solvefor x,-cos(x)=0

{ "query": { "display": "solve for $$x,\\:-\\cos\\left(x\\right)=0$$", "symbolab_question": "SOLVE_FOR#solvefor x,-\\cos(x)=0" }, "solution": { "level": "PERFORMED", "subject": "Trigonometry", "topic": "Trig Equations", "subTopic": "Trig Equations", "default": "x=\\frac{π}{2}+2πn,x=\\frac{3π}{2}+2πn", "degrees": "x=90^{\\circ }+360^{\\circ }n,x=270^{\\circ }+360^{\\circ }n", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$solve\\:for\\:x,\\:-\\cos\\left(x\\right)=0{\\quad:\\quad}x=\\frac{π}{2}+2πn,\\:x=\\frac{3π}{2}+2πn$$", "input": "-\\cos\\left(x\\right)=0", "steps": [ { "type": "interim", "title": "Divide both sides by $$-1$$", "input": "-\\cos\\left(x\\right)=0", "result": "\\cos\\left(x\\right)=0", "steps": [ { "type": "step", "primary": "Divide both sides by $$-1$$", "result": "\\frac{-\\cos\\left(x\\right)}{-1}=\\frac{0}{-1}" }, { "type": "step", "primary": "Simplify", "result": "\\cos\\left(x\\right)=0" } ], "meta": { "interimType": "Divide Both Sides Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7sxSOfOl7Um+zqSbarNITtl8cbxFUMoDCVak3t/iKzS5Pnie/F7TDTqTxkucfXTe/aPjNWQqta/TPfIU5KdkDBR5BaNcRvf57ITSIqANtbANRRP/JaC8tC1WMi/z+2Ga3Du6nN7PwvLYQ5uRWlJn0/E3Phs7LzsB07w4E2lV5LIECLH8RFNHPqNWjTGGCrIw4KXGT2ZvqVP5WwVfXop0HoYFIHExjr9FM7rrF2q2YNulclIZfLzRJ3bRksUs1lMp48hXn+PJfFc3Uh1JOPr5wj0iZ3bB0BgUtwTNr3KnZpFWw14FmAV06QSwgd736/nVs5nTxhhX/JMpykzZI6J3VrKGoxMw626x3PqNkZlKxJbSBBTEk/JQ2cZ9WKuRzClU7xtW5QjyZI5HEMqcH8fiPqqohZmmRn6yVOqViMWGzG8g=" } }, { "type": "interim", "title": "General solutions for $$\\cos\\left(x\\right)=0$$", "result": "x=\\frac{π}{2}+2πn,\\:x=\\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": "x=\\frac{π}{2}+2πn,\\:x=\\frac{3π}{2}+2πn" } ], "meta": { "interimType": "Trig General Solutions cos 1Eq" } } ], "meta": { "solvingClass": "Trig Equations", "practiceLink": "/practice/trigonometry-practice#area=main&subtopic=Trig%20Equations", "practiceTopic": "Trig Equations" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "-\\cos(x)" }, "showViewLarger": true } }, "meta": { "showVerify": true } }