sqrt(2)cos(x)-1=0,0<= x<= 2pi

{ "query": { "display": "$$\\sqrt{2}\\cos\\left(x\\right)-1=0,\\:0\\le\\:x\\le\\:2π$$", "symbolab_question": "EQUATION#\\sqrt{2}\\cos(x)-1=0,0\\le x\\le 2π" }, "solution": { "level": "PERFORMED", "subject": "Trigonometry", "topic": "Trig Equations", "subTopic": "Trig Equations", "default": "x=\\frac{π}{4},x=\\frac{7π}{4}", "degrees": "x=45^{\\circ },x=315^{\\circ }", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\sqrt{2}\\cos\\left(x\\right)-1=0,\\:0\\le\\:x\\le\\:2π{\\quad:\\quad}x=\\frac{π}{4},\\:x=\\frac{7π}{4}$$", "input": "\\sqrt{2}\\cos\\left(x\\right)-1=0,\\:0\\le\\:x\\le\\:2π", "steps": [ { "type": "interim", "title": "Move $$1\\:$$to the right side", "input": "\\sqrt{2}\\cos\\left(x\\right)-1=0", "result": "\\sqrt{2}\\cos\\left(x\\right)=1", "steps": [ { "type": "step", "primary": "Add $$1$$ to both sides", "result": "\\sqrt{2}\\cos\\left(x\\right)-1+1=0+1" }, { "type": "step", "primary": "Simplify", "result": "\\sqrt{2}\\cos\\left(x\\right)=1" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": "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" } }, { "type": "interim", "title": "Divide both sides by $$\\sqrt{2}$$", "input": "\\sqrt{2}\\cos\\left(x\\right)=1", "result": "\\cos\\left(x\\right)=\\frac{\\sqrt{2}}{2}", "steps": [ { "type": "step", "primary": "Divide both sides by $$\\sqrt{2}$$", "result": "\\frac{\\sqrt{2}\\cos\\left(x\\right)}{\\sqrt{2}}=\\frac{1}{\\sqrt{2}}" }, { "type": "interim", "title": "Simplify", "input": "\\frac{\\sqrt{2}\\cos\\left(x\\right)}{\\sqrt{2}}=\\frac{1}{\\sqrt{2}}", "result": "\\cos\\left(x\\right)=\\frac{\\sqrt{2}}{2}", "steps": [ { "type": "interim", "title": "Simplify $$\\frac{\\sqrt{2}\\cos\\left(x\\right)}{\\sqrt{2}}:{\\quad}\\cos\\left(x\\right)$$", "input": "\\frac{\\sqrt{2}\\cos\\left(x\\right)}{\\sqrt{2}}", "steps": [ { "type": "step", "primary": "Cancel the common factor: $$\\sqrt{2}$$", "result": "=\\cos\\left(x\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s74DwjiBDVBIb8+yh6fkkMMaaVueIzGPhcS6MNJUrXNW4kKHFGvyocNUizdTmaAyO8q47vuWedXv2WUg94ER8IwasnM+lFdK+lbOA6QVAeavaBBTEk/JQ2cZ9WKuRzClU7QG+dQXHhPmuaaYlTGyzk2a8audweY9KIVydfJkmL1iCn8QoAqXqo32pGMW2McTRgFkVDw5yEF2DEw4msQzKTew==" } }, { "type": "interim", "title": "Simplify $$\\frac{1}{\\sqrt{2}}:{\\quad}\\frac{\\sqrt{2}}{2}$$", "input": "\\frac{1}{\\sqrt{2}}", "steps": [ { "type": "step", "primary": "Multiply by the conjugate $$\\frac{\\sqrt{2}}{\\sqrt{2}}$$", "result": "=\\frac{1\\cdot\\:\\sqrt{2}}{\\sqrt{2}\\sqrt{2}}", "meta": { "title": { "extension": "To rationalize the denominator, multiply numerator and denominator by the conjugate of the radical $$\\sqrt{2}$$" } } }, { "type": "step", "primary": "$$1\\cdot\\:\\sqrt{2}=\\sqrt{2}$$" }, { "type": "interim", "title": "$$\\sqrt{2}\\sqrt{2}=2$$", "input": "\\sqrt{2}\\sqrt{2}", "result": "=\\frac{\\sqrt{2}}{2}", "steps": [ { "type": "step", "primary": "Apply radical rule: $$\\sqrt{a}\\sqrt{a}=a$$", "secondary": [ "$$\\sqrt{2}\\sqrt{2}=2$$" ], "result": "=2", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7OXRSLY4uXiMbvMfICP5T88aFSKFknkW7qg0U6E21MorMwViaLUXkeD+JukROhWdjBJz+17+I6kHbIEeqwhy4mdC5tTYOsrqRy+4iBsRgZTQLmgWGjI523GgzSlA2k6Cv" } } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s78B+yE0cCl8hT7YErNdnZGSQocUa/Khw1SLN1OZoDI7yrju+5Z51e/ZZSD3gRHwjBnvjDY21b5XBQ44AG3rKeoJPz7fKEZJdwb5+eH5xAg+lN5Aod6Hr1Lp2e/29KhSgUgQHNT2nz+ImvmnBFu276+QAWYM1eO6oSaF4qo2LhF0Akt3WiGR7ZaCaXvz77bMjS" } }, { "type": "step", "result": "\\cos\\left(x\\right)=\\frac{\\sqrt{2}}{2}" } ], "meta": { "interimType": "Generic Simplify 0Eq" } } ], "meta": { "interimType": "Divide Both Sides Specific 1Eq", "gptData": "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" } }, { "type": "interim", "title": "General solutions for $$\\cos\\left(x\\right)=\\frac{\\sqrt{2}}{2}$$", "result": "x=\\frac{π}{4}+2πn,\\:x=\\frac{7π}{4}+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}{4}+2πn,\\:x=\\frac{7π}{4}+2πn" } ], "meta": { "interimType": "Trig General Solutions cos 1Eq" } }, { "type": "step", "primary": "Solutions for the range $$0\\le\\:x\\le\\:2π$$", "result": "x=\\frac{π}{4},\\:x=\\frac{7π}{4}" } ], "meta": { "solvingClass": "Trig Equations", "practiceLink": "/practice/trigonometry-practice#area=main&subtopic=Trig%20Equations", "practiceTopic": "Trig Equations" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "\\sqrt{2}\\cos(x)-1,0\\le x\\le 2π" }, "showViewLarger": true } }, "meta": { "showVerify": true } }