3cos(3t)=0

{ "query": { "display": "$$3\\cos\\left(3t\\right)=0$$", "symbolab_question": "EQUATION#3\\cos(3t)=0" }, "solution": { "level": "PERFORMED", "subject": "Trigonometry", "topic": "Trig Equations", "subTopic": "Trig Equations", "default": "t=\\frac{π}{6}+\\frac{2πn}{3},t=\\frac{π}{2}+\\frac{2πn}{3}", "degrees": "t=30^{\\circ }+120^{\\circ }n,t=90^{\\circ }+120^{\\circ }n", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$3\\cos\\left(3t\\right)=0{\\quad:\\quad}t=\\frac{π}{6}+\\frac{2πn}{3},\\:t=\\frac{π}{2}+\\frac{2πn}{3}$$", "input": "3\\cos\\left(3t\\right)=0", "steps": [ { "type": "interim", "title": "Divide both sides by $$3$$", "input": "3\\cos\\left(3t\\right)=0", "result": "\\cos\\left(3t\\right)=0", "steps": [ { "type": "step", "primary": "Divide both sides by $$3$$", "result": "\\frac{3\\cos\\left(3t\\right)}{3}=\\frac{0}{3}" }, { "type": "step", "primary": "Simplify", "result": "\\cos\\left(3t\\right)=0" } ], "meta": { "interimType": "Divide Both Sides Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7U1+2nCPXEar7tTVMd80nJ+Z7FG6FY0FlWeDmdTF+fFPVh90DS7J9J40zZjnm5cN3cnK4iZsgGBXjwFxe9i6qwB5BaNcRvf57ITSIqANtbAO9nVUdbZPFHCqx9x8nEmMHT6E+Z+lYFfRqAnQ26sDvnCwtYv+gYP75aLTZcT7K4J2HjCJOLglkInccUWJVHAEMOkr2eGdxDqvMbGz01KDfKIG6+XMfwW58BnXoJUdnzrpclIZfLzRJ3bRksUs1lMp48hXn+PJfFc3Uh1JOPr5wj0iZ3bB0BgUtwTNr3KnZpFWw14FmAV06QSwgd736/nVs5nTxhhX/JMpykzZI6J3VrKGoxMw626x3PqNkZlKxJbSBBTEk/JQ2cZ9WKuRzClU7xtW5QjyZI5HEMqcH8fiPqoXuhKi+83aYPR9xi16pqTk=" } }, { "type": "interim", "title": "General solutions for $$\\cos\\left(3t\\right)=0$$", "result": "3t=\\frac{π}{2}+2πn,\\:3t=\\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": "3t=\\frac{π}{2}+2πn,\\:3t=\\frac{3π}{2}+2πn" } ], "meta": { "interimType": "Trig General Solutions cos 1Eq" } }, { "type": "interim", "title": "Solve $$3t=\\frac{π}{2}+2πn:{\\quad}t=\\frac{π}{6}+\\frac{2πn}{3}$$", "input": "3t=\\frac{π}{2}+2πn", "steps": [ { "type": "interim", "title": "Divide both sides by $$3$$", "input": "3t=\\frac{π}{2}+2πn", "result": "t=\\frac{π}{6}+\\frac{2πn}{3}", "steps": [ { "type": "step", "primary": "Divide both sides by $$3$$", "result": "\\frac{3t}{3}=\\frac{\\frac{π}{2}}{3}+\\frac{2πn}{3}" }, { "type": "interim", "title": "Simplify", "input": "\\frac{3t}{3}=\\frac{\\frac{π}{2}}{3}+\\frac{2πn}{3}", "result": "t=\\frac{π}{6}+\\frac{2πn}{3}", "steps": [ { "type": "interim", "title": "Simplify $$\\frac{3t}{3}:{\\quad}t$$", "input": "\\frac{3t}{3}", "steps": [ { "type": "step", "primary": "Divide the numbers: $$\\frac{3}{3}=1$$", "result": "=t" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7qCMOfgR5adfYkbXZhloWhi061ljBSPJeENOw2efoSWsLFuJmNSu3scQUBEAUF37pRSpN33oxZMojoqvYhvSJACLkpiVA7S8UT8ieezZKu6qNNHaM4tjKmlZtLx26oXkK" } }, { "type": "interim", "title": "Simplify $$\\frac{\\frac{π}{2}}{3}+\\frac{2πn}{3}:{\\quad}\\frac{π}{6}+\\frac{2πn}{3}$$", "input": "\\frac{\\frac{π}{2}}{3}+\\frac{2πn}{3}", "steps": [ { "type": "interim", "title": "$$\\frac{\\frac{π}{2}}{3}=\\frac{π}{6}$$", "input": "\\frac{\\frac{π}{2}}{3}", "steps": [ { "type": "step", "primary": "Apply the fraction rule: $$\\frac{\\frac{b}{c}}{a}=\\frac{b}{c\\:\\cdot\\:a}$$", "result": "=\\frac{π}{2\\cdot\\:3}" }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:3=6$$", "result": "=\\frac{π}{6}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7tiTmKkmeEqHWILbBb88ajdU2utO1NDv4gIlh/66PqgrNGoPE9TME3q+OPmgkv2RQV7wanvTgjKqnID/UMq66G9DvBptQbBir7GwZ+GlV8S++/XUfFY44oQy7yrU6sldlPl7acUDUp+uHLnpG1rz26ek1nlqWS2nkMNwVvpLj2uw=" } }, { "type": "step", "result": "=\\frac{π}{6}+\\frac{2πn}{3}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7tiTmKkmeEqHWILbBb88ajf4x2k3NLD/hOs5Mdx7C6jSddNoa0JibEfgUXb69v4MPzRqDxPUzBN6vjj5oJL9kUFe8Gp704IyqpyA/1DKuuhsItVpJWhR3QDi6bni0CS279yPegwOu0B0iWFG/q1vMfJjcBIL5pmo83UMFZRSzJsUt8Vz4ri5I/oyZLRAY9v9SVOJ2fOgPewbgX5w12u6Ro/xmJzKLCps2zTe0VCJi6hO/Mg94S0N9we//Py6WzxN6" } }, { "type": "step", "result": "t=\\frac{π}{6}+\\frac{2πn}{3}" } ], "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 $$3t=\\frac{3π}{2}+2πn:{\\quad}t=\\frac{π}{2}+\\frac{2πn}{3}$$", "input": "3t=\\frac{3π}{2}+2πn", "steps": [ { "type": "interim", "title": "Divide both sides by $$3$$", "input": "3t=\\frac{3π}{2}+2πn", "result": "t=\\frac{π}{2}+\\frac{2πn}{3}", "steps": [ { "type": "step", "primary": "Divide both sides by $$3$$", "result": "\\frac{3t}{3}=\\frac{\\frac{3π}{2}}{3}+\\frac{2πn}{3}" }, { "type": "interim", "title": "Simplify", "input": "\\frac{3t}{3}=\\frac{\\frac{3π}{2}}{3}+\\frac{2πn}{3}", "result": "t=\\frac{π}{2}+\\frac{2πn}{3}", "steps": [ { "type": "interim", "title": "Simplify $$\\frac{3t}{3}:{\\quad}t$$", "input": "\\frac{3t}{3}", "steps": [ { "type": "step", "primary": "Divide the numbers: $$\\frac{3}{3}=1$$", "result": "=t" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7qCMOfgR5adfYkbXZhloWhi061ljBSPJeENOw2efoSWsLFuJmNSu3scQUBEAUF37pRSpN33oxZMojoqvYhvSJACLkpiVA7S8UT8ieezZKu6qNNHaM4tjKmlZtLx26oXkK" } }, { "type": "interim", "title": "Simplify $$\\frac{\\frac{3π}{2}}{3}+\\frac{2πn}{3}:{\\quad}\\frac{π}{2}+\\frac{2πn}{3}$$", "input": "\\frac{\\frac{3π}{2}}{3}+\\frac{2πn}{3}", "steps": [ { "type": "interim", "title": "$$\\frac{\\frac{3π}{2}}{3}=\\frac{π}{2}$$", "input": "\\frac{\\frac{3π}{2}}{3}", "steps": [ { "type": "step", "primary": "Apply the fraction rule: $$\\frac{\\frac{b}{c}}{a}=\\frac{b}{c\\:\\cdot\\:a}$$", "result": "=\\frac{3π}{2\\cdot\\:3}" }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:3=6$$", "result": "=\\frac{3π}{6}" }, { "type": "step", "primary": "Cancel the common factor: $$3$$", "result": "=\\frac{π}{2}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7tiTmKkmeEqHWILbBb88ajaaxrMgtxqxItjWpVJaLAobdd47a0hQ8flDbGsI5To1d+bRvmdH+XexJq0YRmlk3drKAEC1immJ8buNmGLd/2Z+IDZ1VqNwwAzGjxOWVq2SRJ2ijmmTW7DUH5cQghTxZhzAsLO/twYclA9e0Ao3wf0k=" } }, { "type": "step", "result": "=\\frac{π}{2}+\\frac{2πn}{3}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7tiTmKkmeEqHWILbBb88ajdrNDsI7OTwDDhLH4ub/7pk2BuryoOOEqPAk8viwO4hR3XeO2tIUPH5Q2xrCOU6NXfm0b5nR/l3sSatGEZpZN3bF85/p6iRd7AWrGKOnhJOgE6v794USxh0EoAWD1ttBCosaY2cs9YZ/08feWmaOgkhPpaTWPScnc3H2GfYFvoQGjzD7MOXPIGfDOjIDyfQd7LxnamvDLMW8R+aRNfa0VTmJqVxX90jlMfh9fKn6dzC4" } }, { "type": "step", "result": "t=\\frac{π}{2}+\\frac{2πn}{3}" } ], "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": "step", "result": "t=\\frac{π}{6}+\\frac{2πn}{3},\\:t=\\frac{π}{2}+\\frac{2πn}{3}" } ], "meta": { "solvingClass": "Trig Equations", "practiceLink": "/practice/trigonometry-practice#area=main&subtopic=Trig%20Equations", "practiceTopic": "Trig Equations" } }, "plot_output": { "meta": { "plotInfo": { "variable": "t", "plotRequest": "3\\cos(3t)" }, "showViewLarger": true } }, "meta": { "showVerify": true } }