sin(9x+6)=cos(3x-4)

{ "query": { "display": "$$\\sin\\left(9x+6\\right)=\\cos\\left(3x-4\\right)$$", "symbolab_question": "EQUATION#\\sin(9x+6)=\\cos(3x-4)" }, "solution": { "level": "PERFORMED", "subject": "Trigonometry", "topic": "Trig Equations", "subTopic": "Trig Equations", "default": "x=\\frac{4πn+π-4}{24},x=\\frac{π+4πn-20}{12}", "degrees": "x=-2.04929…^{\\circ }+30^{\\circ }n,x=-80.49296…^{\\circ }+60^{\\circ }n", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\sin\\left(9x+6\\right)=\\cos\\left(3x-4\\right){\\quad:\\quad}x=\\frac{4πn+π-4}{24},\\:x=\\frac{π+4πn-20}{12}$$", "input": "\\sin\\left(9x+6\\right)=\\cos\\left(3x-4\\right)", "steps": [ { "type": "interim", "title": "Rewrite using trig identities", "input": "\\sin\\left(9x+6\\right)=\\cos\\left(3x-4\\right)", "result": "\\sin\\left(9x+6\\right)=\\sin\\left(\\frac{π}{2}-\\left(3x-4\\right)\\right)", "steps": [ { "type": "step", "primary": "Use the following identity: $$\\cos\\left(x\\right)=\\sin\\left(\\frac{π}{2}-x\\right)$$", "result": "\\sin\\left(9x+6\\right)=\\sin\\left(\\frac{π}{2}-\\left(3x-4\\right)\\right)" } ], "meta": { "interimType": "Trig Rewrite Using Trig identities 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7Awm8OHaUUE5X73KqyTFjFCQMEIYvrf0LEy1Y52fuHf0UOLSbMox6vSWmoo6JD5Rw6oDeFBEXdI8LVv9QsNOFoRacrm3LsJ1aIjJN65AlUxeJDKsu+SuqQAE/BN/8mXKTV6BkdjDK0hCx54mKMTfucAKjd4a2pVq4DEaVPyiV22w+Uvjj3eTo8pQ0YW7i32PqvXQ2ZFwrOfgsYNWDbyTT1HvcmBdkOakr94A5S0KARY4+chlAMIGsKWEx/1b7YN7x" } }, { "type": "interim", "title": "Apply trig inverse properties", "input": "\\sin\\left(9x+6\\right)=\\sin\\left(\\frac{π}{2}-\\left(3x-4\\right)\\right)", "result": "9x+6=\\frac{π}{2}-\\left(3x-4\\right)+2πn,\\:9x+6=π-\\left(\\frac{π}{2}-\\left(3x-4\\right)\\right)+2πn", "steps": [ { "type": "step", "primary": "$$\\sin\\left(x\\right)=\\sin\\left(y\\right)\\quad\\Rightarrow\\quad\\:x=y+2{\\pi}n,\\:x=\\pi-y+2{\\pi}n$$", "result": "9x+6=\\frac{π}{2}-\\left(3x-4\\right)+2πn,\\:9x+6=π-\\left(\\frac{π}{2}-\\left(3x-4\\right)\\right)+2πn" } ], "meta": { "interimType": "Trig Apply Inverse Props 0Eq" } }, { "type": "interim", "title": "$$9x+6=\\frac{π}{2}-\\left(3x-4\\right)+2πn{\\quad:\\quad}x=\\frac{4πn+π-4}{24}$$", "input": "9x+6=\\frac{π}{2}-\\left(3x-4\\right)+2πn", "steps": [ { "type": "interim", "title": "Expand $$\\frac{π}{2}-\\left(3x-4\\right)+2πn:{\\quad}\\frac{π}{2}-3x+4+2πn$$", "input": "\\frac{π}{2}-\\left(3x-4\\right)+2πn", "steps": [ { "type": "interim", "title": "$$-\\left(3x-4\\right):{\\quad}-3x+4$$", "input": "-\\left(3x-4\\right)", "result": "=\\frac{π}{2}-3x+4+2πn", "steps": [ { "type": "step", "primary": "Distribute parentheses", "result": "=-\\left(3x\\right)-\\left(-4\\right)" }, { "type": "step", "primary": "Apply minus-plus rules", "secondary": [ "$$-\\left(-a\\right)=a,\\:\\:\\:-\\left(a\\right)=-a$$" ], "result": "=-3x+4" } ], "meta": { "interimType": "N/A" } } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Expand Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYniMGJoCgq5OssuroM6paXIuvzYp16iv1vHl5x+pFTNTQJgsHNdowDvNWB5jHy+mFfbYRM9rkvj4p3T+KeGRU7lN5Aod6Hr1Lp2e/29KhSgUt3JYzoyp8ESeIHTUw0y56TO8vXF5v5ufbt3REmOJBg+5bt+0ckfCSEFYBIdGLjra" } }, { "type": "step", "result": "9x+6=\\frac{π}{2}-3x+4+2πn" }, { "type": "interim", "title": "Move $$6\\:$$to the right side", "input": "9x+6=\\frac{π}{2}-3x+4+2πn", "result": "9x=-3x+2πn+\\frac{π}{2}-2", "steps": [ { "type": "step", "primary": "Subtract $$6$$ from both sides", "result": "9x+6-6=\\frac{π}{2}-3x+4+2πn-6" }, { "type": "interim", "title": "Simplify", "input": "9x+6-6=\\frac{π}{2}-3x+4+2πn-6", "result": "9x=-3x+2πn+\\frac{π}{2}-2", "steps": [ { "type": "interim", "title": "Simplify $$9x+6-6:{\\quad}9x$$", "input": "9x+6-6", "steps": [ { "type": "step", "primary": "Add similar elements: $$6-6=0$$" }, { "type": "step", "result": "=9x" } ], "meta": { "interimType": "Generic Simplify Specific 1Eq" } }, { "type": "interim", "title": "Simplify $$\\frac{π}{2}-3x+4+2πn-6:{\\quad}-3x+2πn+\\frac{π}{2}-2$$", "input": "\\frac{π}{2}-3x+4+2πn-6", "steps": [ { "type": "step", "primary": "Group like terms", "result": "=-3x+2πn+\\frac{π}{2}+4-6" }, { "type": "step", "primary": "Add/Subtract the numbers: $$4-6=-2$$", "result": "=-3x+2πn+\\frac{π}{2}-2" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7yMpAUr6i+tlsbA4M/UXxTVV+hTJ3HVKlenxZZ0NIgM3dd47a0hQ8flDbGsI5To1dbb8Y1D3F0f5cCbNsFaLSbujxqKJ+mcqmkBx3bXXSh1g/y9DKGIPglJ+qMi9xDu2KaRI7GCp0HQz+zDw23axddEL3VHW8RTR7El0wxF4ItTkj+1A5Mk2y/rC6vAzDoCwN" } }, { "type": "step", "result": "9x=-3x+2πn+\\frac{π}{2}-2" } ], "meta": { "interimType": "Generic Simplify 0Eq" } } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7IYaFKrId4BlbIbMPbS/uP7hwEuMILQAIFu5utLt3nCHiT/RgWmxufEqiE4woRAisyI3IEx+66nz2VU9wYA1wi/7OlGoLFf1X4hrgcqZHy87qSsZnbixwRVjmzCEwRHqkJU3UCOoD1mMCBv0XrPjCn/wiajpQZ7MDvrnD2Ue9u8CsHuCWujTcybm8y4lstc3AiiuTvHZMNMidREaJCrkWGfYkSjgjNOC3FBGwg+YohGVNkYfxKoe1Ra5hrZ2nDRu6mSHVZSuLZCyNEJOYOZ83vJlsDSw0VDjooHMJOX+4xJVuPNpEBMSuTyeVrHv2KNCL1v3DeFtYZWNghtvd0aopQgJMnr81tgv4O3Zmayl+TFs9KqaqL38nCVgZLONNaxsid7zWeEaNgq8oxH3J5ErgtnbIllu/0k0LUpCbYoE0M45OC1XxR5OU6NuvCKo2efCi0mGkD3+4P3JH8+HqppYV9TkrIeytDe07pErV7r9V36yVrUYSIPtChMsh4ws3iNwf0GoqR269Z5nVqV2M25/NwcrOCQn16lcBUSDyTTKFAYd62GGnKtBN1rhK9Rxmb8qw78BvdxLNf+AHitinsQpd63mFH005xAwfzLPUQYTibRuA6EVwpco2FU2OEEd9ueof8GB5H5hx083tyV5Ny44iaQ==" } }, { "type": "interim", "title": "Move $$3x\\:$$to the left side", "input": "9x=-3x+2πn+\\frac{π}{2}-2", "result": "12x=2πn+\\frac{π}{2}-2", "steps": [ { "type": "step", "primary": "Add $$3x$$ to both sides", "result": "9x+3x=-3x+2πn+\\frac{π}{2}-2+3x" }, { "type": "step", "primary": "Simplify", "result": "12x=2πn+\\frac{π}{2}-2" } ], "meta": { "interimType": "Move to the Left Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7df1aRZXJWJuOKBTUNaUvg5pPe3hdtPbeoe7vcLDGnA2QMwFmbcuZ9iF5JcL2wvr3SWczt+HbCP/9GNXKsbgGs/JU44kK14Gvt5oHRh1P17ofbifmzMz73oFWYE2mBXHx5w8lM1ZoU0soMD/+wRi8K9hpgNhne93xMCMNgIx6CMRT2KTlYg0auxai0vNLv8GMyKUbYNSoys/1ejnnHsq2sJwjbMUlM46SI+o7kD/PM9kzCWZC/+o+Y7VVDiCUgTiqt926tAXmOALNzqssjc7qaFnBYmptev6bL8FOJDh8cPARUV9JpOiwh0CRJPjLgJhuFAJHdsciUDxm4TPs+qxLiFRcTIzggRpe/YOa9myw1G1kxoyII1ltBb2pdMhg0YkJ7dvPqn4K4I4sW+eYzJsksOjNGWPNDOSwRjqumGEUVk7ggwh9g3AzYmpTLr8uwZ8J5PlGPSQROb/2EmerPhnM852KpwArzUqjj8qnIU+IJnx2m2cIuScrTf9bn5kIkoppBjIEkvaPTvlHoUdWt09SE1p2DDO5jolvRyiBIh4pWq5RxuKcrXgaifl/dAFyf8kqjRMba424ZtSf4yCZErnExLJj/qjqCqlOQVp6bMfROCcp7n6Pen8rW7RauD45MwlsJLd1ohke2Wgml78++2zI0g==" } }, { "type": "interim", "title": "Divide both sides by $$12$$", "input": "12x=2πn+\\frac{π}{2}-2", "result": "x=\\frac{4πn+π-4}{24}", "steps": [ { "type": "step", "primary": "Divide both sides by $$12$$", "result": "\\frac{12x}{12}=\\frac{2πn}{12}+\\frac{\\frac{π}{2}}{12}-\\frac{2}{12}" }, { "type": "interim", "title": "Simplify", "input": "\\frac{12x}{12}=\\frac{2πn}{12}+\\frac{\\frac{π}{2}}{12}-\\frac{2}{12}", "result": "x=\\frac{4πn+π-4}{24}", "steps": [ { "type": "interim", "title": "Simplify $$\\frac{12x}{12}:{\\quad}x$$", "input": "\\frac{12x}{12}", "steps": [ { "type": "step", "primary": "Divide the numbers: $$\\frac{12}{12}=1$$", "result": "=x" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7jx0bPoTfDSyxds17kYoNyXyRHuGw7+tM5METTDj6vVEed1oZgJr3Rrt+25B4RF18ZEt3ZXAiqUE0HIXrrrezJHZKl2mV4AlztiApTAFigz28oEBSPDPQa5rxqD6vLXZlialcV/dI5TH4fXyp+ncwuA==" } }, { "type": "interim", "title": "Simplify $$\\frac{2πn}{12}+\\frac{\\frac{π}{2}}{12}-\\frac{2}{12}:{\\quad}\\frac{4πn+π-4}{24}$$", "input": "\\frac{2πn}{12}+\\frac{\\frac{π}{2}}{12}-\\frac{2}{12}", "steps": [ { "type": "step", "primary": "Apply rule $$\\frac{a}{c}\\pm\\frac{b}{c}=\\frac{a\\pm\\:b}{c}$$", "result": "=\\frac{2πn+\\frac{π}{2}-2}{12}" }, { "type": "interim", "title": "Join $$2πn+\\frac{π}{2}-2:{\\quad}\\frac{4πn+π-4}{2}$$", "input": "2πn+\\frac{π}{2}-2", "result": "=\\frac{\\frac{4πn+π-4}{2}}{12}", "steps": [ { "type": "step", "primary": "Convert element to fraction: $$2πn=\\frac{2πn2}{2},\\:2=\\frac{2\\cdot\\:2}{2}$$", "result": "=\\frac{2πn\\cdot\\:2}{2}+\\frac{π}{2}-\\frac{2\\cdot\\:2}{2}" }, { "type": "step", "primary": "Since the denominators are equal, combine the fractions: $$\\frac{a}{c}\\pm\\frac{b}{c}=\\frac{a\\pm\\:b}{c}$$", "result": "=\\frac{2πn\\cdot\\:2+π-2\\cdot\\:2}{2}" }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:2=4$$", "result": "=\\frac{4πn+π-4}{2}" } ], "meta": { "interimType": "Algebraic Manipulation Join Concise Title 1Eq" } }, { "type": "step", "primary": "Apply the fraction rule: $$\\frac{\\frac{b}{c}}{a}=\\frac{b}{c\\:\\cdot\\:a}$$", "result": "=\\frac{4πn+π-4}{2\\cdot\\:12}" }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:12=24$$", "result": "=\\frac{4πn+π-4}{24}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7mm9OvdXNF32a8zno+EUmU7yy5cb6x2FmBFFFQ4LmCEUnf1JB1QfM4cGCT2MOFOkO4XtR5ZdcmfHzKAJuLYxCts0ag8T1MwTer44+aCS/ZFBZOiK0MRSPV+NAV+q5TmJd5R10lMDCCSWUj5kQi8ZLL+9sGZu5A1MXROmEpnxG69ryk4whNiKT03wC0hzj1bDVXW+UIL5CjPKPPPjcLOCm0TCsio/RtQQ3RD3LfUwLffVAXd9FbNgppIKzOQVGu8j6Jdl9TzOEdYw+KwfBwsfnYw==" } }, { "type": "step", "result": "x=\\frac{4πn+π-4}{24}" } ], "meta": { "interimType": "Generic Simplify 0Eq" } } ], "meta": { "interimType": "Divide Both Sides Specific 1Eq", "gptData": "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" } } ], "meta": { "solvingClass": "Equations", "interimType": "Equations" } }, { "type": "interim", "title": "$$9x+6=π-\\left(\\frac{π}{2}-\\left(3x-4\\right)\\right)+2πn{\\quad:\\quad}x=\\frac{π+4πn-20}{12}$$", "input": "9x+6=π-\\left(\\frac{π}{2}-\\left(3x-4\\right)\\right)+2πn", "steps": [ { "type": "interim", "title": "Expand $$π-\\left(\\frac{π}{2}-\\left(3x-4\\right)\\right)+2πn:{\\quad}π-\\frac{π}{2}+3x-4+2πn$$", "input": "π-\\left(\\frac{π}{2}-\\left(3x-4\\right)\\right)+2πn", "steps": [ { "type": "interim", "title": "$$-\\left(3x-4\\right):{\\quad}-3x+4$$", "input": "-\\left(3x-4\\right)", "steps": [ { "type": "step", "primary": "Distribute parentheses", "result": "=-\\left(3x\\right)-\\left(-4\\right)" }, { "type": "step", "primary": "Apply minus-plus rules", "secondary": [ "$$-\\left(-a\\right)=a,\\:\\:\\:-\\left(a\\right)=-a$$" ], "result": "=-3x+4" } ], "meta": { "interimType": "N/A" } }, { "type": "step", "result": "=π-\\left(-3x+4+\\frac{π}{2}\\right)+2πn" }, { "type": "interim", "title": "$$-\\left(\\frac{π}{2}-3x+4\\right):{\\quad}-\\frac{π}{2}+3x-4$$", "input": "-\\left(\\frac{π}{2}-3x+4\\right)", "result": "=π-\\frac{π}{2}+3x-4+2πn", "steps": [ { "type": "step", "primary": "Distribute parentheses", "result": "=-\\left(\\frac{π}{2}\\right)-\\left(-3x\\right)-\\left(4\\right)" }, { "type": "step", "primary": "Apply minus-plus rules", "secondary": [ "$$-\\left(-a\\right)=a,\\:\\:\\:-\\left(a\\right)=-a$$" ], "result": "=-\\frac{π}{2}+3x-4" } ], "meta": { "interimType": "N/A" } } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Expand Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7y8RdtQbqcqkCTkMN3ZLu8+NDbGRnSuoEIval8H71KiLJoEES4ZA/m0zbDwmuPYJy6EwgAt4XDVtRVLKgiN9Z37doOsALPuyc3sKn3rGJKfjvbBmbuQNTF0TphKZ8Ruva7JGdqcaYPk004EMFomkqo1CPNxGdO1+kxXfp5+W4dmxJ215GfK7nkmgv+0csTs1J" } }, { "type": "step", "result": "9x+6=π-\\frac{π}{2}+3x-4+2πn" }, { "type": "interim", "title": "Move $$6\\:$$to the right side", "input": "9x+6=π-\\frac{π}{2}+3x-4+2πn", "result": "9x=3x+2πn+π-10-\\frac{π}{2}", "steps": [ { "type": "step", "primary": "Subtract $$6$$ from both sides", "result": "9x+6-6=π-\\frac{π}{2}+3x-4+2πn-6" }, { "type": "interim", "title": "Simplify", "input": "9x+6-6=π-\\frac{π}{2}+3x-4+2πn-6", "result": "9x=3x+2πn+π-10-\\frac{π}{2}", "steps": [ { "type": "interim", "title": "Simplify $$9x+6-6:{\\quad}9x$$", "input": "9x+6-6", "steps": [ { "type": "step", "primary": "Add similar elements: $$6-6=0$$" }, { "type": "step", "result": "=9x" } ], "meta": { "interimType": "Generic Simplify Specific 1Eq" } }, { "type": "interim", "title": "Simplify $$π-\\frac{π}{2}+3x-4+2πn-6:{\\quad}3x+2πn+π-10-\\frac{π}{2}$$", "input": "π-\\frac{π}{2}+3x-4+2πn-6", "steps": [ { "type": "step", "primary": "Group like terms", "result": "=3x+π+2πn-\\frac{π}{2}-4-6" }, { "type": "step", "primary": "Subtract the numbers: $$-4-6=-10$$", "result": "=3x+2πn+π-10-\\frac{π}{2}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7cXkxxIJhdmQPmbj1hY14lepR7l4VnfuHtqHGZPcO37dV00rpv8+ZC6TM10tVCSHsOku0T8udIvBjbGGtyUJ32VVWe9wpXEJND/CLkx7Z3XZ+XQj20M97oOMfjaynXbUAHjb2+5NLFZrsH9fcPWg/TX/lDrN/HKETOY1zib59TTDlZYXiFqdK0D+kt+Xeajg/k78hVGpXUyZ7pkwHbdfxsg==" } }, { "type": "step", "result": "9x=3x+2πn+π-10-\\frac{π}{2}" } ], "meta": { "interimType": "Generic Simplify 0Eq" } } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": "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" } }, { "type": "interim", "title": "Move $$3x\\:$$to the left side", "input": "9x=3x+2πn+π-10-\\frac{π}{2}", "result": "6x=2πn+π-10-\\frac{π}{2}", "steps": [ { "type": "step", "primary": "Subtract $$3x$$ from both sides", "result": "9x-3x=3x+2πn+π-10-\\frac{π}{2}-3x" }, { "type": "step", "primary": "Simplify", "result": "6x=2πn+π-10-\\frac{π}{2}" } ], "meta": { "interimType": "Move to the Left Title 1Eq", "gptData": "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" } }, { "type": "interim", "title": "Divide both sides by $$6$$", "input": "6x=2πn+π-10-\\frac{π}{2}", "result": "x=\\frac{π+4πn-20}{12}", "steps": [ { "type": "step", "primary": "Divide both sides by $$6$$", "result": "\\frac{6x}{6}=\\frac{2πn}{6}+\\frac{π}{6}-\\frac{10}{6}-\\frac{\\frac{π}{2}}{6}" }, { "type": "interim", "title": "Simplify", "input": "\\frac{6x}{6}=\\frac{2πn}{6}+\\frac{π}{6}-\\frac{10}{6}-\\frac{\\frac{π}{2}}{6}", "result": "x=\\frac{π+4πn-20}{12}", "steps": [ { "type": "interim", "title": "Simplify $$\\frac{6x}{6}:{\\quad}x$$", "input": "\\frac{6x}{6}", "steps": [ { "type": "step", "primary": "Divide the numbers: $$\\frac{6}{6}=1$$", "result": "=x" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7Ef5HWtgtSheecLgzWn9cbC061ljBSPJeENOw2efoSWt2cZ6LAXMcKee6PSyjLEFrRSpN33oxZMojoqvYhvSJACLkpiVA7S8UT8ieezZKu6ozYDWEKLcgLlQkwDuRSw+k" } }, { "type": "interim", "title": "Simplify $$\\frac{2πn}{6}+\\frac{π}{6}-\\frac{10}{6}-\\frac{\\frac{π}{2}}{6}:{\\quad}\\frac{π+4πn-20}{12}$$", "input": "\\frac{2πn}{6}+\\frac{π}{6}-\\frac{10}{6}-\\frac{\\frac{π}{2}}{6}", "steps": [ { "type": "step", "primary": "Apply rule $$\\frac{a}{c}\\pm\\frac{b}{c}=\\frac{a\\pm\\:b}{c}$$", "result": "=\\frac{2πn+π-10-\\frac{π}{2}}{6}" }, { "type": "interim", "title": "Join $$2πn+π-10-\\frac{π}{2}:{\\quad}\\frac{π+4πn-20}{2}$$", "input": "2πn+π-10-\\frac{π}{2}", "result": "=\\frac{\\frac{π+4πn-20}{2}}{6}", "steps": [ { "type": "step", "primary": "Convert element to fraction: $$2πn=\\frac{2πn2}{2},\\:π=\\frac{π2}{2},\\:10=\\frac{10\\cdot\\:2}{2}$$", "result": "=\\frac{2πn\\cdot\\:2}{2}+\\frac{π2}{2}-\\frac{10\\cdot\\:2}{2}-\\frac{π}{2}" }, { "type": "step", "primary": "Since the denominators are equal, combine the fractions: $$\\frac{a}{c}\\pm\\frac{b}{c}=\\frac{a\\pm\\:b}{c}$$", "result": "=\\frac{2πn\\cdot\\:2+π2-10\\cdot\\:2-π}{2}" }, { "type": "interim", "title": "$$2πn\\cdot\\:2+π2-10\\cdot\\:2-π=π+4πn-20$$", "input": "2πn\\cdot\\:2+π2-10\\cdot\\:2-π", "steps": [ { "type": "step", "primary": "Group like terms", "result": "=2π-π+2\\cdot\\:2πn-10\\cdot\\:2" }, { "type": "step", "primary": "Add similar elements: $$2π-π=π$$", "result": "=π+2\\cdot\\:2πn-10\\cdot\\:2" }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:2=4$$", "result": "=π+4πn-10\\cdot\\:2" }, { "type": "step", "primary": "Multiply the numbers: $$10\\cdot\\:2=20$$", "result": "=π+4πn-20" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7IRg8zNvHAxvPh71gHgzGNV8miICHxVj9cUzk/3uv4aMAlilG71elit3w1IBbYN0PF4Oq9aV+4dtWtk7xrJRcY7qTNcNPaFhZI1gEJtbFGfiPCZX1LCCWZV5U81JkKqM1WEuqOPPzWRhtzdVtYJpryU6em2a53YCQQjBtC+iNuNKwiNrEngO+NNvZ9sqNu+2V" } }, { "type": "step", "result": "=\\frac{π+4πn-20}{2}" } ], "meta": { "interimType": "Algebraic Manipulation Join Concise Title 1Eq" } }, { "type": "step", "primary": "Apply the fraction rule: $$\\frac{\\frac{b}{c}}{a}=\\frac{b}{c\\:\\cdot\\:a}$$", "result": "=\\frac{π+4πn-20}{2\\cdot\\:6}" }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:6=12$$", "result": "=\\frac{π+4πn-20}{12}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7J17HOf2aCZ5UFhwSrtZOWXT5CeDlDJsj4cVYSOhvfdyoZThzr3rzXH7i2bYBItRswRIG2JInyDqbzqIimJ7dmmTVUs/UF48LLaFt3193bX2rju+5Z51e/ZZSD3gRHwjB9lAMUeeMUjfzqc/ZcBKWhlelAlc+Yn14PHGAwXe3EgLWwPs1+Gw97t4MeuaNjSYTwPBBw3byB5RPtpSy6YLt2uSDndlCbUE3uhx+Eg88WlFaayWKJu2ODI3j5O7IlqJDv5fkbwgjAlcvbMvhKRI6FsWjq6PH7iwN7IC50S/X/hKwiNrEngO+NNvZ9sqNu+2V" } }, { "type": "step", "result": "x=\\frac{π+4πn-20}{12}" } ], "meta": { "interimType": "Generic Simplify 0Eq" } } ], "meta": { "interimType": "Divide Both Sides Specific 1Eq", "gptData": "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" } }, { "type": "step", "result": "x=\\frac{4πn+π-4}{24},\\:x=\\frac{π+4πn-20}{12}" } ], "meta": { "solvingClass": "Equations", "interimType": "Equations" } }, { "type": "step", "result": "x=\\frac{4πn+π-4}{24},\\:x=\\frac{π+4πn-20}{12}" } ], "meta": { "solvingClass": "Trig Equations", "practiceLink": "/practice/trigonometry-practice#area=main&subtopic=Trig%20Equations", "practiceTopic": "Trig Equations" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "\\sin(9x+6)-\\cos(3x-4)" }, "showViewLarger": true } }, "meta": { "showVerify": true } }