,k\\in \\mathbb{Z}\u00a0 \\\\
\n\\end{array}\u00a0 \\\\
\n{} & \\Leftrightarrow\u00a0 & \\theta =\\frac{k\\pi }{2}\\,,\\,\\,k\\in \\mathbb{Z}\u00a0 \\\\
\n\\end{array}\\]
\n
\nOra,
\n\\[\\begin{array}{*{35}{l}}
\n\\cos \\,(2\\theta +\\frac{\\pi }{6})=\\frac{\\sqrt{3}}{2} & \\Leftrightarrow\u00a0 & 2\\theta +\\frac{\\pi }{6}=\\mp \\frac{\\pi }{6}+2k\\pi \\,,\\,\\,k\\in \\mathbb{Z}\u00a0 \\\\
\n{} & \\Leftrightarrow\u00a0 & \\begin{array}{*{35}{l}}
\n2\\theta =2k\\pi\u00a0 & \\vee\u00a0 & 2\\theta =-\\frac{\\pi }{3}+2k\\pi \\,,\\,\\,k\\in \\mathbb{Z}\u00a0 \\\\
\n\\end{array}\u00a0 \\\\
\n{} & \\Leftrightarrow\u00a0 & \\begin{array}{*{35}{l}}
\n\\theta =k\\pi\u00a0 & \\vee\u00a0 & \\theta =-\\frac{\\pi }{6}+k\\pi \\,,\\,\\,k\\in \\mathbb{Z}\u00a0 \\\\
\n\\end{array}\u00a0 \\\\
\n\\end{array}\\]
\n
\nOra,
\n\\[\\begin{array}{*{35}{l}}
\n{{\\cos }^{2}}\\theta =1 & \\Leftrightarrow\u00a0 & \\begin{array}{*{35}{l}}
\n\\cos \\,\\theta =-1 & \\vee\u00a0 & \\cos \\,\\theta =1\u00a0 \\\\
\n\\end{array}\u00a0 \\\\
\n{} & \\Leftrightarrow\u00a0 & \\begin{array}{*{35}{l}}
\n\\theta =\\pi +2k\\pi\u00a0 & \\vee\u00a0 & \\theta =0+2k\\pi \\,,\\,\\,k\\in \\mathbb{Z}\u00a0 \\\\
\n\\end{array}\u00a0 \\\\
\n{} & \\Leftrightarrow\u00a0 & \\theta =k\\pi \\,,\\,\\,k\\in \\mathbb{Z}\\,\\,\\,\\,\\,\\text{(Porqu }\\!\\!\\hat{\\mathrm{e}}\\!\\!\\text{ ?)}\u00a0 \\\\
\n\\end{array}\\]
\n
\nOra,
\n\\[\\begin{array}{*{35}{l}}
\n-1+\\sqrt{2}\\,sen\\,\\theta =2 & \\Leftrightarrow\u00a0 & \\sqrt{2}\\,sen\\,\\theta =3\u00a0 \\\\
\n{} & \\Leftrightarrow\u00a0 & sen\\,\\theta =\\frac{3}{\\sqrt{2}}\u00a0 \\\\
\n{} & \\Leftrightarrow\u00a0 & sen\\,\\theta =\\frac{3\\sqrt{2}}{2}\u00a0 \\\\
\n{} & \\Leftrightarrow\u00a0 & \\theta \\in \\left\\{ {} \\right\\}\u00a0 \\\\
\n\\end{array}\\]<\/p>\n