sin^2(2x)+cos^2(x)-1=0

{ "query": { "display": "$$\\sin^{2}\\left(2x\\right)+\\cos^{2}\\left(x\\right)-1=0$$", "symbolab_question": "EQUATION#\\sin^{2}(2x)+\\cos^{2}(x)-1=0" }, "solution": { "level": "PERFORMED", "subject": "Trigonometry", "topic": "Trig Equations", "subTopic": "Trig Equations", "default": "x=\\frac{π}{3}+\\frac{4πn}{3},x=π+\\frac{4πn}{3},x=\\frac{4πn}{3},x=\\frac{2π}{3}+\\frac{4πn}{3}", "degrees": "x=60^{\\circ }+240^{\\circ }n,x=180^{\\circ }+240^{\\circ }n,x=0^{\\circ }+240^{\\circ }n,x=120^{\\circ }+240^{\\circ }n", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\sin^{2}\\left(2x\\right)+\\cos^{2}\\left(x\\right)-1=0{\\quad:\\quad}x=\\frac{π}{3}+\\frac{4πn}{3},\\:x=π+\\frac{4πn}{3},\\:x=\\frac{4πn}{3},\\:x=\\frac{2π}{3}+\\frac{4πn}{3}$$", "input": "\\sin^{2}\\left(2x\\right)+\\cos^{2}\\left(x\\right)-1=0", "steps": [ { "type": "interim", "title": "Rewrite using trig identities", "input": "-1+\\cos^{2}\\left(x\\right)+\\sin^{2}\\left(2x\\right)", "result": "-\\cos^{2}\\left(2x\\right)+\\cos^{2}\\left(x\\right)=0", "steps": [ { "type": "step", "primary": "Use the Pythagorean identity: $$1=\\cos^{2}\\left(x\\right)+\\sin^{2}\\left(x\\right)$$", "secondary": [ "$$1-\\sin^{2}\\left(x\\right)=\\cos^{2}\\left(x\\right)$$", "$$-1+\\sin^{2}\\left(x\\right)=-\\cos^{2}\\left(x\\right)$$" ], "result": "=\\cos^{2}\\left(x\\right)-\\cos^{2}\\left(2x\\right)" } ], "meta": { "interimType": "Trig Rewrite Using Trig identities 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7XQoqDkagBdkaGF3BP1M6VCtz9r12SJ1VUCm39rRHKkHSjyK9s/OUQCgLq3IDEU4H8wQyZ2Br8agbY8qkLUt6WPm0tu7qq7Iur/j8bUReEEOq5I4KKx/r65u7kTSvPypsHvDy11fJKM17UZGqO12p0jKZJoTPnWgEvvlyrrU52pMhpPLyCYrLk9jd6X5FtSaWSYU/rlY9QcIgZOnNP8QBxx62DrT5lSWo/mMH5F6V1QDkAxHFda36329rSpnJqDHJ" } }, { "type": "interim", "title": "Factor $$-\\cos^{2}\\left(2x\\right)+\\cos^{2}\\left(x\\right):{\\quad}\\left(\\cos\\left(x\\right)+\\cos\\left(2x\\right)\\right)\\left(\\cos\\left(x\\right)-\\cos\\left(2x\\right)\\right)$$", "input": "-\\cos^{2}\\left(2x\\right)+\\cos^{2}\\left(x\\right)", "steps": [ { "type": "step", "primary": "Apply Difference of Two Squares Formula: $$x^{2}-y^{2}=\\left(x+y\\right)\\left(x-y\\right)$$", "secondary": [ "$$\\cos^{2}\\left(x\\right)-\\cos^{2}\\left(2x\\right)=\\left(\\cos\\left(x\\right)+\\cos\\left(2x\\right)\\right)\\left(\\cos\\left(x\\right)-\\cos\\left(2x\\right)\\right)$$" ], "result": "=\\left(\\cos\\left(x\\right)+\\cos\\left(2x\\right)\\right)\\left(\\cos\\left(x\\right)-\\cos\\left(2x\\right)\\right)", "meta": { "practiceLink": "/practice/factoring-practice#area=main&subtopic=Difference%20of%20Two%20Squares", "practiceTopic": "Factor Difference of Squares" } } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Factor Title 1Eq" } }, { "type": "step", "result": "\\left(\\cos\\left(x\\right)+\\cos\\left(2x\\right)\\right)\\left(\\cos\\left(x\\right)-\\cos\\left(2x\\right)\\right)=0" }, { "type": "step", "primary": "Solving each part separately", "result": "\\cos\\left(x\\right)+\\cos\\left(2x\\right)=0\\lor\\:\\cos\\left(x\\right)-\\cos\\left(2x\\right)=0" }, { "type": "interim", "title": "$$\\cos\\left(x\\right)+\\cos\\left(2x\\right)=0{\\quad:\\quad}x=\\frac{π}{3}+\\frac{4πn}{3},\\:x=π+\\frac{4πn}{3}$$", "input": "\\cos\\left(x\\right)+\\cos\\left(2x\\right)=0", "steps": [ { "type": "interim", "title": "Rewrite using trig identities", "input": "\\cos\\left(2x\\right)+\\cos\\left(x\\right)", "result": "2\\cos\\left(\\frac{3x}{2}\\right)\\cos\\left(\\frac{x}{2}\\right)=0", "steps": [ { "type": "step", "primary": "Use the Sum to Product identity: $$\\cos\\left(s\\right)+\\cos\\left(t\\right)=2\\cos\\left(\\frac{s+t}{2}\\right)\\cos\\left(\\frac{s-t}{2}\\right)$$", "result": "=2\\cos\\left(\\frac{2x+x}{2}\\right)\\cos\\left(\\frac{2x-x}{2}\\right)" }, { "type": "interim", "title": "Simplify $$2\\cos\\left(\\frac{2x+x}{2}\\right)\\cos\\left(\\frac{2x-x}{2}\\right):{\\quad}2\\cos\\left(\\frac{3x}{2}\\right)\\cos\\left(\\frac{x}{2}\\right)$$", "input": "2\\cos\\left(\\frac{2x+x}{2}\\right)\\cos\\left(\\frac{2x-x}{2}\\right)", "result": "=2\\cos\\left(\\frac{3x}{2}\\right)\\cos\\left(\\frac{x}{2}\\right)", "steps": [ { "type": "step", "primary": "Add similar elements: $$2x+x=3x$$", "result": "=2\\cos\\left(\\frac{3x}{2}\\right)\\cos\\left(\\frac{2x-x}{2}\\right)" }, { "type": "step", "primary": "Add similar elements: $$2x-x=x$$", "result": "=2\\cos\\left(\\frac{3x}{2}\\right)\\cos\\left(\\frac{x}{2}\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7M3U1zttpUVfifMFzC3pOLD4XKwlpDsm5hM4RZ/wvNA5MoRaWkEhmnoE/aJv7EBjULTrWWMFI8l4Q07DZ5+hJa4jgfIU8WN8DfGp2psrsB7f73f3LxFj9jEKe/yLPEM66H5K0KUT95LS+dPFB4g0U6ewM/K1NiI3qqTnLscWlEe2Y3ASC+aZqPN1DBWUUsybFLfqJnZhCq3nEZywazuGdqkYT49SanSk3yvo2CnlmggjMpfNybzWClgLarK9ZAEaIHEKNduINWbMSV9/77/WPMw==" } } ], "meta": { "interimType": "Trig Rewrite Using Trig identities 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7lRPp6QndqgNT+L6IZutAbKewjMmu8eNWkuEiB5jiiz2yuRcnFgnh8AijhYNjhqyRbyZYyX6EGMAcAMgjOADqPLajIxP2OrjkBCECQd8a6w6+tyFakYh/d3WLkaecZ89b+939y8RY/YxCnv8izxDOuh+StClE/eS0vnTxQeINFOnfmjCC56PXxkd+OmwDX5iIGHLxKfgKyeyIEEIrMkmAx+L2uKOmWULehCuWXItXeeiuoO3GmZ6yUXlhK+L1PXpv" } }, { "type": "step", "primary": "Solving each part separately", "result": "\\cos\\left(\\frac{3x}{2}\\right)=0\\lor\\:\\cos\\left(\\frac{x}{2}\\right)=0" }, { "type": "interim", "title": "$$\\cos\\left(\\frac{3x}{2}\\right)=0{\\quad:\\quad}x=\\frac{π}{3}+\\frac{4πn}{3},\\:x=π+\\frac{4πn}{3}$$", "input": "\\cos\\left(\\frac{3x}{2}\\right)=0", "steps": [ { "type": "interim", "title": "General solutions for $$\\cos\\left(\\frac{3x}{2}\\right)=0$$", "result": "\\frac{3x}{2}=\\frac{π}{2}+2πn,\\:\\frac{3x}{2}=\\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": "\\frac{3x}{2}=\\frac{π}{2}+2πn,\\:\\frac{3x}{2}=\\frac{3π}{2}+2πn" } ], "meta": { "interimType": "Trig General Solutions cos 1Eq" } }, { "type": "interim", "title": "Solve $$\\frac{3x}{2}=\\frac{π}{2}+2πn:{\\quad}x=\\frac{π}{3}+\\frac{4πn}{3}$$", "input": "\\frac{3x}{2}=\\frac{π}{2}+2πn", "steps": [ { "type": "interim", "title": "Multiply both sides by $$2$$", "input": "\\frac{3x}{2}=\\frac{π}{2}+2πn", "result": "3x=π+4πn", "steps": [ { "type": "step", "primary": "Multiply both sides by $$2$$", "result": "\\frac{2\\cdot\\:3x}{2}=2\\cdot\\:\\frac{π}{2}+2\\cdot\\:2πn" }, { "type": "interim", "title": "Simplify", "input": "\\frac{2\\cdot\\:3x}{2}=2\\cdot\\:\\frac{π}{2}+2\\cdot\\:2πn", "result": "3x=π+4πn", "steps": [ { "type": "interim", "title": "Simplify $$\\frac{2\\cdot\\:3x}{2}:{\\quad}3x$$", "input": "\\frac{2\\cdot\\:3x}{2}", "steps": [ { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:3=6$$", "result": "=\\frac{6x}{2}" }, { "type": "step", "primary": "Divide the numbers: $$\\frac{6}{2}=3$$", "result": "=3x" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s76wwMXmpSw8uHSAcJv7PDws8oiPD+GUDTzuygccTgeNwJQJZuTAY5js+oqjdT8kslNWpjDv3Ur5eTU4c7Funhyx429vuTSxWa7B/X3D1oP03AWQmX+FAZQ57eQ8HwbCJC2SukbwkT8HU9SdwWLKUyRSS3daIZHtloJpe/PvtsyNI=" } }, { "type": "interim", "title": "Simplify $$2\\cdot\\:\\frac{π}{2}+2\\cdot\\:2πn:{\\quad}π+4πn$$", "input": "2\\cdot\\:\\frac{π}{2}+2\\cdot\\:2πn", "steps": [ { "type": "interim", "title": "$$2\\cdot\\:\\frac{π}{2}=π$$", "input": "2\\cdot\\:\\frac{π}{2}", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{π2}{2}" }, { "type": "step", "primary": "Cancel the common factor: $$2$$", "result": "=π" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7/OsC643lXZbU+VEjF1qvijDjqYwMGUrS2638vECqwzQJQJZuTAY5js+oqjdT8ksl58bIKwWzS3dUPXOsTsmj0kC6uVDh6a2y9mv8KbdwZ1i2qd84DzTVIGoMiX9UU4UgJLd1ohke2Wgml78++2zI0g==" } }, { "type": "interim", "title": "$$2\\cdot\\:2πn=4πn$$", "input": "2\\cdot\\:2πn", "steps": [ { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:2=4$$", "result": "=4πn" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s73+wEiPbaLObTqbUS2D3M3FXTSum/z5kLpMzXS1UJIewNLJBFHVNJXcF3Tu8Q3iPc3sgi2dqxE8XWitN20kuh9g/+xmM08uF4rRxptcnmtxywiNrEngO+NNvZ9sqNu+2V" } }, { "type": "step", "result": "=π+4πn" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7/OsC643lXZbU+VEjF1qvilCoPdO5x2o8Cj7ECxrihR7+n/+FjN833TADO2gxs6I7zMFYmi1F5Hg/ibpEToVnY2i/aUbhj4z+AETj0umEW8Z6pfF1z6umzUJTJvt+ojYZa+XIqT7+sGIIONXYNLACJP1a2w6Oc+LUUZTFRBwhXOsXmilMI8vITnF82h12HmD+" } }, { "type": "step", "result": "3x=π+4πn" } ], "meta": { "interimType": "Generic Simplify 0Eq" } } ], "meta": { "interimType": "Multiply Both Sides Specific 1Eq", "gptData": "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" } }, { "type": "interim", "title": "Divide both sides by $$3$$", "input": "3x=π+4πn", "result": "x=\\frac{π}{3}+\\frac{4πn}{3}", "steps": [ { "type": "step", "primary": "Divide both sides by $$3$$", "result": "\\frac{3x}{3}=\\frac{π}{3}+\\frac{4πn}{3}" }, { "type": "step", "primary": "Simplify", "result": "x=\\frac{π}{3}+\\frac{4πn}{3}" } ], "meta": { "interimType": "Divide Both Sides Specific 1Eq", "gptData": "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" } } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve Title 1Eq" } }, { "type": "interim", "title": "Solve $$\\frac{3x}{2}=\\frac{3π}{2}+2πn:{\\quad}x=π+\\frac{4πn}{3}$$", "input": "\\frac{3x}{2}=\\frac{3π}{2}+2πn", "steps": [ { "type": "interim", "title": "Multiply both sides by $$2$$", "input": "\\frac{3x}{2}=\\frac{3π}{2}+2πn", "result": "3x=3π+4πn", "steps": [ { "type": "step", "primary": "Multiply both sides by $$2$$", "result": "\\frac{2\\cdot\\:3x}{2}=2\\cdot\\:\\frac{3π}{2}+2\\cdot\\:2πn" }, { "type": "interim", "title": "Simplify", "input": "\\frac{2\\cdot\\:3x}{2}=2\\cdot\\:\\frac{3π}{2}+2\\cdot\\:2πn", "result": "3x=3π+4πn", "steps": [ { "type": "interim", "title": "Simplify $$\\frac{2\\cdot\\:3x}{2}:{\\quad}3x$$", "input": "\\frac{2\\cdot\\:3x}{2}", "steps": [ { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:3=6$$", "result": "=\\frac{6x}{2}" }, { "type": "step", "primary": "Divide the numbers: $$\\frac{6}{2}=3$$", "result": "=3x" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s76wwMXmpSw8uHSAcJv7PDws8oiPD+GUDTzuygccTgeNwJQJZuTAY5js+oqjdT8kslNWpjDv3Ur5eTU4c7Funhyx429vuTSxWa7B/X3D1oP03AWQmX+FAZQ57eQ8HwbCJC2SukbwkT8HU9SdwWLKUyRSS3daIZHtloJpe/PvtsyNI=" } }, { "type": "interim", "title": "Simplify $$2\\cdot\\:\\frac{3π}{2}+2\\cdot\\:2πn:{\\quad}3π+4πn$$", "input": "2\\cdot\\:\\frac{3π}{2}+2\\cdot\\:2πn", "steps": [ { "type": "interim", "title": "$$2\\cdot\\:\\frac{3π}{2}=3π$$", "input": "2\\cdot\\:\\frac{3π}{2}", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{3π2}{2}" }, { "type": "step", "primary": "Cancel the common factor: $$2$$", "result": "=3π" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7/OsC643lXZbU+VEjF1qvilwlCNo677MqXRSQeT6brHt1g99dC9fj9sg0EHzBIRDRIbch5xBxdffnEEe6KINgATHzb05+K8EdiiILkCIoydvQfK+rj8X8D5Sw8xzzK531G6Jn6ojVoz2IDd0XmrUm3Q==" } }, { "type": "interim", "title": "$$2\\cdot\\:2πn=4πn$$", "input": "2\\cdot\\:2πn", "steps": [ { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:2=4$$", "result": "=4πn" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s73+wEiPbaLObTqbUS2D3M3FXTSum/z5kLpMzXS1UJIewNLJBFHVNJXcF3Tu8Q3iPc3sgi2dqxE8XWitN20kuh9g/+xmM08uF4rRxptcnmtxywiNrEngO+NNvZ9sqNu+2V" } }, { "type": "step", "result": "=3π+4πn" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7/OsC643lXZbU+VEjF1qvigvPZSXEbrW4Y1PfvOoQsK3j+O38RAvKhZolrKec34mYcJChiVhDxT5N/LHSTLMjyBnKOeSWLlXYFjTYpevDOQnwt9LEn7QCBUukJKctfSJKdMfIq1syr76aSjFL6k656wdLZvTx8r3rpSXgXxuDYZXaWogMZQD3CFDEUHEJUAgTsIjaxJ4DvjTb2fbKjbvtlQ==" } }, { "type": "step", "result": "3x=3π+4πn" } ], "meta": { "interimType": "Generic Simplify 0Eq" } } ], "meta": { "interimType": "Multiply Both Sides Specific 1Eq", "gptData": "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" } }, { "type": "interim", "title": "Divide both sides by $$3$$", "input": "3x=3π+4πn", "result": "x=π+\\frac{4πn}{3}", "steps": [ { "type": "step", "primary": "Divide both sides by $$3$$", "result": "\\frac{3x}{3}=\\frac{3π}{3}+\\frac{4πn}{3}" }, { "type": "step", "primary": "Simplify", "result": "x=π+\\frac{4πn}{3}" } ], "meta": { "interimType": "Divide Both Sides Specific 1Eq", "gptData": "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" } } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve Title 1Eq" } }, { "type": "step", "result": "x=\\frac{π}{3}+\\frac{4πn}{3},\\:x=π+\\frac{4πn}{3}" } ], "meta": { "solvingClass": "Trig Equations", "interimType": "Trig Equations" } }, { "type": "interim", "title": "$$\\cos\\left(\\frac{x}{2}\\right)=0{\\quad:\\quad}x=π+4πn,\\:x=3π+4πn$$", "input": "\\cos\\left(\\frac{x}{2}\\right)=0", "steps": [ { "type": "interim", "title": "General solutions for $$\\cos\\left(\\frac{x}{2}\\right)=0$$", "result": "\\frac{x}{2}=\\frac{π}{2}+2πn,\\:\\frac{x}{2}=\\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": "\\frac{x}{2}=\\frac{π}{2}+2πn,\\:\\frac{x}{2}=\\frac{3π}{2}+2πn" } ], "meta": { "interimType": "Trig General Solutions cos 1Eq" } }, { "type": "interim", "title": "Solve $$\\frac{x}{2}=\\frac{π}{2}+2πn:{\\quad}x=π+4πn$$", "input": "\\frac{x}{2}=\\frac{π}{2}+2πn", "steps": [ { "type": "interim", "title": "Multiply both sides by $$2$$", "input": "\\frac{x}{2}=\\frac{π}{2}+2πn", "result": "x=π+4πn", "steps": [ { "type": "step", "primary": "Multiply both sides by $$2$$", "result": "\\frac{2x}{2}=2\\cdot\\:\\frac{π}{2}+2\\cdot\\:2πn" }, { "type": "interim", "title": "Simplify", "input": "\\frac{2x}{2}=2\\cdot\\:\\frac{π}{2}+2\\cdot\\:2πn", "result": "x=π+4πn", "steps": [ { "type": "interim", "title": "Simplify $$\\frac{2x}{2}:{\\quad}x$$", "input": "\\frac{2x}{2}", "steps": [ { "type": "step", "primary": "Divide the numbers: $$\\frac{2}{2}=1$$", "result": "=x" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7wyZUoCmW9j1Yiq04nFm+Ei061ljBSPJeENOw2efoSWt2cZ6LAXMcKee6PSyjLEFrRSpN33oxZMojoqvYhvSJACLkpiVA7S8UT8ieezZKu6rogJeWDMKCUBljFmQOLy/H" } }, { "type": "interim", "title": "Simplify $$2\\cdot\\:\\frac{π}{2}+2\\cdot\\:2πn:{\\quad}π+4πn$$", "input": "2\\cdot\\:\\frac{π}{2}+2\\cdot\\:2πn", "steps": [ { "type": "interim", "title": "$$2\\cdot\\:\\frac{π}{2}=π$$", "input": "2\\cdot\\:\\frac{π}{2}", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{π2}{2}" }, { "type": "step", "primary": "Cancel the common factor: $$2$$", "result": "=π" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7/OsC643lXZbU+VEjF1qvijDjqYwMGUrS2638vECqwzQJQJZuTAY5js+oqjdT8ksl58bIKwWzS3dUPXOsTsmj0kC6uVDh6a2y9mv8KbdwZ1i2qd84DzTVIGoMiX9UU4UgJLd1ohke2Wgml78++2zI0g==" } }, { "type": "interim", "title": "$$2\\cdot\\:2πn=4πn$$", "input": "2\\cdot\\:2πn", "steps": [ { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:2=4$$", "result": "=4πn" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s73+wEiPbaLObTqbUS2D3M3FXTSum/z5kLpMzXS1UJIewNLJBFHVNJXcF3Tu8Q3iPc3sgi2dqxE8XWitN20kuh9g/+xmM08uF4rRxptcnmtxywiNrEngO+NNvZ9sqNu+2V" } }, { "type": "step", "result": "=π+4πn" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": 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"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" } } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve Title 1Eq" } }, { "type": "interim", "title": "Solve $$\\frac{x}{2}=\\frac{3π}{2}+2πn:{\\quad}x=3π+4πn$$", "input": "\\frac{x}{2}=\\frac{3π}{2}+2πn", "steps": [ { "type": "interim", "title": "Multiply both sides by $$2$$", "input": "\\frac{x}{2}=\\frac{3π}{2}+2πn", "result": "x=3π+4πn", "steps": [ { "type": "step", "primary": "Multiply both sides by $$2$$", "result": "\\frac{2x}{2}=2\\cdot\\:\\frac{3π}{2}+2\\cdot\\:2πn" }, { "type": "interim", "title": "Simplify", "input": "\\frac{2x}{2}=2\\cdot\\:\\frac{3π}{2}+2\\cdot\\:2πn", "result": "x=3π+4πn", "steps": [ { "type": "interim", "title": "Simplify $$\\frac{2x}{2}:{\\quad}x$$", "input": "\\frac{2x}{2}", "steps": [ { "type": "step", "primary": "Divide the numbers: $$\\frac{2}{2}=1$$", "result": "=x" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7wyZUoCmW9j1Yiq04nFm+Ei061ljBSPJeENOw2efoSWt2cZ6LAXMcKee6PSyjLEFrRSpN33oxZMojoqvYhvSJACLkpiVA7S8UT8ieezZKu6rogJeWDMKCUBljFmQOLy/H" } }, { "type": "interim", "title": "Simplify $$2\\cdot\\:\\frac{3π}{2}+2\\cdot\\:2πn:{\\quad}3π+4πn$$", "input": "2\\cdot\\:\\frac{3π}{2}+2\\cdot\\:2πn", "steps": [ { "type": "interim", "title": "$$2\\cdot\\:\\frac{3π}{2}=3π$$", "input": "2\\cdot\\:\\frac{3π}{2}", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{3π2}{2}" }, { "type": "step", "primary": "Cancel the common factor: $$2$$", "result": "=3π" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7/OsC643lXZbU+VEjF1qvilwlCNo677MqXRSQeT6brHt1g99dC9fj9sg0EHzBIRDRIbch5xBxdffnEEe6KINgATHzb05+K8EdiiILkCIoydvQfK+rj8X8D5Sw8xzzK531G6Jn6ojVoz2IDd0XmrUm3Q==" } }, { "type": "interim", "title": "$$2\\cdot\\:2πn=4πn$$", "input": "2\\cdot\\:2πn", "steps": [ { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:2=4$$", "result": "=4πn" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s73+wEiPbaLObTqbUS2D3M3FXTSum/z5kLpMzXS1UJIewNLJBFHVNJXcF3Tu8Q3iPc3sgi2dqxE8XWitN20kuh9g/+xmM08uF4rRxptcnmtxywiNrEngO+NNvZ9sqNu+2V" } }, { "type": "step", "result": "=3π+4πn" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7/OsC643lXZbU+VEjF1qvigvPZSXEbrW4Y1PfvOoQsK3j+O38RAvKhZolrKec34mYcJChiVhDxT5N/LHSTLMjyBnKOeSWLlXYFjTYpevDOQnwt9LEn7QCBUukJKctfSJKdMfIq1syr76aSjFL6k656wdLZvTx8r3rpSXgXxuDYZXaWogMZQD3CFDEUHEJUAgTsIjaxJ4DvjTb2fbKjbvtlQ==" } }, { "type": "step", "result": "x=3π+4πn" } ], "meta": { "interimType": "Generic Simplify 0Eq" } } ], "meta": { "interimType": "Multiply Both Sides Specific 1Eq", "gptData": "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" } } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve Title 1Eq" } }, { "type": "step", "result": "x=π+4πn,\\:x=3π+4πn" } ], "meta": { "solvingClass": "Trig Equations", "interimType": "Trig Equations" } }, { "type": "step", "primary": "Combine all the solutions", "result": "x=\\frac{π}{3}+\\frac{4πn}{3},\\:x=π+\\frac{4πn}{3},\\:x=π+4πn,\\:x=3π+4πn" }, { "type": "step", "primary": "Merge Overlapping Intervals", "result": "x=\\frac{π}{3}+\\frac{4πn}{3},\\:x=π+\\frac{4πn}{3}" } ], "meta": { "solvingClass": "Trig Equations", "interimType": "Trig Equations" } }, { "type": "interim", "title": "$$\\cos\\left(x\\right)-\\cos\\left(2x\\right)=0{\\quad:\\quad}x=\\frac{4πn}{3},\\:x=\\frac{2π}{3}+\\frac{4πn}{3}$$", "input": "\\cos\\left(x\\right)-\\cos\\left(2x\\right)=0", "steps": [ { "type": "interim", "title": "Rewrite using trig identities", "input": "-\\cos\\left(2x\\right)+\\cos\\left(x\\right)", "result": "2\\sin\\left(\\frac{3x}{2}\\right)\\sin\\left(\\frac{x}{2}\\right)=0", "steps": [ { "type": "step", "primary": "Use the Sum to Product identity: $$\\cos\\left(s\\right)-\\cos\\left(t\\right)=-2\\sin\\left(\\frac{s+t}{2}\\right)\\sin\\left(\\frac{s-t}{2}\\right)$$", "result": "=-2\\sin\\left(\\frac{x+2x}{2}\\right)\\sin\\left(\\frac{x-2x}{2}\\right)" }, { "type": "interim", "title": "Simplify $$-2\\sin\\left(\\frac{x+2x}{2}\\right)\\sin\\left(\\frac{x-2x}{2}\\right):{\\quad}2\\sin\\left(\\frac{x}{2}\\right)\\sin\\left(\\frac{3x}{2}\\right)$$", "input": "-2\\sin\\left(\\frac{x+2x}{2}\\right)\\sin\\left(\\frac{x-2x}{2}\\right)", "result": "=2\\sin\\left(\\frac{x}{2}\\right)\\sin\\left(\\frac{3x}{2}\\right)", "steps": [ { "type": "step", "primary": "Add similar elements: $$x+2x=3x$$", "result": "=-2\\sin\\left(\\frac{3x}{2}\\right)\\sin\\left(\\frac{x-2x}{2}\\right)" }, { "type": "interim", "title": "$$\\frac{x-2x}{2}=-\\frac{x}{2}$$", "input": "\\frac{x-2x}{2}", "steps": [ { "type": "step", "primary": "Add similar elements: $$x-2x=-x$$", "result": "=\\frac{-x}{2}" }, { "type": "step", "primary": "Apply the fraction rule: $$\\frac{-a}{b}=-\\frac{a}{b}$$", "result": "=-\\frac{x}{2}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7T4ChNOKDvlsS/DCW65OF8HyRHuGw7+tM5METTDj6vVHDFYNeJ0RX0LOwY+dS7jXMODW1YCsEDgcC0VHue0Jwiw4bfwiV6iMLJ5sC1nL7dObiQMmQxrN4S5keejuQCq4xW/ClxyEcCHs2YhRAHFs6fg==" } }, { "type": "step", "result": "=-2\\sin\\left(\\frac{3x}{2}\\right)\\sin\\left(-\\frac{x}{2}\\right)" }, { "type": "step", "primary": "Use the negative angle identity: $$\\sin\\left(-x\\right)=-\\sin\\left(x\\right)$$", "secondary": [], "result": "=-2\\left(-\\sin\\left(\\frac{x}{2}\\right)\\right)\\sin\\left(\\frac{3x}{2}\\right)" }, { "type": "step", "primary": "Apply rule $$-\\left(-a\\right)=a$$", "result": "=2\\sin\\left(\\frac{x}{2}\\right)\\sin\\left(\\frac{3x}{2}\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7WKQDtlNbtfOFijcUoWrkOtHupgEl44uw9UkAUuyqGUHfi3ZpsqR+cQO+bAPItZJfAJYpRu9XpYrd8NSAW2DdD1qcvkGoOmj7vUKbPyNbZzE6Yy2We67C5B09oCKDpYKHjL7LoDW6Mn+UwgnqVJVzBkay5clvjQgLLPXUAXecNaGLGmNnLPWGf9PH3lpmjoJIT4cAG4APDSy83xPxZ7xkVE3GZWRX2Y9N+oJNX5kKaL4ACEz8c1r6jSExsIoROXa/JdZmavklwua3EVJPJxIXHA==" } } ], "meta": { "interimType": "Trig Rewrite Using Trig identities 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7sxSOfOl7Um+zqSbarNITtiXXI00/DNv6eQP2B0D4CX4FMS0G0HgjUXwc3RQOmn9BYznLdIBaHD4ibBnEILJTDXUHW1mE5UQUTIwuQd6ag+xNQUndUJlCvVvSnNUC8d6vOmMtlnuuwuQdPaAig6WCh4y+y6A1ujJ/lMIJ6lSVcwbXqlClB0B5NQCp3MtO0YP323Xi3IvT385kAYFK2CQOxBkJGB3SwIAzffMMCti9BhrWM+R/5vzPLegxuhGAmoA+" } }, { "type": "step", "primary": "Solving each part separately", "result": "\\sin\\left(\\frac{3x}{2}\\right)=0\\lor\\:\\sin\\left(\\frac{x}{2}\\right)=0" }, { "type": "interim", "title": "$$\\sin\\left(\\frac{3x}{2}\\right)=0{\\quad:\\quad}x=\\frac{4πn}{3},\\:x=\\frac{2π}{3}+\\frac{4πn}{3}$$", "input": "\\sin\\left(\\frac{3x}{2}\\right)=0", "steps": [ { "type": "interim", "title": "General solutions for $$\\sin\\left(\\frac{3x}{2}\\right)=0$$", "result": "\\frac{3x}{2}=0+2πn,\\:\\frac{3x}{2}=π+2πn", "steps": [ { "type": "step", "primary": "$$\\sin\\left(x\\right)$$ periodicity table with $$2πn$$ cycle:<br/>$$\\begin{array}{|c|c|c|c|}\\hline x&\\sin(x)&x&\\sin(x)\\\\\\hline 0&0&π&0\\\\\\hline \\frac{π}{6}&\\frac{1}{2}&\\frac{7π}{6}&-\\frac{1}{2}\\\\\\hline \\frac{π}{4}&\\frac{\\sqrt{2}}{2}&\\frac{5π}{4}&-\\frac{\\sqrt{2}}{2}\\\\\\hline \\frac{π}{3}&\\frac{\\sqrt{3}}{2}&\\frac{4π}{3}&-\\frac{\\sqrt{3}}{2}\\\\\\hline \\frac{π}{2}&1&\\frac{3π}{2}&-1\\\\\\hline \\frac{2π}{3}&\\frac{\\sqrt{3}}{2}&\\frac{5π}{3}&-\\frac{\\sqrt{3}}{2}\\\\\\hline \\frac{3π}{4}&\\frac{\\sqrt{2}}{2}&\\frac{7π}{4}&-\\frac{\\sqrt{2}}{2}\\\\\\hline \\frac{5π}{6}&\\frac{1}{2}&\\frac{11π}{6}&-\\frac{1}{2}\\\\\\hline \\end{array}$$" }, { "type": "step", "result": "\\frac{3x}{2}=0+2πn,\\:\\frac{3x}{2}=π+2πn" } ], "meta": { "interimType": "Trig General Solutions sin 1Eq" } }, { "type": "interim", "title": "Solve $$\\frac{3x}{2}=0+2πn:{\\quad}x=\\frac{4πn}{3}$$", "input": "\\frac{3x}{2}=0+2πn", "steps": [ { "type": "step", "primary": "$$0+2πn=2πn$$", "result": "\\frac{3x}{2}=2πn" }, { "type": "interim", "title": "Multiply both sides by $$2$$", "input": "\\frac{3x}{2}=2πn", "result": "3x=4πn", "steps": [ { "type": "step", "primary": "Multiply both sides by $$2$$", "result": "\\frac{2\\cdot\\:3x}{2}=2\\cdot\\:2πn" }, { "type": "step", "primary": "Simplify", "result": "3x=4πn" } ], "meta": { "interimType": "Multiply Both Sides Specific 1Eq", "gptData": 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"3x=2π+4πn", "steps": [ { "type": "step", "primary": "Multiply both sides by $$2$$", "result": "\\frac{2\\cdot\\:3x}{2}=2π+2\\cdot\\:2πn" }, { "type": "step", "primary": "Simplify", "result": "3x=2π+4πn" } ], "meta": { "interimType": "Multiply Both Sides Specific 1Eq", "gptData": "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" } }, { "type": "interim", "title": "Divide both sides by $$3$$", "input": "3x=2π+4πn", "result": "x=\\frac{2π}{3}+\\frac{4πn}{3}", "steps": [ { "type": "step", "primary": "Divide both sides by $$3$$", "result": "\\frac{3x}{3}=\\frac{2π}{3}+\\frac{4πn}{3}" }, { "type": "step", "primary": "Simplify", "result": "x=\\frac{2π}{3}+\\frac{4πn}{3}" } ], "meta": { "interimType": "Divide Both Sides Specific 1Eq", "gptData": 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"$$\\sin\\left(\\frac{x}{2}\\right)=0{\\quad:\\quad}x=4πn,\\:x=2π+4πn$$", "input": "\\sin\\left(\\frac{x}{2}\\right)=0", "steps": [ { "type": "interim", "title": "General solutions for $$\\sin\\left(\\frac{x}{2}\\right)=0$$", "result": "\\frac{x}{2}=0+2πn,\\:\\frac{x}{2}=π+2πn", "steps": [ { "type": "step", "primary": "$$\\sin\\left(x\\right)$$ periodicity table with $$2πn$$ cycle:<br/>$$\\begin{array}{|c|c|c|c|}\\hline x&\\sin(x)&x&\\sin(x)\\\\\\hline 0&0&π&0\\\\\\hline \\frac{π}{6}&\\frac{1}{2}&\\frac{7π}{6}&-\\frac{1}{2}\\\\\\hline \\frac{π}{4}&\\frac{\\sqrt{2}}{2}&\\frac{5π}{4}&-\\frac{\\sqrt{2}}{2}\\\\\\hline \\frac{π}{3}&\\frac{\\sqrt{3}}{2}&\\frac{4π}{3}&-\\frac{\\sqrt{3}}{2}\\\\\\hline \\frac{π}{2}&1&\\frac{3π}{2}&-1\\\\\\hline \\frac{2π}{3}&\\frac{\\sqrt{3}}{2}&\\frac{5π}{3}&-\\frac{\\sqrt{3}}{2}\\\\\\hline \\frac{3π}{4}&\\frac{\\sqrt{2}}{2}&\\frac{7π}{4}&-\\frac{\\sqrt{2}}{2}\\\\\\hline \\frac{5π}{6}&\\frac{1}{2}&\\frac{11π}{6}&-\\frac{1}{2}\\\\\\hline \\end{array}$$" }, { "type": "step", "result": "\\frac{x}{2}=0+2πn,\\:\\frac{x}{2}=π+2πn" } ], "meta": { "interimType": "Trig General Solutions sin 1Eq" } }, { "type": "interim", "title": "Solve $$\\frac{x}{2}=0+2πn:{\\quad}x=4πn$$", "input": "\\frac{x}{2}=0+2πn", "steps": [ { "type": "step", "primary": "$$0+2πn=2πn$$", "result": "\\frac{x}{2}=2πn" }, { "type": "interim", "title": "Multiply both sides by $$2$$", "input": "\\frac{x}{2}=2πn", "result": "x=4πn", "steps": [ { "type": "step", "primary": "Multiply both sides by $$2$$", "result": "\\frac{2x}{2}=2\\cdot\\:2πn" }, { "type": "step", "primary": "Simplify", "result": "x=4πn" } ], "meta": { "interimType": "Multiply Both Sides Specific 1Eq", "gptData": 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"primary": "Multiply both sides by $$2$$", "result": "\\frac{2x}{2}=2π+2\\cdot\\:2πn" }, { "type": "step", "primary": "Simplify", "result": "x=2π+4πn" } ], "meta": { "interimType": "Multiply Both Sides Specific 1Eq", "gptData": "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" } } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve Title 1Eq" } }, { "type": "step", "result": "x=4πn,\\:x=2π+4πn" } ], "meta": { "solvingClass": "Trig Equations", "interimType": "Trig Equations" } }, { "type": "step", "primary": "Combine all the solutions", "result": "x=\\frac{4πn}{3},\\:x=\\frac{2π}{3}+\\frac{4πn}{3},\\:x=4πn,\\:x=2π+4πn" }, { "type": "step", "primary": "Merge Overlapping Intervals", "result": "x=\\frac{4πn}{3},\\:x=\\frac{2π}{3}+\\frac{4πn}{3}" } ], "meta": { "solvingClass": "Trig Equations", "interimType": "Trig Equations" } }, { "type": "step", "primary": "Combine all the solutions", "result": "x=\\frac{π}{3}+\\frac{4πn}{3},\\:x=π+\\frac{4πn}{3},\\:x=\\frac{4πn}{3},\\:x=\\frac{2π}{3}+\\frac{4πn}{3}" } ], "meta": { "solvingClass": "Trig Equations", "practiceLink": "/practice/trigonometry-practice#area=main&subtopic=Trig%20Equations", "practiceTopic": "Trig Equations" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "\\sin^{2}(2x)+\\cos^{2}(x)-1" }, "showViewLarger": true } }, "meta": { "showVerify": true } }