2cos(5x)=0

{ "query": { "display": "$$2\\cos\\left(5x\\right)=0$$", "symbolab_question": "EQUATION#2\\cos(5x)=0" }, "solution": { "level": "PERFORMED", "subject": "Trigonometry", "topic": "Trig Equations", "subTopic": "Trig Equations", "default": "x=\\frac{π}{10}+\\frac{2πn}{5},x=\\frac{3π}{10}+\\frac{2πn}{5}", "degrees": "x=18^{\\circ }+72^{\\circ }n,x=54^{\\circ }+72^{\\circ }n", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$2\\cos\\left(5x\\right)=0{\\quad:\\quad}x=\\frac{π}{10}+\\frac{2πn}{5},\\:x=\\frac{3π}{10}+\\frac{2πn}{5}$$", "input": "2\\cos\\left(5x\\right)=0", "steps": [ { "type": "interim", "title": "Divide both sides by $$2$$", "input": "2\\cos\\left(5x\\right)=0", "result": "\\cos\\left(5x\\right)=0", "steps": [ { "type": "step", "primary": "Divide both sides by $$2$$", "result": "\\frac{2\\cos\\left(5x\\right)}{2}=\\frac{0}{2}" }, { "type": "step", "primary": "Simplify", "result": "\\cos\\left(5x\\right)=0" } ], "meta": { "interimType": "Divide Both Sides Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s791AXYceO9Pc4CRAMY99YBXkGNJ3uZ4uLtQ2W2O4XsNrVh90DS7J9J40zZjnm5cN365VeVDIdizrNAR8IgglmAx5BaNcRvf57ITSIqANtbAPqwuu9Arryw70X55FIDLLiT6E+Z+lYFfRqAnQ26sDvnCwtYv+gYP75aLTZcT7K4J2HjCJOLglkInccUWJVHAEMOkr2eGdxDqvMbGz01KDfKINHJwudpsYKtj0vKoD6/FxclIZfLzRJ3bRksUs1lMp48hXn+PJfFc3Uh1JOPr5wj0iZ3bB0BgUtwTNr3KnZpFWw14FmAV06QSwgd736/nVs5nTxhhX/JMpykzZI6J3VrKGoxMw626x3PqNkZlKxJbSBBTEk/JQ2cZ9WKuRzClU7xtW5QjyZI5HEMqcH8fiPqrZpt3i/0kd0fBhbYBZdE0E=" } }, { "type": "interim", "title": "General solutions for $$\\cos\\left(5x\\right)=0$$", "result": "5x=\\frac{π}{2}+2πn,\\:5x=\\frac{3π}{2}+2πn", "steps": [ { "type": "step", "primary": "$$\\cos\\left(x\\right)$$ periodicity table with $$2πn$$ cycle:<br/>$$\\begin{array}{|c|c|c|c|}\\hline x&\\cos(x)&x&\\cos(x)\\\\\\hline 0&1&π&-1\\\\\\hline \\frac{π}{6}&\\frac{\\sqrt{3}}{2}&\\frac{7π}{6}&-\\frac{\\sqrt{3}}{2}\\\\\\hline \\frac{π}{4}&\\frac{\\sqrt{2}}{2}&\\frac{5π}{4}&-\\frac{\\sqrt{2}}{2}\\\\\\hline \\frac{π}{3}&\\frac{1}{2}&\\frac{4π}{3}&-\\frac{1}{2}\\\\\\hline \\frac{π}{2}&0&\\frac{3π}{2}&0\\\\\\hline \\frac{2π}{3}&-\\frac{1}{2}&\\frac{5π}{3}&\\frac{1}{2}\\\\\\hline \\frac{3π}{4}&-\\frac{\\sqrt{2}}{2}&\\frac{7π}{4}&\\frac{\\sqrt{2}}{2}\\\\\\hline \\frac{5π}{6}&-\\frac{\\sqrt{3}}{2}&\\frac{11π}{6}&\\frac{\\sqrt{3}}{2}\\\\\\hline \\end{array}$$" }, { "type": "step", "result": "5x=\\frac{π}{2}+2πn,\\:5x=\\frac{3π}{2}+2πn" } ], "meta": { "interimType": "Trig General Solutions cos 1Eq" } }, { "type": "interim", "title": "Solve $$5x=\\frac{π}{2}+2πn:{\\quad}x=\\frac{π}{10}+\\frac{2πn}{5}$$", "input": "5x=\\frac{π}{2}+2πn", "steps": [ { "type": "interim", "title": "Divide both sides by $$5$$", "input": "5x=\\frac{π}{2}+2πn", "result": "x=\\frac{π}{10}+\\frac{2πn}{5}", "steps": [ { "type": "step", "primary": "Divide both sides by $$5$$", "result": "\\frac{5x}{5}=\\frac{\\frac{π}{2}}{5}+\\frac{2πn}{5}" }, { "type": "interim", "title": "Simplify", "input": "\\frac{5x}{5}=\\frac{\\frac{π}{2}}{5}+\\frac{2πn}{5}", "result": "x=\\frac{π}{10}+\\frac{2πn}{5}", "steps": [ { "type": "interim", "title": "Simplify $$\\frac{5x}{5}:{\\quad}x$$", "input": "\\frac{5x}{5}", "steps": [ { "type": "step", "primary": "Divide the numbers: $$\\frac{5}{5}=1$$", "result": "=x" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7UbPonvUjYYAFpt9k9qwz7S061ljBSPJeENOw2efoSWt2cZ6LAXMcKee6PSyjLEFrRSpN33oxZMojoqvYhvSJACLkpiVA7S8UT8ieezZKu6oJZcORqLkTYBI6AfJrEln+" } }, { "type": "interim", "title": "Simplify $$\\frac{\\frac{π}{2}}{5}+\\frac{2πn}{5}:{\\quad}\\frac{π}{10}+\\frac{2πn}{5}$$", "input": "\\frac{\\frac{π}{2}}{5}+\\frac{2πn}{5}", "steps": [ { "type": "interim", "title": "$$\\frac{\\frac{π}{2}}{5}=\\frac{π}{10}$$", "input": "\\frac{\\frac{π}{2}}{5}", "steps": [ { "type": "step", "primary": "Apply the fraction rule: $$\\frac{\\frac{b}{c}}{a}=\\frac{b}{c\\:\\cdot\\:a}$$", "result": "=\\frac{π}{2\\cdot\\:5}" }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:5=10$$", "result": "=\\frac{π}{10}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7tiTmKkmeEqHWILbBb88ajX9yL9JUl6RYWn7SUDzLA3nNGoPE9TME3q+OPmgkv2RQV7wanvTgjKqnID/UMq66G78qkW+ln13/bPevjVBs4xCIDZ1VqNwwAzGjxOWVq2SR0IqBdNuEG0Vz6Z2SyvSMBJsYe6Aui24C95joY5d6FAU=" } }, { "type": "step", "result": "=\\frac{π}{10}+\\frac{2πn}{5}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7tiTmKkmeEqHWILbBb88ajXVJDouDPGryhBm3p8nv/nCndWOzNcm2xwdcdA7QJhcqzRqDxPUzBN6vjj5oJL9kUFe8Gp704IyqpyA/1DKuuhv7h+bnwBFcROMyVuGYebEumkFcN9ad21BLANVASTrE5osaY2cs9YZ/08feWmaOgkhPpaTWPScnc3H2GfYFvoQGrH/aKCFXqlOx3c4+RPJW/qldG/poWDTOAXrILx0RCv2wiNrEngO+NNvZ9sqNu+2V" } }, { "type": "step", "result": "x=\\frac{π}{10}+\\frac{2πn}{5}" } ], "meta": { "interimType": "Generic Simplify 0Eq" } } ], "meta": { "interimType": "Divide Both Sides Specific 1Eq", "gptData": "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" } } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve Title 1Eq" } }, { "type": "interim", "title": "Solve $$5x=\\frac{3π}{2}+2πn:{\\quad}x=\\frac{3π}{10}+\\frac{2πn}{5}$$", "input": "5x=\\frac{3π}{2}+2πn", "steps": [ { "type": "interim", "title": "Divide both sides by $$5$$", "input": "5x=\\frac{3π}{2}+2πn", "result": "x=\\frac{3π}{10}+\\frac{2πn}{5}", "steps": [ { "type": "step", "primary": "Divide both sides by $$5$$", "result": "\\frac{5x}{5}=\\frac{\\frac{3π}{2}}{5}+\\frac{2πn}{5}" }, { "type": "interim", "title": "Simplify", "input": "\\frac{5x}{5}=\\frac{\\frac{3π}{2}}{5}+\\frac{2πn}{5}", "result": "x=\\frac{3π}{10}+\\frac{2πn}{5}", "steps": [ { "type": "interim", "title": "Simplify $$\\frac{5x}{5}:{\\quad}x$$", "input": "\\frac{5x}{5}", "steps": [ { "type": "step", "primary": "Divide the numbers: $$\\frac{5}{5}=1$$", "result": "=x" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7UbPonvUjYYAFpt9k9qwz7S061ljBSPJeENOw2efoSWt2cZ6LAXMcKee6PSyjLEFrRSpN33oxZMojoqvYhvSJACLkpiVA7S8UT8ieezZKu6oJZcORqLkTYBI6AfJrEln+" } }, { "type": "interim", "title": "Simplify $$\\frac{\\frac{3π}{2}}{5}+\\frac{2πn}{5}:{\\quad}\\frac{3π}{10}+\\frac{2πn}{5}$$", "input": "\\frac{\\frac{3π}{2}}{5}+\\frac{2πn}{5}", "steps": [ { "type": "interim", "title": "$$\\frac{\\frac{3π}{2}}{5}=\\frac{3π}{10}$$", "input": "\\frac{\\frac{3π}{2}}{5}", "steps": [ { "type": "step", "primary": "Apply the fraction rule: $$\\frac{\\frac{b}{c}}{a}=\\frac{b}{c\\:\\cdot\\:a}$$", "result": "=\\frac{3π}{2\\cdot\\:5}" }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:5=10$$", "result": "=\\frac{3π}{10}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7tiTmKkmeEqHWILbBb88ajaq0OOW8OLlL92dkFMYv0DTdd47a0hQ8flDbGsI5To1drfMRPynLSfP3VYe1OQldRLlxMv8vXOWjSyJazwtmWNwjigtccxNWJk7pEnO8OW1UYAt8h5hJ0qMwhX7GJjGFQte7PfaLFbqiRxc/AQzAf5iwiNrEngO+NNvZ9sqNu+2V" } }, { "type": "step", "result": "=\\frac{3π}{10}+\\frac{2πn}{5}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": 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"practiceTopic": "Trig Equations" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "2\\cos(5x)" }, "showViewLarger": true } }, "meta": { "showVerify": true } }