sin(4k-22)=cos(6k-13)

{ "query": { "display": "$$\\sin\\left(4k-22\\right)=\\cos\\left(6k-13\\right)$$", "symbolab_question": "EQUATION#\\sin(4k-22)=\\cos(6k-13)" }, "solution": { "level": "PERFORMED", "subject": "Trigonometry", "topic": "Trig Equations", "subTopic": "Trig Equations", "default": "k=\\frac{4πn+70+π}{20},k=-\\frac{4πn+18+π}{4}", "degrees": "k=209.53522…^{\\circ }+36^{\\circ }n,k=-302.83100…^{\\circ }-180^{\\circ }n", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\sin\\left(4k-22\\right)=\\cos\\left(6k-13\\right){\\quad:\\quad}k=\\frac{4πn+70+π}{20},\\:k=-\\frac{4πn+18+π}{4}$$", "input": "\\sin\\left(4k-22\\right)=\\cos\\left(6k-13\\right)", "steps": [ { "type": "interim", "title": "Rewrite using trig identities", "input": "\\sin\\left(4k-22\\right)=\\cos\\left(6k-13\\right)", "result": "\\sin\\left(4k-22\\right)=\\sin\\left(\\frac{π}{2}-\\left(6k-13\\right)\\right)", "steps": [ { "type": "step", "primary": "Use the following identity: $$\\cos\\left(x\\right)=\\sin\\left(\\frac{π}{2}-x\\right)$$", "result": "\\sin\\left(4k-22\\right)=\\sin\\left(\\frac{π}{2}-\\left(6k-13\\right)\\right)" } ], "meta": { "interimType": "Trig Rewrite Using Trig identities 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7Awm8OHaUUE5X73KqyTFjFPcXCbm3WxOysXE7W7oSEPzPEKKodBz0iY1Gjq40UtjwCL1WefiKaC4ihafc3V6qN0YiRvdBbbDvLg3zdFKS1tz+RS6nKnvFfEHxiYoRDkegsyEzxQiTjXdrYUsk9Xkq6G1DnVhU7cPoA+NNczCSWtNQcb3SKnq43PZ96Gcwxa2f8LfSxJ+0AgVLpCSnLX0iSpNlUHajxrJmE2mMnrJZR00XOCl7PMYCTYi5CcK9G/+TialcV/dI5TH4fXyp+ncwuA==" } }, { "type": "interim", "title": "Apply trig inverse properties", "input": "\\sin\\left(4k-22\\right)=\\sin\\left(\\frac{π}{2}-\\left(6k-13\\right)\\right)", "result": "4k-22=\\frac{π}{2}-\\left(6k-13\\right)+2πn,\\:4k-22=π-\\left(\\frac{π}{2}-\\left(6k-13\\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": "4k-22=\\frac{π}{2}-\\left(6k-13\\right)+2πn,\\:4k-22=π-\\left(\\frac{π}{2}-\\left(6k-13\\right)\\right)+2πn" } ], "meta": { "interimType": "Trig Apply Inverse Props 0Eq" } }, { "type": "interim", "title": "$$4k-22=\\frac{π}{2}-\\left(6k-13\\right)+2πn{\\quad:\\quad}k=\\frac{4πn+70+π}{20}$$", "input": "4k-22=\\frac{π}{2}-\\left(6k-13\\right)+2πn", "steps": [ { "type": "interim", "title": "Expand $$\\frac{π}{2}-\\left(6k-13\\right)+2πn:{\\quad}\\frac{π}{2}-6k+13+2πn$$", "input": "\\frac{π}{2}-\\left(6k-13\\right)+2πn", "steps": [ { "type": "interim", "title": "$$-\\left(6k-13\\right):{\\quad}-6k+13$$", "input": "-\\left(6k-13\\right)", "result": "=\\frac{π}{2}-6k+13+2πn", "steps": [ { "type": "step", "primary": "Distribute parentheses", "result": "=-\\left(6k\\right)-\\left(-13\\right)" }, { "type": "step", "primary": "Apply minus-plus rules", "secondary": [ "$$-\\left(-a\\right)=a,\\:\\:\\:-\\left(a\\right)=-a$$" ], "result": "=-6k+13" } ], "meta": { "interimType": "N/A" } } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Expand Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYi7X3NHKTE7CIErK5/zbnMuIydchH6mcwpw4uJvqv2+PdQxShwOg3W+URNAMu5sPa6VYwrogLl29RT6HYd2NJ316pfF1z6umzUJTJvt+ojYZEgMzQ6KqRUeJ0bFe7kX7pQVPfoyRxWHb7pOYlUFfEI0ceFyf+uA6Q2Ysf1eSxiTL" } }, { "type": "step", "result": "4k-22=\\frac{π}{2}-6k+13+2πn" }, { "type": "interim", "title": "Move $$22\\:$$to the right side", "input": "4k-22=\\frac{π}{2}-6k+13+2πn", "result": "4k=-6k+2πn+35+\\frac{π}{2}", "steps": [ { "type": "step", "primary": "Add $$22$$ to both sides", "result": "4k-22+22=\\frac{π}{2}-6k+13+2πn+22" }, { "type": "interim", "title": "Simplify", "input": "4k-22+22=\\frac{π}{2}-6k+13+2πn+22", "result": "4k=-6k+2πn+35+\\frac{π}{2}", "steps": [ { "type": "interim", "title": "Simplify $$4k-22+22:{\\quad}4k$$", "input": "4k-22+22", "steps": [ { "type": "step", "primary": "Add similar elements: $$-22+22=0$$" }, { "type": "step", "result": "=4k" } ], "meta": { "interimType": "Generic Simplify Specific 1Eq" } }, { "type": "interim", "title": "Simplify $$\\frac{π}{2}-6k+13+2πn+22:{\\quad}-6k+2πn+35+\\frac{π}{2}$$", "input": "\\frac{π}{2}-6k+13+2πn+22", "steps": [ { "type": "step", "primary": "Group like terms", "result": "=-6k+2πn+\\frac{π}{2}+13+22" }, { "type": "step", "primary": "Add the numbers: $$13+22=35$$", "result": "=-6k+2πn+35+\\frac{π}{2}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7yMpAUr6i+tlsbA4M/UXxTWsPnhe57ph/pb0yT6uAm2PehkKrn0era9rz8TlL+x/vPp0kRkc83M4lu3elp44S54eK/ao+fBfEYUt8fGgQpEheTZMykILbMC5S4vTIC/oK7kAjP76qW66lOUsURwT0nVO7gTACeCLS+oqUDl50bY3Ih8Fv+sFWIPEqdQu05ZV3dY1aNzSrSBkiKqz8eYDm8Q==" } }, { "type": "step", "result": "4k=-6k+2πn+35+\\frac{π}{2}" } ], "meta": { "interimType": "Generic Simplify 0Eq" } } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7C7H8OT1ENjA7/vnUrs29zcwJH2kz5GjE2C/f6IoDetVaxuxKXCR9DeSoyOqAeZ0XKjt5tZPC53b/4UeR0qwP1ZAfd6GCL3c7/qgAIswOkjGAqomnF50Zs2t8jA+QD47lyrLWqEhfwnyfg8gW8kCh2iSIB+5rZzjN9rZB584s2S28VmH5b+IMwMCjiIIju0VR3CmpoWEZ6Xx6oiIfKYR080iu613x5IyBSj4elHIXAAwgIfmxhP/AWs9oWAlWRg0n4e96+tdx2vFumkUCewGHeOA7QfNko7HT+RkPLHjHffxhca41gTeJaDZRIG/WxVZgKNTKiv8FWiQkB2ndvZ/Xgsd1ybSLzE1W3zepwHc3olgA2AxkzdzOhVTRkP1v37yB17I7hE3TeFS3BbJ0R8049LJ/xkd+bNmCDxpcW9XeuIXbhiBuISu5U0zuwSRDg8CMTxiuZJF6Oi5XpwWE3BNQwNTcoBjiQMeKUzqvjloxiaFFPoW8imL5Ihzz6jeBHy3Vi126KflIGXPj0T66SV3pmAInWVDUUDBWNHqYNa25MfkOAOWTWL/jIC0Sf89wVXjAo3oe/oyhMy2+1TQhDBd2fxNGhPaOe1mzJ7n2xS3HBgS6LBtT4H6f7c74IgzFg/ox2oea1dT8rJycbEDAqgbDcA==" } }, { "type": "interim", "title": "Move $$6k\\:$$to the left side", "input": "4k=-6k+2πn+35+\\frac{π}{2}", "result": "10k=2πn+35+\\frac{π}{2}", "steps": [ { "type": "step", "primary": "Add $$6k$$ to both sides", "result": "4k+6k=-6k+2πn+35+\\frac{π}{2}+6k" }, { "type": "step", "primary": "Simplify", "result": "10k=2πn+35+\\frac{π}{2}" } ], "meta": { "interimType": "Move to the Left Title 1Eq", "gptData": "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" } }, { "type": "interim", "title": "Divide both sides by $$10$$", "input": "10k=2πn+35+\\frac{π}{2}", "result": "k=\\frac{4πn+70+π}{20}", "steps": [ { "type": "step", "primary": "Divide both sides by $$10$$", "result": "\\frac{10k}{10}=\\frac{2πn}{10}+\\frac{35}{10}+\\frac{\\frac{π}{2}}{10}" }, { "type": "interim", "title": "Simplify", "input": "\\frac{10k}{10}=\\frac{2πn}{10}+\\frac{35}{10}+\\frac{\\frac{π}{2}}{10}", "result": "k=\\frac{4πn+70+π}{20}", "steps": [ { "type": "interim", "title": "Simplify $$\\frac{10k}{10}:{\\quad}k$$", "input": "\\frac{10k}{10}", "steps": [ { "type": "step", "primary": "Divide the numbers: $$\\frac{10}{10}=1$$", "result": "=k" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7IPueq4BoI7HI0W+T+Fs0fXyRHuGw7+tM5METTDj6vVGBPx2+a2lHg0oydxbFHoIWZEt3ZXAiqUE0HIXrrrezJHZKl2mV4AlztiApTAFigz0jzMQvQIK/0X7IThzsVurDialcV/dI5TH4fXyp+ncwuA==" } }, { "type": "interim", "title": "Simplify $$\\frac{2πn}{10}+\\frac{35}{10}+\\frac{\\frac{π}{2}}{10}:{\\quad}\\frac{4πn+70+π}{20}$$", "input": "\\frac{2πn}{10}+\\frac{35}{10}+\\frac{\\frac{π}{2}}{10}", "steps": [ { "type": "step", "primary": "Apply rule $$\\frac{a}{c}\\pm\\frac{b}{c}=\\frac{a\\pm\\:b}{c}$$", "result": "=\\frac{2πn+35+\\frac{π}{2}}{10}" }, { "type": "interim", "title": "Join $$2πn+35+\\frac{π}{2}:{\\quad}\\frac{4πn+70+π}{2}$$", "input": "2πn+35+\\frac{π}{2}", "result": "=\\frac{\\frac{4πn+70+π}{2}}{10}", "steps": [ { "type": "step", "primary": "Convert element to fraction: $$2πn=\\frac{2πn2}{2},\\:35=\\frac{35\\cdot\\:2}{2}$$", "result": "=\\frac{2πn\\cdot\\:2}{2}+\\frac{35\\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+35\\cdot\\:2+π}{2}" }, { "type": "interim", "title": "$$2πn\\cdot\\:2+35\\cdot\\:2+π=4πn+70+π$$", "input": "2πn\\cdot\\:2+35\\cdot\\:2+π", "steps": [ { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:2=4$$", "result": "=4πn+35\\cdot\\:2+π" }, { "type": "step", "primary": "Multiply the numbers: $$35\\cdot\\:2=70$$", "result": "=4πn+70+π" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s722rs/UtTzhh0/11VX7Cb4/q/2okUI6gJC7Pdm5qyNOcgJ/ZZA32ZInFBpDtxBfiKSQa81OmZKqPqNAFWPJMxvHBL6UdhVd6AiqtRImFj5WIoeNmmfAlWrUtbw1ZVoKYvGu3e297/TORiWh81AVbh7pi7XTMP075Q4L+afXS54WU=" } }, { "type": "step", "result": "=\\frac{4πn+70+π}{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+70+π}{2\\cdot\\:10}" }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:10=20$$", "result": "=\\frac{4πn+70+π}{20}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7mm9OvdXNF32a8zno+EUmU/VkWS+WBRHbKs4WsZyvkVLhn4vzFA5Z82kZ+QISqyjtnIxagpLVPh1oI4aQUMAS2d13jtrSFDx+UNsawjlOjV0pt6Dc/Y/AG/hhFDuDiiGd7yXtHIuYDBVPKLyV62cU+6N6Hv6MoTMtvtU0IQwXdn9szOhN37mcRdV5CgGGkVwg6QuKXEJy/0cfNg1qxCN0nWkgP6il++qEB5oSbYkfScMfXM3pkXTKTvPOuvZK62F1AqkX2ZuMeKqNpsx5J5NXdA==" } }, { "type": "step", "result": "k=\\frac{4πn+70+π}{20}" } ], "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": "$$4k-22=π-\\left(\\frac{π}{2}-\\left(6k-13\\right)\\right)+2πn{\\quad:\\quad}k=-\\frac{4πn+18+π}{4}$$", "input": "4k-22=π-\\left(\\frac{π}{2}-\\left(6k-13\\right)\\right)+2πn", "steps": [ { "type": "interim", "title": "Expand $$π-\\left(\\frac{π}{2}-\\left(6k-13\\right)\\right)+2πn:{\\quad}π-\\frac{π}{2}+6k-13+2πn$$", "input": "π-\\left(\\frac{π}{2}-\\left(6k-13\\right)\\right)+2πn", "steps": [ { "type": "interim", "title": "$$-\\left(6k-13\\right):{\\quad}-6k+13$$", "input": "-\\left(6k-13\\right)", "steps": [ { "type": "step", "primary": "Distribute parentheses", "result": "=-\\left(6k\\right)-\\left(-13\\right)" }, { "type": "step", "primary": "Apply minus-plus rules", "secondary": [ "$$-\\left(-a\\right)=a,\\:\\:\\:-\\left(a\\right)=-a$$" ], "result": "=-6k+13" } ], "meta": { "interimType": "N/A" } }, { "type": "step", "result": "=π-\\left(-6k+13+\\frac{π}{2}\\right)+2πn" }, { "type": "interim", "title": "$$-\\left(\\frac{π}{2}-6k+13\\right):{\\quad}-\\frac{π}{2}+6k-13$$", "input": "-\\left(\\frac{π}{2}-6k+13\\right)", "result": "=π-\\frac{π}{2}+6k-13+2πn", "steps": [ { "type": "step", "primary": "Distribute parentheses", "result": "=-\\left(\\frac{π}{2}\\right)-\\left(-6k\\right)-\\left(13\\right)" }, { "type": "step", "primary": "Apply minus-plus rules", "secondary": [ "$$-\\left(-a\\right)=a,\\:\\:\\:-\\left(a\\right)=-a$$" ], "result": "=-\\frac{π}{2}+6k-13" } ], "meta": { "interimType": "N/A" } } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Expand Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7y8RdtQbqcqkCTkMN3ZLu83XmbItKHzsKF15icKaiQkNBjVVD4RkOtbuKMppP6H5Em8Cy3/+KhUDPMF7zx3LrTDrItGTK5senSF/kdAFuKtRFKk3fejFkyiOiq9iG9IkAg+TSL+/6ntLOpwwfva9DT9BBED7eEz/5dwfOroArw2U1ewVf3JoUCRbFtKPHPOSW" } }, { "type": "step", "result": "4k-22=π-\\frac{π}{2}+6k-13+2πn" }, { "type": "interim", "title": "Move $$22\\:$$to the right side", "input": "4k-22=π-\\frac{π}{2}+6k-13+2πn", "result": "4k=6k+2πn+9+π-\\frac{π}{2}", "steps": [ { "type": "step", "primary": "Add $$22$$ to both sides", "result": "4k-22+22=π-\\frac{π}{2}+6k-13+2πn+22" }, { "type": "interim", "title": "Simplify", "input": "4k-22+22=π-\\frac{π}{2}+6k-13+2πn+22", "result": "4k=6k+2πn+9+π-\\frac{π}{2}", "steps": [ { "type": "interim", "title": "Simplify $$4k-22+22:{\\quad}4k$$", "input": "4k-22+22", "steps": [ { "type": "step", "primary": "Add similar elements: $$-22+22=0$$" }, { "type": "step", "result": "=4k" } ], "meta": { "interimType": "Generic Simplify Specific 1Eq" } }, { "type": "interim", "title": "Simplify $$π-\\frac{π}{2}+6k-13+2πn+22:{\\quad}6k+2πn+9+π-\\frac{π}{2}$$", "input": "π-\\frac{π}{2}+6k-13+2πn+22", "steps": [ { "type": "step", "primary": "Group like terms", "result": "=6k+π+2πn-\\frac{π}{2}-13+22" }, { "type": "step", "primary": "Add/Subtract the numbers: $$-13+22=9$$", "result": "=6k+2πn+9+π-\\frac{π}{2}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7cXkxxIJhdmQPmbj1hY14lbfNVFvHbAU5uESriWrGtZ4AlilG71elit3w1IBbYN0PK3/rrIgRHBx10Bho/grBtnzBnyO4acN93zVGfyUsiyCxhEN+qWXkRtTRqNBdv+BOTeQKHeh69S6dnv9vSoUoFMYgODOEypui9lQ8s3G5vIlTWZarNrQvf6tDoE9NL47dj/dAlIaVE90Oq8BGdhHRbQ==" } }, { "type": "step", "result": "4k=6k+2πn+9+π-\\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 $$6k\\:$$to the left side", "input": "4k=6k+2πn+9+π-\\frac{π}{2}", "result": "-2k=2πn+9+π-\\frac{π}{2}", "steps": [ { "type": "step", "primary": "Subtract $$6k$$ from both sides", "result": "4k-6k=6k+2πn+9+π-\\frac{π}{2}-6k" }, { "type": "step", "primary": "Simplify", "result": "-2k=2πn+9+π-\\frac{π}{2}" } ], "meta": { "interimType": "Move to the Left Title 1Eq", "gptData": "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" } }, { "type": "interim", "title": "Divide both sides by $$-2$$", "input": "-2k=2πn+9+π-\\frac{π}{2}", "result": "k=-\\frac{4πn+18+π}{4}", "steps": [ { "type": "step", "primary": "Divide both sides by $$-2$$", "result": "\\frac{-2k}{-2}=\\frac{2πn}{-2}+\\frac{9}{-2}+\\frac{π}{-2}-\\frac{\\frac{π}{2}}{-2}" }, { "type": "interim", "title": "Simplify", "input": "\\frac{-2k}{-2}=\\frac{2πn}{-2}+\\frac{9}{-2}+\\frac{π}{-2}-\\frac{\\frac{π}{2}}{-2}", "result": "k=-\\frac{4πn+18+π}{4}", "steps": [ { "type": "interim", "title": "Simplify $$\\frac{-2k}{-2}:{\\quad}k$$", "input": "\\frac{-2k}{-2}", "steps": [ { "type": "step", "primary": "Apply the fraction rule: $$\\frac{-a}{-b}=\\frac{a}{b}$$", "result": "=\\frac{2k}{2}" }, { "type": "step", "primary": "Divide the numbers: $$\\frac{2}{2}=1$$", "result": "=k" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s77PtZCL4LuTm507GPvUlqA3yRHuGw7+tM5METTDj6vVGBPx2+a2lHg0oydxbFHoIWZEt3ZXAiqUE0HIXrrrezJHZKl2mV4AlztiApTAFigz2aW6Mq1D6KrnLMpVwZUKHKialcV/dI5TH4fXyp+ncwuA==" } }, { "type": "interim", "title": "Simplify $$\\frac{2πn}{-2}+\\frac{9}{-2}+\\frac{π}{-2}-\\frac{\\frac{π}{2}}{-2}:{\\quad}-\\frac{4πn+18+π}{4}$$", "input": "\\frac{2πn}{-2}+\\frac{9}{-2}+\\frac{π}{-2}-\\frac{\\frac{π}{2}}{-2}", "steps": [ { "type": "step", "primary": "Apply rule $$\\frac{a}{c}\\pm\\frac{b}{c}=\\frac{a\\pm\\:b}{c}$$", "result": "=\\frac{2πn+9+π-\\frac{π}{2}}{-2}" }, { "type": "step", "primary": "Apply the fraction rule: $$\\frac{a}{-b}=-\\frac{a}{b}$$", "result": "=-\\frac{2πn+9+π-\\frac{π}{2}}{2}" }, { "type": "interim", "title": "Join $$2πn+9+π-\\frac{π}{2}:{\\quad}\\frac{4πn+18+π}{2}$$", "input": "2πn+9+π-\\frac{π}{2}", "result": "=-\\frac{\\frac{4πn+π+18}{2}}{2}", "steps": [ { "type": "step", "primary": "Convert element to fraction: $$2πn=\\frac{2πn2}{2},\\:9=\\frac{9\\cdot\\:2}{2},\\:π=\\frac{π2}{2}$$", "result": "=\\frac{2πn\\cdot\\:2}{2}+\\frac{9\\cdot\\:2}{2}+\\frac{π2}{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+9\\cdot\\:2+π2-π}{2}" }, { "type": "interim", "title": "$$2πn\\cdot\\:2+9\\cdot\\:2+π2-π=4πn+18+π$$", "input": "2πn\\cdot\\:2+9\\cdot\\:2+π2-π", "steps": [ { "type": "step", "primary": "Add similar elements: $$2π-π=π$$", "result": "=2\\cdot\\:2πn+9\\cdot\\:2+π" }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:2=4$$", "result": "=4πn+9\\cdot\\:2+π" }, { "type": "step", "primary": "Multiply the numbers: $$9\\cdot\\:2=18$$", "result": "=4πn+18+π" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7E/dj4CaUABcf8j8h2k1srSbO4qdcMrY0MX32hceMXHwtOtZYwUjyXhDTsNnn6ElrzdaHCbSDaKafeOSN7exCSISgepjXagyIYoB75mg7d6V4ccKGo8zkK1RnBb5QdtajD0C4LZMR5eUEh5h4dzdgsPDLbBwkrohb5TxdzvugamQkt3WiGR7ZaCaXvz77bMjS" } }, { "type": "step", "result": "=\\frac{4πn+18+π}{2}" } ], "meta": { "interimType": "Algebraic Manipulation Join Concise Title 1Eq" } }, { "type": "interim", "title": "Simplify $$\\frac{\\frac{4πn+18+π}{2}}{2}:{\\quad}\\frac{4πn+18+π}{4}$$", "input": "\\frac{\\frac{4πn+18+π}{2}}{2}", "result": "=-\\frac{4πn+π+18}{4}", "steps": [ { "type": "step", "primary": "Apply the fraction rule: $$\\frac{\\frac{b}{c}}{a}=\\frac{b}{c\\:\\cdot\\:a}$$", "result": "=\\frac{4πn+18+π}{2\\cdot\\:2}" }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:2=4$$", "result": "=\\frac{4πn+18+π}{4}" } ], "meta": { "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "result": "=-\\frac{4πn+18+π}{4}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7rWv8Qj96y2UCfi+p66aEe5VWtoF5YIcSpmBKgi2gfhD6JmQmGhN5T09nu0ryBE30MKyKj9G1BDdEPct9TAt99RZlKnyhH/m7dbwTSHIRjaIDnzlbPZjyKgy1eUCFsLd5Hu5Tg5GDbKK5+ie6pksA+gnWAcApDGZtIblEounpGVPvbBmbuQNTF0TphKZ8Ruva8pOMITYik9N8AtIc49Ww1Y/9vhIs3KMCK3Uc3OIm/8rNDKCf8LTYPPE19Yg3NVgnn/anXQASVNGw6wx/UDwI+4t2AmtwFT61NSc18SZpzdkXNhBqqjki7uMbe9XK2IFD" } }, { "type": "step", "result": "k=-\\frac{4πn+18+π}{4}" } ], "meta": { "interimType": "Generic Simplify 0Eq" } } ], "meta": { "interimType": "Divide Both Sides Specific 1Eq", "gptData": "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" } }, { "type": "step", "result": "k=\\frac{4πn+70+π}{20},\\:k=-\\frac{4πn+18+π}{4}" } ], "meta": { "solvingClass": "Equations", "interimType": "Equations" } }, { "type": "step", "result": "k=\\frac{4πn+70+π}{20},\\:k=-\\frac{4πn+18+π}{4}" } ], "meta": { "solvingClass": "Trig Equations", "practiceLink": "/practice/trigonometry-practice#area=main&subtopic=Trig%20Equations", "practiceTopic": "Trig Equations" } }, "plot_output": { "meta": { "plotInfo": { "variable": "k", "plotRequest": "\\sin(4k-22)-\\cos(6k-13)" }, "showViewLarger": true } }, "meta": { "showVerify": true } }