4cos^2(2x)=3

{ "query": { "display": "$$4\\cos^{2}\\left(2x\\right)=3$$", "symbolab_question": "EQUATION#4\\cos^{2}(2x)=3" }, "solution": { "level": "PERFORMED", "subject": "Trigonometry", "topic": "Trig Equations", "subTopic": "Trig Equations", "default": "x=\\frac{π}{12}+πn,x=\\frac{11π}{12}+πn,x=\\frac{5π}{12}+πn,x=\\frac{7π}{12}+πn", "degrees": "x=15^{\\circ }+180^{\\circ }n,x=165^{\\circ }+180^{\\circ }n,x=75^{\\circ }+180^{\\circ }n,x=105^{\\circ }+180^{\\circ }n", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$4\\cos^{2}\\left(2x\\right)=3{\\quad:\\quad}x=\\frac{π}{12}+πn,\\:x=\\frac{11π}{12}+πn,\\:x=\\frac{5π}{12}+πn,\\:x=\\frac{7π}{12}+πn$$", "input": "4\\cos^{2}\\left(2x\\right)=3", "steps": [ { "type": "interim", "title": "Solve by substitution", "input": "4\\cos^{2}\\left(2x\\right)=3", "result": "\\cos\\left(2x\\right)=\\frac{\\sqrt{3}}{2},\\:\\cos\\left(2x\\right)=-\\frac{\\sqrt{3}}{2}", "steps": [ { "type": "step", "primary": "Let: $$\\cos\\left(2x\\right)=u$$", "result": "4u^{2}=3" }, { "type": "interim", "title": "$$4u^{2}=3{\\quad:\\quad}u=\\frac{\\sqrt{3}}{2},\\:u=-\\frac{\\sqrt{3}}{2}$$", "input": "4u^{2}=3", "steps": [ { "type": "interim", "title": "Divide both sides by $$4$$", "input": "4u^{2}=3", "result": "u^{2}=\\frac{3}{4}", "steps": [ { "type": "step", "primary": "Divide both sides by $$4$$", "result": "\\frac{4u^{2}}{4}=\\frac{3}{4}" }, { "type": "step", "primary": "Simplify", "result": "u^{2}=\\frac{3}{4}" } ], "meta": { "interimType": "Divide Both Sides Specific 1Eq", "gptData": "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" } }, { "type": "step", "primary": "For $$x^{2}=f\\left(a\\right)$$ the solutions are $$x=\\sqrt{f\\left(a\\right)},\\:\\:-\\sqrt{f\\left(a\\right)}$$" }, { "type": "step", "result": "u=\\sqrt{\\frac{3}{4}},\\:u=-\\sqrt{\\frac{3}{4}}" }, { "type": "interim", "title": "$$\\sqrt{\\frac{3}{4}}=\\frac{\\sqrt{3}}{2}$$", "input": "\\sqrt{\\frac{3}{4}}", "steps": [ { "type": "step", "primary": "Apply radical rule: $$\\sqrt[n]{\\frac{a}{b}}=\\frac{\\sqrt[n]{a}}{\\sqrt[n]{b}},\\:\\quad$$ assuming $$a\\ge0,\\:b\\ge0$$", "result": "=\\frac{\\sqrt{3}}{\\sqrt{4}}", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } }, { "type": "interim", "title": "$$\\sqrt{4}=2$$", "input": "\\sqrt{4}", "result": "=\\frac{\\sqrt{3}}{2}", "steps": [ { "type": "step", "primary": "Factor the number: $$4=2^{2}$$", "result": "=\\sqrt{2^{2}}" }, { "type": "step", "primary": "Apply radical rule: $$\\sqrt[n]{a^n}=a$$", "secondary": [ "$$\\sqrt{2^{2}}=2$$" ], "result": "=2", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } } ], "meta": { "interimType": "N/A" } } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7FJIkkmi1CWuhEmDQGlA0N481GBhaFmRuYU8ivCdOkRGrju+5Z51e/ZZSD3gRHwjBnvjDY21b5XBQ44AG3rKeoERJNpqBgWYJo7POzHRJNgWKy+4oc4WkS69iOkw+tuw7OdycaE+yHbER8Bud6bqjpd6vq8ch6QORd3MMFeFuyk+wiNrEngO+NNvZ9sqNu+2V" } }, { "type": "interim", "title": "$$-\\sqrt{\\frac{3}{4}}=-\\frac{\\sqrt{3}}{2}$$", "input": "-\\sqrt{\\frac{3}{4}}", "steps": [ { "type": "interim", "title": "Simplify $$\\sqrt{\\frac{3}{4}}:{\\quad}\\frac{\\sqrt{3}}{2}$$", "input": "\\sqrt{\\frac{3}{4}}", "result": "=-\\frac{\\sqrt{3}}{2}", "steps": [ { "type": "step", "primary": "Apply radical rule: $$\\sqrt[n]{\\frac{a}{b}}=\\frac{\\sqrt[n]{a}}{\\sqrt[n]{b}},\\:\\quad$$ assuming $$a\\ge0,\\:b\\ge0$$", "result": "=\\frac{\\sqrt{3}}{\\sqrt{4}}", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } }, { "type": "interim", "title": "$$\\sqrt{4}=2$$", "input": "\\sqrt{4}", "result": "=\\frac{\\sqrt{3}}{2}", "steps": [ { "type": "step", "primary": "Factor the number: $$4=2^{2}$$", "result": "=\\sqrt{2^{2}}" }, { "type": "step", "primary": "Apply radical rule: $$\\sqrt[n]{a^n}=a$$", "secondary": [ "$$\\sqrt{2^{2}}=2$$" ], "result": "=2", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } } ], "meta": { "interimType": "N/A" } } ], "meta": { "interimType": "Algebraic Manipulation Simplify Title 1Eq" } } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7xVDfakA4g1ZczTHh+Arxm/DDmuD5U4GDJIF9k+W0I1gJQJZuTAY5js+oqjdT8kslDa1hUvtzgBcwD7zey8pjX+7J0HqxMc0mt58lG7zw6WRrXh3OSOXhogkv14GjeoU6kzLzxByUErH9eEiuIkZhcyb97z4ciRBzgXtcnYl8cilurCc0v6+SezXpJu+Sddf4" } }, { "type": "step", "result": "u=\\frac{\\sqrt{3}}{2},\\:u=-\\frac{\\sqrt{3}}{2}" } ], "meta": { "solvingClass": "Equations", "interimType": "Equations" } }, { "type": "step", "primary": "Substitute back $$u=\\cos\\left(2x\\right)$$", "result": "\\cos\\left(2x\\right)=\\frac{\\sqrt{3}}{2},\\:\\cos\\left(2x\\right)=-\\frac{\\sqrt{3}}{2}" } ], "meta": { "interimType": "Substitution Method 0Eq" } }, { "type": "interim", "title": "$$\\cos\\left(2x\\right)=\\frac{\\sqrt{3}}{2}{\\quad:\\quad}x=\\frac{π}{12}+πn,\\:x=\\frac{11π}{12}+πn$$", "input": "\\cos\\left(2x\\right)=\\frac{\\sqrt{3}}{2}", "steps": [ { "type": "interim", "title": "General solutions for $$\\cos\\left(2x\\right)=\\frac{\\sqrt{3}}{2}$$", "result": "2x=\\frac{π}{6}+2πn,\\:2x=\\frac{11π}{6}+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": "2x=\\frac{π}{6}+2πn,\\:2x=\\frac{11π}{6}+2πn" } ], "meta": { "interimType": "Trig General Solutions cos 1Eq" } }, { "type": "interim", "title": "Solve $$2x=\\frac{π}{6}+2πn:{\\quad}x=\\frac{π}{12}+πn$$", "input": "2x=\\frac{π}{6}+2πn", "steps": [ { "type": "interim", "title": "Divide both sides by $$2$$", "input": "2x=\\frac{π}{6}+2πn", "result": "x=\\frac{π}{12}+πn", "steps": [ { "type": "step", "primary": "Divide both sides by $$2$$", "result": "\\frac{2x}{2}=\\frac{\\frac{π}{6}}{2}+\\frac{2πn}{2}" }, { "type": "interim", "title": "Simplify", "input": "\\frac{2x}{2}=\\frac{\\frac{π}{6}}{2}+\\frac{2πn}{2}", "result": "x=\\frac{π}{12}+πn", "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 $$\\frac{\\frac{π}{6}}{2}+\\frac{2πn}{2}:{\\quad}\\frac{π}{12}+πn$$", "input": "\\frac{\\frac{π}{6}}{2}+\\frac{2πn}{2}", "steps": [ { "type": "interim", "title": "$$\\frac{\\frac{π}{6}}{2}=\\frac{π}{12}$$", "input": "\\frac{\\frac{π}{6}}{2}", "steps": [ { "type": "step", "primary": "Apply the fraction rule: $$\\frac{\\frac{b}{c}}{a}=\\frac{b}{c\\:\\cdot\\:a}$$", "result": "=\\frac{π}{6\\cdot\\:2}" }, { "type": "step", "primary": "Multiply the numbers: $$6\\cdot\\:2=12$$", "result": "=\\frac{π}{12}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7tiTmKkmeEqHWILbBb88ajWqCk54oAXku+C9eC3AYg+HNGoPE9TME3q+OPmgkv2RQV7wanvTgjKqnID/UMq66GyOk8ADv3Qq+ATrkEXcaomSIDZ1VqNwwAzGjxOWVq2SRIeT+01jSenFHVJVrApCtnrwW5Nhd3K/P3qgp381wL9s=" } }, { "type": "interim", "title": "$$\\frac{2πn}{2}=πn$$", "input": "\\frac{2πn}{2}", "steps": [ { "type": "step", "primary": "Divide the numbers: $$\\frac{2}{2}=1$$", "result": "=πn" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7YVyjdFRW0jf8Bxzp61JzqHyRHuGw7+tM5METTDj6vVECnBmppnUV+797UqKeeyBQmGP7cF4dEaUuqswlNsH4qzvxhqyDNGf7EUmy3SORhvXPUvVtHPbCNTQhn1k5UbDm" } }, { "type": "step", "result": "=\\frac{π}{12}+πn" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7tiTmKkmeEqHWILbBb88ajXgE6C3CSriiKdwyx/Ba2X7Cl1oDQxTLS01lmG4qDSnCzRqDxPUzBN6vjj5oJL9kUFe8Gp704IyqpyA/1DKuuhu2zDrJYatHsKrXlsSOX1ji1sD7NfhsPe7eDHrmjY0mE8DwQcN28geUT7aUsumC7doaryAYIFbz6VGRGlOmTNL9ZTEpIsWx2g1tU2drz2Msym6sJzS/r5J7Nekm75J11/g=" } }, { "type": "step", "result": "x=\\frac{π}{12}+πn" } ], "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 $$2x=\\frac{11π}{6}+2πn:{\\quad}x=\\frac{11π}{12}+πn$$", "input": "2x=\\frac{11π}{6}+2πn", "steps": [ { "type": "interim", "title": "Divide both sides by $$2$$", "input": "2x=\\frac{11π}{6}+2πn", "result": "x=\\frac{11π}{12}+πn", "steps": [ { "type": "step", "primary": "Divide both sides by $$2$$", "result": "\\frac{2x}{2}=\\frac{\\frac{11π}{6}}{2}+\\frac{2πn}{2}" }, { "type": "interim", "title": "Simplify", "input": "\\frac{2x}{2}=\\frac{\\frac{11π}{6}}{2}+\\frac{2πn}{2}", "result": "x=\\frac{11π}{12}+πn", "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 $$\\frac{\\frac{11π}{6}}{2}+\\frac{2πn}{2}:{\\quad}\\frac{11π}{12}+πn$$", "input": "\\frac{\\frac{11π}{6}}{2}+\\frac{2πn}{2}", "steps": [ { "type": "interim", "title": "$$\\frac{\\frac{11π}{6}}{2}=\\frac{11π}{12}$$", "input": "\\frac{\\frac{11π}{6}}{2}", "steps": [ { "type": "step", "primary": "Apply the fraction rule: $$\\frac{\\frac{b}{c}}{a}=\\frac{b}{c\\:\\cdot\\:a}$$", "result": "=\\frac{11π}{6\\cdot\\:2}" }, { "type": "step", "primary": "Multiply the numbers: $$6\\cdot\\:2=12$$", "result": "=\\frac{11π}{12}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7tiTmKkmeEqHWILbBb88ajcm5FXQ50IQsy9V6MeVoA3UgJ/ZZA32ZInFBpDtxBfiKXYGCmiBF99lesmXZ9iIfJ4AbY1ZZlr0ES7ShAvPM/yhTW26qciuyUBGXQExCUedYvS8N5VCze8qz+LNUf5QAR8NqfIbgCoEQ5iqOpxvXCtHuJ1iXawr0xn/4zW2wj+oZ" } }, { "type": "interim", "title": "$$\\frac{2πn}{2}=πn$$", "input": "\\frac{2πn}{2}", "steps": [ { "type": "step", "primary": "Divide the numbers: $$\\frac{2}{2}=1$$", "result": "=πn" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7YVyjdFRW0jf8Bxzp61JzqHyRHuGw7+tM5METTDj6vVECnBmppnUV+797UqKeeyBQmGP7cF4dEaUuqswlNsH4qzvxhqyDNGf7EUmy3SORhvXPUvVtHPbCNTQhn1k5UbDm" } }, { "type": "step", "result": "=\\frac{11π}{12}+πn" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7tiTmKkmeEqHWILbBb88aja8YN6VzLM/g2fVQvpHYB3AYKXMStDfJcV1gA+Z/JDXIICf2WQN9mSJxQaQ7cQX4il2BgpogRffZXrJl2fYiHyfmx8MSjQWWYpQEOqS7iNsYRSpN33oxZMojoqvYhvSJACLkpiVA7S8UT8ieezZKu6q99iKcg75ivhzP3grxubPxeAToLcJKuKIp3DLH8FrZfjkRYT57po0fVp+kBBk2foE=" } }, { "type": "step", "result": "x=\\frac{11π}{12}+πn" } ], "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": "x=\\frac{π}{12}+πn,\\:x=\\frac{11π}{12}+πn" } ], "meta": { "interimType": "N/A" } }, { "type": "interim", "title": "$$\\cos\\left(2x\\right)=-\\frac{\\sqrt{3}}{2}{\\quad:\\quad}x=\\frac{5π}{12}+πn,\\:x=\\frac{7π}{12}+πn$$", "input": "\\cos\\left(2x\\right)=-\\frac{\\sqrt{3}}{2}", "steps": [ { "type": "interim", "title": "General solutions for $$\\cos\\left(2x\\right)=-\\frac{\\sqrt{3}}{2}$$", "result": "2x=\\frac{5π}{6}+2πn,\\:2x=\\frac{7π}{6}+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": "2x=\\frac{5π}{6}+2πn,\\:2x=\\frac{7π}{6}+2πn" } ], "meta": { "interimType": "Trig General Solutions cos 1Eq" } }, { "type": "interim", "title": "Solve $$2x=\\frac{5π}{6}+2πn:{\\quad}x=\\frac{5π}{12}+πn$$", "input": "2x=\\frac{5π}{6}+2πn", "steps": [ { "type": "interim", "title": "Divide both sides by $$2$$", "input": "2x=\\frac{5π}{6}+2πn", "result": "x=\\frac{5π}{12}+πn", "steps": [ { "type": "step", "primary": "Divide both sides by $$2$$", "result": "\\frac{2x}{2}=\\frac{\\frac{5π}{6}}{2}+\\frac{2πn}{2}" }, { "type": "interim", "title": "Simplify", "input": "\\frac{2x}{2}=\\frac{\\frac{5π}{6}}{2}+\\frac{2πn}{2}", "result": "x=\\frac{5π}{12}+πn", "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 $$\\frac{\\frac{5π}{6}}{2}+\\frac{2πn}{2}:{\\quad}\\frac{5π}{12}+πn$$", "input": "\\frac{\\frac{5π}{6}}{2}+\\frac{2πn}{2}", "steps": [ { "type": "interim", "title": "$$\\frac{\\frac{5π}{6}}{2}=\\frac{5π}{12}$$", "input": "\\frac{\\frac{5π}{6}}{2}", "steps": [ { "type": "step", "primary": "Apply the fraction rule: $$\\frac{\\frac{b}{c}}{a}=\\frac{b}{c\\:\\cdot\\:a}$$", "result": "=\\frac{5π}{6\\cdot\\:2}" }, { "type": "step", "primary": "Multiply the numbers: $$6\\cdot\\:2=12$$", "result": "=\\frac{5π}{12}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7tiTmKkmeEqHWILbBb88ajQHcSlptnkW86/ptUJq7cMzdd47a0hQ8flDbGsI5To1dYDPq1Ru8svsR9Rij2+7AAELhcJuE223isn9fcbq91bYjigtccxNWJk7pEnO8OW1UJKiMQdQp0z+hxSj5lYQBYQlqyL7GJZULtRYLbJSNAhCwiNrEngO+NNvZ9sqNu+2V" } }, { "type": "interim", "title": "$$\\frac{2πn}{2}=πn$$", "input": "\\frac{2πn}{2}", "steps": [ { "type": "step", "primary": "Divide the numbers: $$\\frac{2}{2}=1$$", "result": "=πn" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7YVyjdFRW0jf8Bxzp61JzqHyRHuGw7+tM5METTDj6vVECnBmppnUV+797UqKeeyBQmGP7cF4dEaUuqswlNsH4qzvxhqyDNGf7EUmy3SORhvXPUvVtHPbCNTQhn1k5UbDm" } }, { "type": "step", "result": "=\\frac{5π}{12}+πn" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7tiTmKkmeEqHWILbBb88ajZrl2Z2+wpZonfILpbPnt/+2z3vSzoDYIK8LwhuQjcbO3XeO2tIUPH5Q2xrCOU6NXWAz6tUbvLL7EfUYo9vuwADmV2RD9WIotqtX60WMCq0CgQUxJPyUNnGfVirkcwpVO0BvnUFx4T5rmmmJUxss5NlQDZjeb2c9djTe3kKtLq722RlP6nbKbEC/HotWxNZG8pW9+QW0/o5wsTggtcymdGQ=" } }, { "type": "step", "result": "x=\\frac{5π}{12}+πn" } ], "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 $$2x=\\frac{7π}{6}+2πn:{\\quad}x=\\frac{7π}{12}+πn$$", "input": "2x=\\frac{7π}{6}+2πn", "steps": [ { "type": "interim", "title": "Divide both sides by $$2$$", "input": "2x=\\frac{7π}{6}+2πn", "result": "x=\\frac{7π}{12}+πn", "steps": [ { "type": "step", "primary": "Divide both sides by $$2$$", "result": "\\frac{2x}{2}=\\frac{\\frac{7π}{6}}{2}+\\frac{2πn}{2}" }, { "type": "interim", "title": "Simplify", "input": "\\frac{2x}{2}=\\frac{\\frac{7π}{6}}{2}+\\frac{2πn}{2}", "result": "x=\\frac{7π}{12}+πn", "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 $$\\frac{\\frac{7π}{6}}{2}+\\frac{2πn}{2}:{\\quad}\\frac{7π}{12}+πn$$", "input": "\\frac{\\frac{7π}{6}}{2}+\\frac{2πn}{2}", "steps": [ { "type": "interim", "title": "$$\\frac{\\frac{7π}{6}}{2}=\\frac{7π}{12}$$", "input": "\\frac{\\frac{7π}{6}}{2}", "steps": [ { "type": "step", "primary": "Apply the fraction rule: $$\\frac{\\frac{b}{c}}{a}=\\frac{b}{c\\:\\cdot\\:a}$$", "result": "=\\frac{7π}{6\\cdot\\:2}" }, { "type": "step", "primary": "Multiply the numbers: $$6\\cdot\\:2=12$$", "result": "=\\frac{7π}{12}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7tiTmKkmeEqHWILbBb88ajXlEtnBRHsSpEpm9woUlEmbdd47a0hQ8flDbGsI5To1dVESQA48qouUS12usnKDA2ELhcJuE223isn9fcbq91bYjigtccxNWJk7pEnO8OW1UTwuOeq45ZybiEP9otuZjpKuNnDpNe1DOk1tOxIo/dxawiNrEngO+NNvZ9sqNu+2V" } }, { "type": "interim", "title": "$$\\frac{2πn}{2}=πn$$", "input": "\\frac{2πn}{2}", "steps": [ { "type": "step", "primary": "Divide the numbers: $$\\frac{2}{2}=1$$", "result": "=πn" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7YVyjdFRW0jf8Bxzp61JzqHyRHuGw7+tM5METTDj6vVECnBmppnUV+797UqKeeyBQmGP7cF4dEaUuqswlNsH4qzvxhqyDNGf7EUmy3SORhvXPUvVtHPbCNTQhn1k5UbDm" } }, { "type": "step", "result": "=\\frac{7π}{12}+πn" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7tiTmKkmeEqHWILbBb88ajeqNE4F/56WSLxMRCLVPLIi2z3vSzoDYIK8LwhuQjcbO3XeO2tIUPH5Q2xrCOU6NXVREkAOPKqLlEtdrrJygwNjmV2RD9WIotqtX60WMCq0CgQUxJPyUNnGfVirkcwpVO0BvnUFx4T5rmmmJUxss5NnvAdonft/6Id5YgksLuZ342RlP6nbKbEC/HotWxNZG8pW9+QW0/o5wsTggtcymdGQ=" } }, { "type": "step", "result": "x=\\frac{7π}{12}+πn" } ], "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": "x=\\frac{5π}{12}+πn,\\:x=\\frac{7π}{12}+πn" } ], "meta": { "interimType": "N/A" } }, { "type": "step", "primary": "Combine all the solutions", "result": "x=\\frac{π}{12}+πn,\\:x=\\frac{11π}{12}+πn,\\:x=\\frac{5π}{12}+πn,\\:x=\\frac{7π}{12}+πn" } ], "meta": { "solvingClass": "Trig Equations", "practiceLink": "/practice/trigonometry-practice#area=main&subtopic=Trig%20Equations", "practiceTopic": "Trig Equations" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "4\\cos^{2}(2x)-3" }, "showViewLarger": true } }, "meta": { "showVerify": true } }