sin^4(x/3)+cos^4(x/3)= 5/8

{ "query": { "display": "$$\\sin^{4}\\left(\\frac{x}{3}\\right)+\\cos^{4}\\left(\\frac{x}{3}\\right)=\\frac{5}{8}$$", "symbolab_question": "EQUATION#\\sin^{4}(\\frac{x}{3})+\\cos^{4}(\\frac{x}{3})=\\frac{5}{8}" }, "solution": { "level": "PERFORMED", "subject": "Trigonometry", "topic": "Trig Equations", "subTopic": "Trig Equations", "default": "x=\\frac{π}{2}+6πn,x=\\frac{11π}{2}+6πn,x=\\frac{5π}{2}+6πn,x=\\frac{7π}{2}+6πn,x=π+6πn,x=5π+6πn,x=2π+6πn,x=4π+6πn", "degrees": "x=90^{\\circ }+1080^{\\circ }n,x=990^{\\circ }+1080^{\\circ }n,x=450^{\\circ }+1080^{\\circ }n,x=630^{\\circ }+1080^{\\circ }n,x=180^{\\circ }+1080^{\\circ }n,x=900^{\\circ }+1080^{\\circ }n,x=360^{\\circ }+1080^{\\circ }n,x=720^{\\circ }+1080^{\\circ }n", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\sin^{4}\\left(\\frac{x}{3}\\right)+\\cos^{4}\\left(\\frac{x}{3}\\right)=\\frac{5}{8}{\\quad:\\quad}x=\\frac{π}{2}+6πn,\\:x=\\frac{11π}{2}+6πn,\\:x=\\frac{5π}{2}+6πn,\\:x=\\frac{7π}{2}+6πn,\\:x=π+6πn,\\:x=5π+6πn,\\:x=2π+6πn,\\:x=4π+6πn$$", "input": "\\sin^{4}\\left(\\frac{x}{3}\\right)+\\cos^{4}\\left(\\frac{x}{3}\\right)=\\frac{5}{8}", "steps": [ { "type": "step", "primary": "Subtract $$\\frac{5}{8}$$ from both sides", "result": "\\sin^{4}\\left(\\frac{x}{3}\\right)+\\cos^{4}\\left(\\frac{x}{3}\\right)-\\frac{5}{8}=0" }, { "type": "interim", "title": "Simplify $$\\sin^{4}\\left(\\frac{x}{3}\\right)+\\cos^{4}\\left(\\frac{x}{3}\\right)-\\frac{5}{8}:{\\quad}\\frac{8\\sin^{4}\\left(\\frac{x}{3}\\right)+8\\cos^{4}\\left(\\frac{x}{3}\\right)-5}{8}$$", "input": "\\sin^{4}\\left(\\frac{x}{3}\\right)+\\cos^{4}\\left(\\frac{x}{3}\\right)-\\frac{5}{8}", "steps": [ { "type": "step", "primary": "Convert element to fraction: $$\\sin^{4}\\left(\\frac{x}{3}\\right)=\\frac{\\sin^{4}\\left(\\frac{x}{3}\\right)8}{8},\\:\\cos^{4}\\left(\\frac{x}{3}\\right)=\\frac{\\cos^{4}\\left(\\frac{x}{3}\\right)8}{8}$$", "result": "=\\frac{\\sin^{4}\\left(\\frac{x}{3}\\right)\\cdot\\:8}{8}+\\frac{\\cos^{4}\\left(\\frac{x}{3}\\right)\\cdot\\:8}{8}-\\frac{5}{8}" }, { "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{\\sin^{4}\\left(\\frac{x}{3}\\right)\\cdot\\:8+\\cos^{4}\\left(\\frac{x}{3}\\right)\\cdot\\:8-5}{8}" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "result": "\\frac{8\\sin^{4}\\left(\\frac{x}{3}\\right)+8\\cos^{4}\\left(\\frac{x}{3}\\right)-5}{8}=0" }, { "type": "step", "primary": "$$\\frac{f\\left(x\\right)}{g\\left(x\\right)}=0{\\quad\\Rightarrow\\quad}f\\left(x\\right)=0$$", "result": "8\\sin^{4}\\left(\\frac{x}{3}\\right)+8\\cos^{4}\\left(\\frac{x}{3}\\right)-5=0" }, { "type": "step", "primary": "Let: $$u=\\frac{x}{3}$$", "result": "8\\sin^{4}\\left(u\\right)+8\\cos^{4}\\left(u\\right)-5=0" }, { "type": "step", "primary": "Apply exponent rule: $$a^{b}=a^{2}a^{b-2}$$", "result": "-5+8\\cos^{4}\\left(u\\right)+8\\sin^{2}\\left(u\\right)\\sin^{2}\\left(u\\right)=0" }, { "type": "interim", "title": "Rewrite using trig identities", "input": "-5+8\\cos^{4}\\left(u\\right)+8\\sin^{2}\\left(u\\right)\\sin^{2}\\left(u\\right)", "result": "3-16\\cos^{2}\\left(u\\right)+16\\cos^{4}\\left(u\\right)=0", "steps": [ { "type": "step", "primary": "Use the Pythagorean identity: $$\\cos^{2}\\left(x\\right)+\\sin^{2}\\left(x\\right)=1$$", "secondary": [ "$$\\sin^{2}\\left(x\\right)=1-\\cos^{2}\\left(x\\right)$$" ], "result": "=-5+8\\cos^{4}\\left(u\\right)+8\\left(1-\\cos^{2}\\left(u\\right)\\right)\\left(1-\\cos^{2}\\left(u\\right)\\right)" }, { "type": "interim", "title": "Simplify $$-5+8\\cos^{4}\\left(u\\right)+8\\left(1-\\cos^{2}\\left(u\\right)\\right)\\left(1-\\cos^{2}\\left(u\\right)\\right):{\\quad}16\\cos^{4}\\left(u\\right)-16\\cos^{2}\\left(u\\right)+3$$", "input": "-5+8\\cos^{4}\\left(u\\right)+8\\left(1-\\cos^{2}\\left(u\\right)\\right)\\left(1-\\cos^{2}\\left(u\\right)\\right)", "result": "=16\\cos^{4}\\left(u\\right)-16\\cos^{2}\\left(u\\right)+3", "steps": [ { "type": "interim", "title": "$$8\\left(1-\\cos^{2}\\left(u\\right)\\right)\\left(1-\\cos^{2}\\left(u\\right)\\right)=8\\left(1-\\cos^{2}\\left(u\\right)\\right)^{2}$$", "input": "8\\left(1-\\cos^{2}\\left(u\\right)\\right)\\left(1-\\cos^{2}\\left(u\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\left(1-\\cos^{2}\\left(u\\right)\\right)\\left(1-\\cos^{2}\\left(u\\right)\\right)=\\:\\left(1-\\cos^{2}\\left(u\\right)\\right)^{1+1}$$" ], "result": "=8\\left(1-\\cos^{2}\\left(u\\right)\\right)^{1+1}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$1+1=2$$", "result": "=8\\left(1-\\cos^{2}\\left(u\\right)\\right)^{2}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79AfdudIjT5esvPtfgV2y+Hh3w23R7oi2pV3C8rmXtN6osElCGYFbNWSzHmPUiwP6o5FYteSPKwXny4uCMrdsK7HT+IvuRVs04zz/YfER+8lASszb04cGcSIApdmJd37zmB73WzQdbUJB9Pa3rLEVuP2I4hL4xps40F7YYJ3kWos76pw1UqsPnWdhHm/IQ1uIH52f7jzVCtWVbn5BHkU8Vg==" } }, { "type": "step", "result": "=-5+8\\cos^{4}\\left(u\\right)+8\\left(-\\cos^{2}\\left(u\\right)+1\\right)^{2}" }, { "type": "interim", "title": "$$\\left(1-\\cos^{2}\\left(u\\right)\\right)^{2}:{\\quad}1-2\\cos^{2}\\left(u\\right)+\\cos^{4}\\left(u\\right)$$", "result": "=-5+8\\cos^{4}\\left(u\\right)+8\\left(1-2\\cos^{2}\\left(u\\right)+\\cos^{4}\\left(u\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply Perfect Square Formula: $$\\left(a-b\\right)^{2}=a^{2}-2ab+b^{2}$$", "secondary": [ "$$a=1,\\:\\:b=\\cos^{2}\\left(u\\right)$$" ], "meta": { "practiceLink": "/practice/expansion-practice#area=main&subtopic=Perfect%20Square", "practiceTopic": "Expand Perfect Square" } }, { "type": "step", "result": "=1^{2}-2\\cdot\\:1\\cdot\\:\\cos^{2}\\left(u\\right)+\\left(\\cos^{2}\\left(u\\right)\\right)^{2}" }, { "type": "interim", "title": "Simplify $$1^{2}-2\\cdot\\:1\\cdot\\:\\cos^{2}\\left(u\\right)+\\left(\\cos^{2}\\left(u\\right)\\right)^{2}:{\\quad}1-2\\cos^{2}\\left(u\\right)+\\cos^{4}\\left(u\\right)$$", "input": "1^{2}-2\\cdot\\:1\\cdot\\:\\cos^{2}\\left(u\\right)+\\left(\\cos^{2}\\left(u\\right)\\right)^{2}", "result": "=1-2\\cos^{2}\\left(u\\right)+\\cos^{4}\\left(u\\right)", "steps": [ { "type": "step", "primary": "Apply rule $$1^{a}=1$$", "secondary": [ "$$1^{2}=1$$" ], "result": "=1-2\\cdot\\:1\\cdot\\:\\cos^{2}\\left(u\\right)+\\left(\\cos^{2}\\left(u\\right)\\right)^{2}" }, { "type": "interim", "title": "$$2\\cdot\\:1\\cdot\\:\\cos^{2}\\left(u\\right)=2\\cos^{2}\\left(u\\right)$$", "input": "2\\cdot\\:1\\cdot\\:\\cos^{2}\\left(u\\right)", "steps": [ { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:1=2$$", "result": "=2\\cos^{2}\\left(u\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7ht1jgkupea8+jUgT+M/VFN3/KG+M6/wbHoFeDHX+ouBV00rpv8+ZC6TM10tVCSHse4qBpIkVy2vvJRCXZt6ryKm0xW8crvy5hQGW+wciGGkx829OfivBHYoiC5AiKMnbAbNA/JCldtsE68mebWGmVc8BW0YgRvvjOf4/MRh4HIyJzJKURtMpp3soxT1N9Bkx" } }, { "type": "interim", "title": "$$\\left(\\cos^{2}\\left(u\\right)\\right)^{2}=\\cos^{4}\\left(u\\right)$$", "input": "\\left(\\cos^{2}\\left(u\\right)\\right)^{2}", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$\\left(a^{b}\\right)^{c}=a^{bc}$$", "result": "=\\cos^{2\\cdot\\:2}\\left(u\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:2=4$$", "result": "=\\cos^{4}\\left(u\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7nqXADW5KIJmIiHZ3YOdDL45IpdliG1E4K4EtDGLN9yvMwViaLUXkeD+JukROhWdj31eYdtJDVCkq8Mn57GAm/laiLgjmyMQYlA0xnylLMScT9hGH/Xgx1ix8W91wX0tcwe3Ucv0agBQ3hMA6IU+Dng==" } }, { "type": "step", "result": "=1-2\\cos^{2}\\left(u\\right)+\\cos^{4}\\left(u\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } } ], "meta": { "interimType": "N/A" } }, { "type": "interim", "title": "Expand $$8\\left(1-2\\cos^{2}\\left(u\\right)+\\cos^{4}\\left(u\\right)\\right):{\\quad}8-16\\cos^{2}\\left(u\\right)+8\\cos^{4}\\left(u\\right)$$", "input": "8\\left(1-2\\cos^{2}\\left(u\\right)+\\cos^{4}\\left(u\\right)\\right)", "result": "=-5+8\\cos^{4}\\left(u\\right)+8-16\\cos^{2}\\left(u\\right)+8\\cos^{4}\\left(u\\right)", "steps": [ { "type": "step", "primary": "Distribute parentheses", "result": "=8\\cdot\\:1+8\\left(-2\\cos^{2}\\left(u\\right)\\right)+8\\cos^{4}\\left(u\\right)", "meta": { "title": { "extension": "Multiply each of the terms within the parentheses<br/>by the term outside the parenthesis" } } }, { "type": "step", "primary": "Apply minus-plus rules", "secondary": [ "$$+\\left(-a\\right)=-a$$" ], "result": "=8\\cdot\\:1-8\\cdot\\:2\\cos^{2}\\left(u\\right)+8\\cos^{4}\\left(u\\right)" }, { "type": "interim", "title": "Simplify $$8\\cdot\\:1-8\\cdot\\:2\\cos^{2}\\left(u\\right)+8\\cos^{4}\\left(u\\right):{\\quad}8-16\\cos^{2}\\left(u\\right)+8\\cos^{4}\\left(u\\right)$$", "input": "8\\cdot\\:1-8\\cdot\\:2\\cos^{2}\\left(u\\right)+8\\cos^{4}\\left(u\\right)", "result": "=8-16\\cos^{2}\\left(u\\right)+8\\cos^{4}\\left(u\\right)", "steps": [ { "type": "step", "primary": "Multiply the numbers: $$8\\cdot\\:1=8$$", "result": "=8-8\\cdot\\:2\\cos^{2}\\left(u\\right)+8\\cos^{4}\\left(u\\right)" }, { "type": "step", "primary": "Multiply the numbers: $$8\\cdot\\:2=16$$", "result": "=8-16\\cos^{2}\\left(u\\right)+8\\cos^{4}\\left(u\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } } ], "meta": { "interimType": "Algebraic Manipulation Expand Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7VpGdtQ86NZycmNI2FNPIbKYzyhP116aQTv/I5bTujvDTLx8mOdHYVzxX643JqKFI9g3IheHaVwlWdCS9svDzVcQj4nMd6LThrH8bTiY/Qu4jJq4m7LBg79TLa+TQN+kw72wZm7kDUxdE6YSmfEbr2hTykqzPie/W6NNtdnKbxS6PUURtyHL06FmEdxfe4Z/g8Hp9RppwUFJ7rH2e8o5Vvw==" } }, { "type": "interim", "title": "Simplify $$-5+8\\cos^{4}\\left(u\\right)+8-16\\cos^{2}\\left(u\\right)+8\\cos^{4}\\left(u\\right):{\\quad}16\\cos^{4}\\left(u\\right)-16\\cos^{2}\\left(u\\right)+3$$", "input": "-5+8\\cos^{4}\\left(u\\right)+8-16\\cos^{2}\\left(u\\right)+8\\cos^{4}\\left(u\\right)", "result": "=16\\cos^{4}\\left(u\\right)-16\\cos^{2}\\left(u\\right)+3", "steps": [ { "type": "step", "primary": "Group like terms", "result": "=8\\cos^{4}\\left(u\\right)-16\\cos^{2}\\left(u\\right)+8\\cos^{4}\\left(u\\right)-5+8" }, { "type": "step", "primary": "Add similar elements: $$8\\cos^{4}\\left(u\\right)+8\\cos^{4}\\left(u\\right)=16\\cos^{4}\\left(u\\right)$$", "result": "=16\\cos^{4}\\left(u\\right)-16\\cos^{2}\\left(u\\right)-5+8" }, { "type": "step", "primary": "Add/Subtract the numbers: $$-5+8=3$$", "result": "=16\\cos^{4}\\left(u\\right)-16\\cos^{2}\\left(u\\right)+3" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7gUthJ9Ou3p9VNC7fihQ4J2cO+w0F+wpE+HTRMLeEtTEESibWJvdCgB1jrQGB4iZ6CuTAP0yD265WWliM6u+TLszBWJotReR4P4m6RE6FZ2PpgQEluUF7S37SZScZ6bB26k+yhAQv23QRJ+Eesj1vkz/L0MoYg+CUn6oyL3EO7YppEjsYKnQdDP7MPDbdrF10ig5NT30brI/sekHdWdFPCJCiAZxVrCyfti2Ia1vCsFeKVsPaa7eNyiDih0rBV5dpazF+HTWE6Qy1X1+vRT9FPA==" } } ], "meta": { "interimType": "Trig Rewrite Using Trig identities 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7VdPaQqyAwhL63EwzmjjGAzNwFwmAzdwrNYz2dYSe5oC7u64Og8Blu+DeC3U6+Ei+nhVLAo+lsQgs9qzV9lE5YxLC2T8NFZvTA+pff5AdXPdX9lRN4mlYq/zI0dgxEXGdPvdnhi0s3lZ+DJ7rEiTeqXQcxyKHWZtk39emlmjA3EJixqfzu6llwwshsZczT5J/4Mrpu3yQ5JdrJjrpOVHXfXZZUbV3G6G3By9zn4CzZOgo753z1XSSxSwjjF/tZKjjAlhg/CJ/s6fmZmxmI0gevA==" } }, { "type": "interim", "title": "Solve by substitution", "input": "3-16\\cos^{2}\\left(u\\right)+16\\cos^{4}\\left(u\\right)=0", "result": "\\cos\\left(u\\right)=\\frac{\\sqrt{3}}{2},\\:\\cos\\left(u\\right)=-\\frac{\\sqrt{3}}{2},\\:\\cos\\left(u\\right)=\\frac{1}{2},\\:\\cos\\left(u\\right)=-\\frac{1}{2}", "steps": [ { "type": "step", "primary": "Let: $$\\cos\\left(u\\right)=u$$", "result": "3-16u^{2}+16u^{4}=0" }, { "type": "interim", "title": "$$3-16u^{2}+16u^{4}=0{\\quad:\\quad}u=\\frac{\\sqrt{3}}{2},\\:u=-\\frac{\\sqrt{3}}{2},\\:u=\\frac{1}{2},\\:u=-\\frac{1}{2}$$", "input": "3-16u^{2}+16u^{4}=0", "steps": [ { "type": "step", "primary": "Write in the standard form $$a_{n}x^{n}+\\ldots\\:+a_{1}x+a_{0}=0$$", "result": "16u^{4}-16u^{2}+3=0" }, { "type": "step", "primary": "Rewrite the equation with $$v=u^{2}$$ and $$v^{2}=u^{4}$$", "result": "16v^{2}-16v+3=0" }, { "type": "interim", "title": "Solve $$16v^{2}-16v+3=0:{\\quad}v=\\frac{3}{4},\\:v=\\frac{1}{4}$$", "input": "16v^{2}-16v+3=0", "steps": [ { "type": "interim", "title": "Solve with the quadratic formula", "input": "16v^{2}-16v+3=0", "result": "{v}_{1,\\:2}=\\frac{-\\left(-16\\right)\\pm\\:\\sqrt{\\left(-16\\right)^{2}-4\\cdot\\:16\\cdot\\:3}}{2\\cdot\\:16}", "steps": [ { "type": "definition", "title": "Quadratic Equation Formula:", "text": "For a quadratic equation of the form $$ax^2+bx+c=0$$ the solutions are <br/>$${\\quad}x_{1,\\:2}=\\frac{-b\\pm\\sqrt{b^2-4ac}}{2a}$$" }, { "type": "step", "primary": "For $${\\quad}a=16,\\:b=-16,\\:c=3$$", "result": "{v}_{1,\\:2}=\\frac{-\\left(-16\\right)\\pm\\:\\sqrt{\\left(-16\\right)^{2}-4\\cdot\\:16\\cdot\\:3}}{2\\cdot\\:16}" } ], "meta": { "interimType": "Solving The Quadratic Equation With Quadratic Formula Definition 0Eq", "gptData": "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" } }, { "type": "interim", "title": "$$\\sqrt{\\left(-16\\right)^{2}-4\\cdot\\:16\\cdot\\:3}=8$$", "input": "\\sqrt{\\left(-16\\right)^{2}-4\\cdot\\:16\\cdot\\:3}", "result": "{v}_{1,\\:2}=\\frac{-\\left(-16\\right)\\pm\\:8}{2\\cdot\\:16}", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$\\left(-a\\right)^{n}=a^{n},\\:$$if $$n$$ is even", "secondary": [ "$$\\left(-16\\right)^{2}=16^{2}$$" ], "result": "=\\sqrt{16^{2}-4\\cdot\\:16\\cdot\\:3}" }, { "type": "step", "primary": "Multiply the numbers: $$4\\cdot\\:16\\cdot\\:3=192$$", "result": "=\\sqrt{16^{2}-192}" }, { "type": "step", "primary": "$$16^{2}=256$$", "result": "=\\sqrt{256-192}" }, { "type": "step", "primary": "Subtract the numbers: $$256-192=64$$", "result": "=\\sqrt{64}" }, { "type": "step", "primary": "Factor the number: $$64=8^{2}$$", "result": "=\\sqrt{8^{2}}" }, { "type": "step", "primary": "Apply radical rule: $$\\sqrt[n]{a^n}=a$$", "secondary": [ "$$\\sqrt{8^{2}}=8$$" ], "result": "=8", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7QPaWiWJ3D1YL+aGXmG/eyn9suxzzvvi2SHxgVRJFAW5VmTWBcJEr5sDcbW5s4ajsq47vuWedXv2WUg94ER8IwRUY16iVSWKlu9zGIhgaexH/ZWLBEWBvGIYT0ZuYSrykVF0OroE7BwvITa1j8+1V4H4COwEgoEuAcEWuxodiIfk=" } }, { "type": "step", "primary": "Separate the solutions", "result": "{v}_{1}=\\frac{-\\left(-16\\right)+8}{2\\cdot\\:16},\\:{v}_{2}=\\frac{-\\left(-16\\right)-8}{2\\cdot\\:16}" }, { "type": "interim", "title": "$$v=\\frac{-\\left(-16\\right)+8}{2\\cdot\\:16}:{\\quad}\\frac{3}{4}$$", "input": "\\frac{-\\left(-16\\right)+8}{2\\cdot\\:16}", "steps": [ { "type": "step", "primary": "Apply rule $$-\\left(-a\\right)=a$$", "result": "=\\frac{16+8}{2\\cdot\\:16}" }, { "type": "step", "primary": "Add the numbers: $$16+8=24$$", "result": "=\\frac{24}{2\\cdot\\:16}" }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:16=32$$", "result": "=\\frac{24}{32}" }, { "type": "step", "primary": "Cancel the common factor: $$8$$", "result": "=\\frac{3}{4}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7cdnwcN0lJeZeYXRkochh62xBBD/Z/G3Nj/QzNRfkSGdV00rpv8+ZC6TM10tVCSHs0xDS+Y5aj0hl+F6LvDaAlgTInJ3gapzcxrpvVzdwWvTGQ+K7WEyY+OUU4xOl6NVM0IzlgrnzOIzPKX5OkmbCLDPnVIpg9/xke/ZpHn+6qpA=" } }, { "type": "interim", "title": "$$v=\\frac{-\\left(-16\\right)-8}{2\\cdot\\:16}:{\\quad}\\frac{1}{4}$$", "input": "\\frac{-\\left(-16\\right)-8}{2\\cdot\\:16}", "steps": [ { "type": "step", "primary": "Apply rule $$-\\left(-a\\right)=a$$", "result": "=\\frac{16-8}{2\\cdot\\:16}" }, { "type": "step", "primary": "Subtract the numbers: $$16-8=8$$", "result": "=\\frac{8}{2\\cdot\\:16}" }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:16=32$$", "result": "=\\frac{8}{32}" }, { "type": "step", "primary": "Cancel the common factor: $$8$$", "result": "=\\frac{1}{4}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7y+kkPVSNIh53vAZywR92UmxBBD/Z/G3Nj/QzNRfkSGdV00rpv8+ZC6TM10tVCSHs0xDS+Y5aj0hl+F6LvDaAlg/mU/6hTfHVOodxq1kvbMzGQ+K7WEyY+OUU4xOl6NVM2Jj3MngEkg1Umv9SAwNwpDPnVIpg9/xke/ZpHn+6qpA=" } }, { "type": "step", "primary": "The solutions to the quadratic equation are:", "result": "v=\\frac{3}{4},\\:v=\\frac{1}{4}" } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve Title 1Eq" } }, { "type": "step", "result": "v=\\frac{3}{4},\\:v=\\frac{1}{4}" }, { "type": "step", "primary": "Substitute back $$v=u^{2},\\:$$solve for $$u$$" }, { "type": "interim", "title": "Solve $$u^{2}=\\frac{3}{4}:{\\quad}u=\\frac{\\sqrt{3}}{2},\\:u=-\\frac{\\sqrt{3}}{2}$$", "input": "u^{2}=\\frac{3}{4}", "steps": [ { "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": "Generic Solve Title 1Eq" } }, { "type": "interim", "title": "Solve $$u^{2}=\\frac{1}{4}:{\\quad}u=\\frac{1}{2},\\:u=-\\frac{1}{2}$$", "input": "u^{2}=\\frac{1}{4}", "steps": [ { "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{1}{4}},\\:u=-\\sqrt{\\frac{1}{4}}" }, { "type": "interim", "title": "$$\\sqrt{\\frac{1}{4}}=\\frac{1}{2}$$", "input": "\\sqrt{\\frac{1}{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{1}}{\\sqrt{4}}", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } }, { "type": "interim", "title": "$$\\sqrt{4}=2$$", "input": "\\sqrt{4}", "result": "=\\frac{\\sqrt{1}}{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" } }, { "type": "step", "primary": "Apply rule $$\\sqrt{1}=1$$", "result": "=\\frac{1}{2}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7FJIkkmi1CWuhEmDQGlA0NzITgDFbnE8wkcXKMHdOwBSrju+5Z51e/ZZSD3gRHwjBZsqxqhl2a6oRKVJk8034tWRLd2VwIqlBNByF6663syTWcLcA3FbS+MZ1fFIklJt5MCuZPgBpwTTzu2tuLa/8abCI2sSeA74029n2yo277ZU=" } }, { "type": "interim", "title": "$$-\\sqrt{\\frac{1}{4}}=-\\frac{1}{2}$$", "input": "-\\sqrt{\\frac{1}{4}}", "steps": [ { "type": "interim", "title": "Simplify $$\\sqrt{\\frac{1}{4}}:{\\quad}\\frac{\\sqrt{1}}{2}$$", "input": "\\sqrt{\\frac{1}{4}}", "result": "=-\\frac{\\sqrt{1}}{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{1}}{\\sqrt{4}}", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } }, { "type": "interim", "title": "$$\\sqrt{4}=2$$", "input": "\\sqrt{4}", "result": "=\\frac{\\sqrt{1}}{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" } }, { "type": "step", "primary": "Apply rule $$\\sqrt{1}=1$$", "result": "=-\\frac{1}{2}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7xVDfakA4g1ZczTHh+ArxmzTrBkvKbZrfJnwMGgx/VgUJQJZuTAY5js+oqjdT8kslx3FTmhk5oBBtojppJ/bq4/8//6/nV5O4fb8Xgwi7mapvsmMaNg8JzlNopDoeZ4sisPHFuuTLWzCcuewLnsue2m6sJzS/r5J7Nekm75J11/g=" } }, { "type": "step", "result": "u=\\frac{1}{2},\\:u=-\\frac{1}{2}" } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve Title 1Eq" } }, { "type": "step", "primary": "The solutions are" }, { "type": "step", "result": "u=\\frac{\\sqrt{3}}{2},\\:u=-\\frac{\\sqrt{3}}{2},\\:u=\\frac{1}{2},\\:u=-\\frac{1}{2}" } ], "meta": { "solvingClass": "Equations", "interimType": "Equations" } }, { "type": "step", "primary": "Substitute back $$u=\\cos\\left(u\\right)$$", "result": "\\cos\\left(u\\right)=\\frac{\\sqrt{3}}{2},\\:\\cos\\left(u\\right)=-\\frac{\\sqrt{3}}{2},\\:\\cos\\left(u\\right)=\\frac{1}{2},\\:\\cos\\left(u\\right)=-\\frac{1}{2}" } ], "meta": { "interimType": "Substitution Method 0Eq" } }, { "type": "interim", "title": "$$\\cos\\left(u\\right)=\\frac{\\sqrt{3}}{2}{\\quad:\\quad}u=\\frac{π}{6}+2πn,\\:u=\\frac{11π}{6}+2πn$$", "input": "\\cos\\left(u\\right)=\\frac{\\sqrt{3}}{2}", "steps": [ { "type": "interim", "title": "General solutions for $$\\cos\\left(u\\right)=\\frac{\\sqrt{3}}{2}$$", "result": "u=\\frac{π}{6}+2πn,\\:u=\\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": "u=\\frac{π}{6}+2πn,\\:u=\\frac{11π}{6}+2πn" } ], "meta": { "interimType": "Trig General Solutions cos 1Eq" } } ], "meta": { "interimType": "N/A" } }, { "type": "interim", "title": "$$\\cos\\left(u\\right)=-\\frac{\\sqrt{3}}{2}{\\quad:\\quad}u=\\frac{5π}{6}+2πn,\\:u=\\frac{7π}{6}+2πn$$", "input": "\\cos\\left(u\\right)=-\\frac{\\sqrt{3}}{2}", "steps": [ { "type": "interim", "title": "General solutions for $$\\cos\\left(u\\right)=-\\frac{\\sqrt{3}}{2}$$", "result": "u=\\frac{5π}{6}+2πn,\\:u=\\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": "u=\\frac{5π}{6}+2πn,\\:u=\\frac{7π}{6}+2πn" } ], "meta": { "interimType": "Trig General Solutions cos 1Eq" } } ], "meta": { "interimType": "N/A" } }, { "type": "interim", "title": "$$\\cos\\left(u\\right)=\\frac{1}{2}{\\quad:\\quad}u=\\frac{π}{3}+2πn,\\:u=\\frac{5π}{3}+2πn$$", "input": "\\cos\\left(u\\right)=\\frac{1}{2}", "steps": [ { "type": "interim", "title": "General solutions for $$\\cos\\left(u\\right)=\\frac{1}{2}$$", "result": "u=\\frac{π}{3}+2πn,\\:u=\\frac{5π}{3}+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": "u=\\frac{π}{3}+2πn,\\:u=\\frac{5π}{3}+2πn" } ], "meta": { "interimType": "Trig General Solutions cos 1Eq" } } ], "meta": { "interimType": "N/A" } }, { "type": "interim", "title": "$$\\cos\\left(u\\right)=-\\frac{1}{2}{\\quad:\\quad}u=\\frac{2π}{3}+2πn,\\:u=\\frac{4π}{3}+2πn$$", "input": "\\cos\\left(u\\right)=-\\frac{1}{2}", "steps": [ { "type": "interim", "title": "General solutions for $$\\cos\\left(u\\right)=-\\frac{1}{2}$$", "result": "u=\\frac{2π}{3}+2πn,\\:u=\\frac{4π}{3}+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": "u=\\frac{2π}{3}+2πn,\\:u=\\frac{4π}{3}+2πn" } ], "meta": { "interimType": "Trig General Solutions cos 1Eq" } } ], "meta": { "interimType": "N/A" } }, { "type": "step", "primary": "Combine all the solutions", "result": "u=\\frac{π}{6}+2πn,\\:u=\\frac{11π}{6}+2πn,\\:u=\\frac{5π}{6}+2πn,\\:u=\\frac{7π}{6}+2πn,\\:u=\\frac{π}{3}+2πn,\\:u=\\frac{5π}{3}+2πn,\\:u=\\frac{2π}{3}+2πn,\\:u=\\frac{4π}{3}+2πn" }, { "type": "step", "primary": "Substitute back $$u=\\frac{x}{3}$$" }, { "type": "interim", "title": "$$\\frac{x}{3}=\\frac{π}{6}+2πn{\\quad:\\quad}x=\\frac{π}{2}+6πn$$", "input": "\\frac{x}{3}=\\frac{π}{6}+2πn", "steps": [ { "type": "interim", "title": "Multiply both sides by $$3$$", "input": "\\frac{x}{3}=\\frac{π}{6}+2πn", "result": "x=\\frac{π}{2}+6πn", "steps": [ { "type": "step", "primary": "Multiply both sides by $$3$$", "result": "\\frac{3x}{3}=3\\cdot\\:\\frac{π}{6}+3\\cdot\\:2πn" }, { "type": "interim", "title": "Simplify", "input": "\\frac{3x}{3}=3\\cdot\\:\\frac{π}{6}+3\\cdot\\:2πn", "result": "x=\\frac{π}{2}+6πn", "steps": [ { "type": "interim", "title": "Simplify $$\\frac{3x}{3}:{\\quad}x$$", "input": "\\frac{3x}{3}", "steps": [ { "type": "step", "primary": "Divide the numbers: $$\\frac{3}{3}=1$$", "result": "=x" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7QJV6FEydUSv+hKOzRRB7mi061ljBSPJeENOw2efoSWt2cZ6LAXMcKee6PSyjLEFrRSpN33oxZMojoqvYhvSJACLkpiVA7S8UT8ieezZKu6ot8gw9eOFDGZzXCUlLc4C0" } }, { "type": "interim", "title": "Simplify $$3\\cdot\\:\\frac{π}{6}+3\\cdot\\:2πn:{\\quad}\\frac{π}{2}+6πn$$", "input": "3\\cdot\\:\\frac{π}{6}+3\\cdot\\:2πn", "steps": [ { "type": "interim", "title": "$$3\\cdot\\:\\frac{π}{6}=\\frac{π}{2}$$", "input": "3\\cdot\\:\\frac{π}{6}", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{π3}{6}" }, { "type": "step", "primary": "Cancel the common factor: $$3$$", "result": "=\\frac{π}{2}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7CzXS+MvaOUdKL6IkXGMFAzqt2Xgtb3t95fujkffGgBwJQJZuTAY5js+oqjdT8kslZguKS1b2cgc0s2D0HaMML/8//6/nV5O4fb8Xgwi7mapx4nUY00HlVZKlUcHy0lFDdDC0pmI5+Hy3BfDLTjeEfm6sJzS/r5J7Nekm75J11/g=" } }, { "type": "interim", "title": "$$3\\cdot\\:2πn=6πn$$", "input": "3\\cdot\\:2πn", "steps": [ { "type": "step", "primary": "Multiply the numbers: $$3\\cdot\\:2=6$$", "result": "=6πn" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7YTBxxpumBqXGn+KTPhB1CFXTSum/z5kLpMzXS1UJIezrYzddS4OvRs60Tm4Kkwg7ZgbKfKW0HJe3lrhv5BsQrVFTnRcAzswxSGd6nlYqYFWwiNrEngO+NNvZ9sqNu+2V" } }, { "type": "step", "result": "=\\frac{π}{2}+6πn" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7CzXS+MvaOUdKL6IkXGMFA9IBrnoSeINRlxb6dPGdZ+z+n/+FjN833TADO2gxs6I7zMFYmi1F5Hg/ibpEToVnYxcYZr+Ycj3h0ad4x1SPItBPl2Rdy6SZiOVUpWNZcMJC7kAjP76qW66lOUsURwT0nXXjp2Lh8oOsWluY6eHxIOTvK8pMCFEIkuc0LdnGufHAe/EW6scPGHMpRVZCtwbXWQ==" } }, { "type": "step", "result": "x=\\frac{π}{2}+6πn" } ], "meta": { "interimType": "Generic Simplify 0Eq" } } ], "meta": { "interimType": "Multiply Both Sides Specific 1Eq", "gptData": "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" } } ], "meta": { "solvingClass": "Equations", "interimType": "Equations" } }, { "type": "interim", "title": "$$\\frac{x}{3}=\\frac{11π}{6}+2πn{\\quad:\\quad}x=\\frac{11π}{2}+6πn$$", "input": "\\frac{x}{3}=\\frac{11π}{6}+2πn", "steps": [ { "type": "interim", "title": "Multiply both sides by $$3$$", "input": "\\frac{x}{3}=\\frac{11π}{6}+2πn", "result": "x=\\frac{11π}{2}+6πn", "steps": [ { "type": "step", "primary": "Multiply both sides by $$3$$", "result": "\\frac{3x}{3}=3\\cdot\\:\\frac{11π}{6}+3\\cdot\\:2πn" }, { "type": "interim", "title": "Simplify", "input": "\\frac{3x}{3}=3\\cdot\\:\\frac{11π}{6}+3\\cdot\\:2πn", "result": "x=\\frac{11π}{2}+6πn", "steps": [ { "type": "interim", "title": "Simplify $$\\frac{3x}{3}:{\\quad}x$$", "input": "\\frac{3x}{3}", "steps": [ { "type": "step", "primary": "Divide the numbers: $$\\frac{3}{3}=1$$", "result": "=x" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7QJV6FEydUSv+hKOzRRB7mi061ljBSPJeENOw2efoSWt2cZ6LAXMcKee6PSyjLEFrRSpN33oxZMojoqvYhvSJACLkpiVA7S8UT8ieezZKu6ot8gw9eOFDGZzXCUlLc4C0" } }, { "type": "interim", "title": "Simplify $$3\\cdot\\:\\frac{11π}{6}+3\\cdot\\:2πn:{\\quad}\\frac{11π}{2}+6πn$$", "input": "3\\cdot\\:\\frac{11π}{6}+3\\cdot\\:2πn", "steps": [ { "type": "interim", "title": "$$3\\cdot\\:\\frac{11π}{6}=\\frac{11π}{2}$$", "input": "3\\cdot\\:\\frac{11π}{6}", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{11π3}{6}" }, { "type": "step", "primary": "Multiply the numbers: $$11\\cdot\\:3=33$$", "result": "=\\frac{33π}{6}" }, { "type": "step", "primary": "Cancel the common factor: $$3$$", "result": "=\\frac{11π}{2}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7CzXS+MvaOUdKL6IkXGMFAxQwGQN9Htd384OoRc4hDZwDnzlbPZjyKgy1eUCFsLd5J3BbtrlvqOdaP0l0jFf1BLKAEC1immJ8buNmGLd/2Z8NvPMHS7d/fKJBclBgIeGRYTWTCSeAO/WMhzBLTcxiokZ7p4MFEUec5FXn0sm0hGQ=" } }, { "type": "interim", "title": "$$3\\cdot\\:2πn=6πn$$", "input": "3\\cdot\\:2πn", "steps": [ { "type": "step", "primary": "Multiply the numbers: $$3\\cdot\\:2=6$$", "result": "=6πn" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7YTBxxpumBqXGn+KTPhB1CFXTSum/z5kLpMzXS1UJIezrYzddS4OvRs60Tm4Kkwg7ZgbKfKW0HJe3lrhv5BsQrVFTnRcAzswxSGd6nlYqYFWwiNrEngO+NNvZ9sqNu+2V" } }, { "type": "step", "result": "=\\frac{11π}{2}+6πn" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7CzXS+MvaOUdKL6IkXGMFA5K0RYC8HN+MjBY0M2IILug8h6BvscJ5OYJ6eAmC7ry3q47vuWedXv2WUg94ER8Iwd9JyD4eZItU4FgwUlYSUHhKDtVFPLexaWSpqL0KmGV6TeQKHeh69S6dnv9vSoUoFJJqm4+lq4siYCCVL/6f3iqF6MXJaydvOFBskfCyhGTWrubwNW1w+462RmftbHrMzyS3daIZHtloJpe/PvtsyNI=" } }, { "type": "step", "result": "x=\\frac{11π}{2}+6πn" } ], "meta": { "interimType": "Generic Simplify 0Eq" } } ], "meta": { "interimType": "Multiply Both Sides Specific 1Eq", "gptData": "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" } } ], "meta": { "solvingClass": "Equations", "interimType": "Equations" } }, { "type": "interim", "title": "$$\\frac{x}{3}=\\frac{5π}{6}+2πn{\\quad:\\quad}x=\\frac{5π}{2}+6πn$$", "input": "\\frac{x}{3}=\\frac{5π}{6}+2πn", "steps": [ { "type": "interim", "title": "Multiply both sides by $$3$$", "input": "\\frac{x}{3}=\\frac{5π}{6}+2πn", "result": "x=\\frac{5π}{2}+6πn", "steps": [ { "type": "step", "primary": "Multiply both sides by $$3$$", "result": "\\frac{3x}{3}=3\\cdot\\:\\frac{5π}{6}+3\\cdot\\:2πn" }, { "type": "interim", "title": "Simplify", "input": "\\frac{3x}{3}=3\\cdot\\:\\frac{5π}{6}+3\\cdot\\:2πn", "result": "x=\\frac{5π}{2}+6πn", "steps": [ { "type": "interim", "title": "Simplify $$\\frac{3x}{3}:{\\quad}x$$", "input": "\\frac{3x}{3}", "steps": [ { "type": "step", "primary": "Divide the numbers: $$\\frac{3}{3}=1$$", "result": "=x" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7QJV6FEydUSv+hKOzRRB7mi061ljBSPJeENOw2efoSWt2cZ6LAXMcKee6PSyjLEFrRSpN33oxZMojoqvYhvSJACLkpiVA7S8UT8ieezZKu6ot8gw9eOFDGZzXCUlLc4C0" } }, { "type": "interim", "title": "Simplify $$3\\cdot\\:\\frac{5π}{6}+3\\cdot\\:2πn:{\\quad}\\frac{5π}{2}+6πn$$", "input": "3\\cdot\\:\\frac{5π}{6}+3\\cdot\\:2πn", "steps": [ { "type": "interim", "title": "$$3\\cdot\\:\\frac{5π}{6}=\\frac{5π}{2}$$", "input": "3\\cdot\\:\\frac{5π}{6}", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{5π3}{6}" }, { "type": "step", "primary": "Multiply the numbers: $$5\\cdot\\:3=15$$", "result": "=\\frac{15π}{6}" }, { "type": "step", "primary": "Cancel the common factor: $$3$$", "result": "=\\frac{5π}{2}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7CzXS+MvaOUdKL6IkXGMFAxfSMm0qrPdKn557Lx5c4c91g99dC9fj9sg0EHzBIRDRbqh7hWjEsCVTbfRLMpHqwV5NkzKQgtswLlLi9MgL+gqUPFKzM1UkgnK7DdbUAz0xrcf38jhlNMMYM1syFPQF+zDdC41sVnMXWat8DX7/QAg=" } }, { "type": "interim", "title": "$$3\\cdot\\:2πn=6πn$$", "input": "3\\cdot\\:2πn", "steps": [ { "type": "step", "primary": "Multiply the numbers: $$3\\cdot\\:2=6$$", "result": "=6πn" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7YTBxxpumBqXGn+KTPhB1CFXTSum/z5kLpMzXS1UJIezrYzddS4OvRs60Tm4Kkwg7ZgbKfKW0HJe3lrhv5BsQrVFTnRcAzswxSGd6nlYqYFWwiNrEngO+NNvZ9sqNu+2V" } }, { "type": "step", "result": "=\\frac{5π}{2}+6πn" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7CzXS+MvaOUdKL6IkXGMFAzU9vLsLKjB+ZQhW7j7CdSbj+O38RAvKhZolrKec34mYcJChiVhDxT5N/LHSTLMjyHIgfVwAC3eagbboV8UFVw+RK51DV3cZmza+DJfBH4KzixpjZyz1hn/Tx95aZo6CSANO82XUOV2Bv9mTFSai6k7XqtWP30oYKqljH6B6Dhl3adl0eCLqgPgD3O2XqRYdVQ==" } }, { "type": "step", "result": "x=\\frac{5π}{2}+6πn" } ], "meta": { "interimType": "Generic Simplify 0Eq" } } ], "meta": { "interimType": "Multiply Both Sides Specific 1Eq", "gptData": 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"$$\\frac{x}{3}=\\frac{7π}{6}+2πn{\\quad:\\quad}x=\\frac{7π}{2}+6πn$$", "input": "\\frac{x}{3}=\\frac{7π}{6}+2πn", "steps": [ { "type": "interim", "title": "Multiply both sides by $$3$$", "input": "\\frac{x}{3}=\\frac{7π}{6}+2πn", "result": "x=\\frac{7π}{2}+6πn", "steps": [ { "type": "step", "primary": "Multiply both sides by $$3$$", "result": "\\frac{3x}{3}=3\\cdot\\:\\frac{7π}{6}+3\\cdot\\:2πn" }, { "type": "interim", "title": "Simplify", "input": "\\frac{3x}{3}=3\\cdot\\:\\frac{7π}{6}+3\\cdot\\:2πn", "result": "x=\\frac{7π}{2}+6πn", "steps": [ { "type": "interim", "title": "Simplify $$\\frac{3x}{3}:{\\quad}x$$", "input": "\\frac{3x}{3}", "steps": [ { "type": "step", "primary": "Divide the numbers: $$\\frac{3}{3}=1$$", "result": "=x" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7QJV6FEydUSv+hKOzRRB7mi061ljBSPJeENOw2efoSWt2cZ6LAXMcKee6PSyjLEFrRSpN33oxZMojoqvYhvSJACLkpiVA7S8UT8ieezZKu6ot8gw9eOFDGZzXCUlLc4C0" } }, { "type": "interim", "title": "Simplify $$3\\cdot\\:\\frac{7π}{6}+3\\cdot\\:2πn:{\\quad}\\frac{7π}{2}+6πn$$", "input": "3\\cdot\\:\\frac{7π}{6}+3\\cdot\\:2πn", "steps": [ { "type": "interim", "title": "$$3\\cdot\\:\\frac{7π}{6}=\\frac{7π}{2}$$", "input": "3\\cdot\\:\\frac{7π}{6}", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{7π3}{6}" }, { "type": "step", "primary": "Multiply the numbers: $$7\\cdot\\:3=21$$", "result": "=\\frac{21π}{6}" }, { "type": "step", "primary": "Cancel the common factor: $$3$$", "result": "=\\frac{7π}{2}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7CzXS+MvaOUdKL6IkXGMFA84LvF+ElSmlOvihyWSUghJ1g99dC9fj9sg0EHzBIRDRRmGZVuFwzliJ8lF0Om5Uxl5NkzKQgtswLlLi9MgL+gqUPFKzM1UkgnK7DdbUAz0xridiKlYBE7eJjFmklajlPOo9IsddNX52YsDhONPFgcc=" } }, { "type": "interim", "title": "$$3\\cdot\\:2πn=6πn$$", "input": "3\\cdot\\:2πn", "steps": [ { "type": "step", "primary": "Multiply the numbers: $$3\\cdot\\:2=6$$", "result": "=6πn" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7YTBxxpumBqXGn+KTPhB1CFXTSum/z5kLpMzXS1UJIezrYzddS4OvRs60Tm4Kkwg7ZgbKfKW0HJe3lrhv5BsQrVFTnRcAzswxSGd6nlYqYFWwiNrEngO+NNvZ9sqNu+2V" } }, { "type": "step", "result": "=\\frac{7π}{2}+6πn" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7CzXS+MvaOUdKL6IkXGMFA3zbqbg9F0WLdO6FblU0nPLj+O38RAvKhZolrKec34mYcJChiVhDxT5N/LHSTLMjyC+SNJET1pcqUHOv6+wWtMeRK51DV3cZmza+DJfBH4KzixpjZyz1hn/Tx95aZo6CSANO82XUOV2Bv9mTFSai6k4JZ3o3y101s3W2FkMeD64tadl0eCLqgPgD3O2XqRYdVQ==" } }, { "type": "step", "result": "x=\\frac{7π}{2}+6πn" } ], "meta": { "interimType": "Generic Simplify 0Eq" } } ], "meta": { "interimType": "Multiply Both Sides Specific 1Eq", "gptData": 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"$$\\frac{x}{3}=\\frac{π}{3}+2πn{\\quad:\\quad}x=π+6πn$$", "input": "\\frac{x}{3}=\\frac{π}{3}+2πn", "steps": [ { "type": "interim", "title": "Multiply both sides by $$3$$", "input": "\\frac{x}{3}=\\frac{π}{3}+2πn", "result": "x=π+6πn", "steps": [ { "type": "step", "primary": "Multiply both sides by $$3$$", "result": "\\frac{3x}{3}=3\\cdot\\:\\frac{π}{3}+3\\cdot\\:2πn" }, { "type": "interim", "title": "Simplify", "input": "\\frac{3x}{3}=3\\cdot\\:\\frac{π}{3}+3\\cdot\\:2πn", "result": "x=π+6πn", "steps": [ { "type": "interim", "title": "Simplify $$\\frac{3x}{3}:{\\quad}x$$", "input": "\\frac{3x}{3}", "steps": [ { "type": "step", "primary": "Divide the numbers: $$\\frac{3}{3}=1$$", "result": "=x" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7QJV6FEydUSv+hKOzRRB7mi061ljBSPJeENOw2efoSWt2cZ6LAXMcKee6PSyjLEFrRSpN33oxZMojoqvYhvSJACLkpiVA7S8UT8ieezZKu6ot8gw9eOFDGZzXCUlLc4C0" } }, { "type": "interim", "title": "Simplify $$3\\cdot\\:\\frac{π}{3}+3\\cdot\\:2πn:{\\quad}π+6πn$$", "input": "3\\cdot\\:\\frac{π}{3}+3\\cdot\\:2πn", "steps": [ { "type": "interim", "title": "$$3\\cdot\\:\\frac{π}{3}=π$$", "input": "3\\cdot\\:\\frac{π}{3}", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{π3}{3}" }, { "type": "step", "primary": "Cancel the common factor: $$3$$", "result": "=π" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7CzXS+MvaOUdKL6IkXGMFA67iz1K2O/khgzmu2OrKOAgJQJZuTAY5js+oqjdT8ksl58bIKwWzS3dUPXOsTsmj0n0gv9/cWlM6Ubp5FWA60c6Wc1aRVrE8QHqEV9z0EiQVJLd1ohke2Wgml78++2zI0g==" } }, { "type": "interim", "title": "$$3\\cdot\\:2πn=6πn$$", "input": "3\\cdot\\:2πn", "steps": [ { "type": "step", "primary": "Multiply the numbers: $$3\\cdot\\:2=6$$", "result": "=6πn" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7YTBxxpumBqXGn+KTPhB1CFXTSum/z5kLpMzXS1UJIezrYzddS4OvRs60Tm4Kkwg7ZgbKfKW0HJe3lrhv5BsQrVFTnRcAzswxSGd6nlYqYFWwiNrEngO+NNvZ9sqNu+2V" } }, { "type": "step", "result": "=π+6πn" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7CzXS+MvaOUdKL6IkXGMFA4f6a2LeYmAaaMVUYjPLTV3+n/+FjN833TADO2gxs6I7zMFYmi1F5Hg/ibpEToVnY4BUBoYnRuQbG0QzXNdtFOd6pfF1z6umzUJTJvt+ojYZ5/OrvdT5hlCCisyv0N3qznCYw/iRhT2WNNCb+bxSQ8baDWWtZztC6VJ8jhnv43h8" } }, { "type": "step", "result": "x=π+6πn" } ], "meta": { "interimType": "Generic Simplify 0Eq" } } ], "meta": { "interimType": "Multiply Both Sides Specific 1Eq", "gptData": 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"\\frac{x}{3}=\\frac{5π}{3}+2πn", "steps": [ { "type": "interim", "title": "Multiply both sides by $$3$$", "input": "\\frac{x}{3}=\\frac{5π}{3}+2πn", "result": "x=5π+6πn", "steps": [ { "type": "step", "primary": "Multiply both sides by $$3$$", "result": "\\frac{3x}{3}=3\\cdot\\:\\frac{5π}{3}+3\\cdot\\:2πn" }, { "type": "interim", "title": "Simplify", "input": "\\frac{3x}{3}=3\\cdot\\:\\frac{5π}{3}+3\\cdot\\:2πn", "result": "x=5π+6πn", "steps": [ { "type": "interim", "title": "Simplify $$\\frac{3x}{3}:{\\quad}x$$", "input": "\\frac{3x}{3}", "steps": [ { "type": "step", "primary": "Divide the numbers: $$\\frac{3}{3}=1$$", "result": "=x" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7QJV6FEydUSv+hKOzRRB7mi061ljBSPJeENOw2efoSWt2cZ6LAXMcKee6PSyjLEFrRSpN33oxZMojoqvYhvSJACLkpiVA7S8UT8ieezZKu6ot8gw9eOFDGZzXCUlLc4C0" } }, { "type": "interim", "title": "Simplify $$3\\cdot\\:\\frac{5π}{3}+3\\cdot\\:2πn:{\\quad}5π+6πn$$", "input": "3\\cdot\\:\\frac{5π}{3}+3\\cdot\\:2πn", "steps": [ { "type": "interim", "title": "$$3\\cdot\\:\\frac{5π}{3}=5π$$", "input": "3\\cdot\\:\\frac{5π}{3}", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{5π3}{3}" }, { "type": "step", "primary": "Cancel the common factor: $$3$$", "result": "=5π" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7CzXS+MvaOUdKL6IkXGMFA03Sgaog1p4NBe0/Tq4T+Rh1g99dC9fj9sg0EHzBIRDR/vFTybheNsvazJfQjKcGUbrgkyCMPB58MhCQWH3BBXSewO9xqEAjTuV8cDoN8D9tG6Jn6ojVoz2IDd0XmrUm3Q==" } }, { "type": "interim", "title": "$$3\\cdot\\:2πn=6πn$$", "input": "3\\cdot\\:2πn", "steps": [ { "type": "step", "primary": "Multiply the numbers: $$3\\cdot\\:2=6$$", "result": "=6πn" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7YTBxxpumBqXGn+KTPhB1CFXTSum/z5kLpMzXS1UJIezrYzddS4OvRs60Tm4Kkwg7ZgbKfKW0HJe3lrhv5BsQrVFTnRcAzswxSGd6nlYqYFWwiNrEngO+NNvZ9sqNu+2V" } }, { "type": "step", "result": "=5π+6πn" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7CzXS+MvaOUdKL6IkXGMFA0TE7DkMyhA+X3SmBrlb0avj+O38RAvKhZolrKec34mYcJChiVhDxT5N/LHSTLMjyEaDltuzY4w6gaQcn+bznuXwt9LEn7QCBUukJKctfSJK6j6Qb3+Pt4A9UGhgwM3TRMLchKqxU1SpX2nUiqjNu1zJhXbwIMTZF2XkNmsW4ZyAsIjaxJ4DvjTb2fbKjbvtlQ==" } }, { "type": "step", "result": "x=5π+6πn" } ], "meta": { "interimType": "Generic Simplify 0Eq" } } ], "meta": { "interimType": "Multiply Both Sides Specific 1Eq", "gptData": 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"\\frac{x}{3}=\\frac{2π}{3}+2πn", "steps": [ { "type": "interim", "title": "Multiply both sides by $$3$$", "input": "\\frac{x}{3}=\\frac{2π}{3}+2πn", "result": "x=2π+6πn", "steps": [ { "type": "step", "primary": "Multiply both sides by $$3$$", "result": "\\frac{3x}{3}=3\\cdot\\:\\frac{2π}{3}+3\\cdot\\:2πn" }, { "type": "interim", "title": "Simplify", "input": "\\frac{3x}{3}=3\\cdot\\:\\frac{2π}{3}+3\\cdot\\:2πn", "result": "x=2π+6πn", "steps": [ { "type": "interim", "title": "Simplify $$\\frac{3x}{3}:{\\quad}x$$", "input": "\\frac{3x}{3}", "steps": [ { "type": "step", "primary": "Divide the numbers: $$\\frac{3}{3}=1$$", "result": "=x" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7QJV6FEydUSv+hKOzRRB7mi061ljBSPJeENOw2efoSWt2cZ6LAXMcKee6PSyjLEFrRSpN33oxZMojoqvYhvSJACLkpiVA7S8UT8ieezZKu6ot8gw9eOFDGZzXCUlLc4C0" } }, { "type": "interim", "title": "Simplify $$3\\cdot\\:\\frac{2π}{3}+3\\cdot\\:2πn:{\\quad}2π+6πn$$", "input": "3\\cdot\\:\\frac{2π}{3}+3\\cdot\\:2πn", "steps": [ { "type": "interim", "title": "$$3\\cdot\\:\\frac{2π}{3}=2π$$", "input": "3\\cdot\\:\\frac{2π}{3}", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{2π3}{3}" }, { "type": "step", "primary": "Cancel the common factor: $$3$$", "result": "=2π" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7CzXS+MvaOUdKL6IkXGMFA0VZfduaDLkYVCBWVXl/ABl1g99dC9fj9sg0EHzBIRDRbb51VZeJNkyCXGJRfBl0DLrgkyCMPB58MhCQWH3BBXTGG9yTog22M7slwPY5rt9gG6Jn6ojVoz2IDd0XmrUm3Q==" } }, { "type": "interim", "title": "$$3\\cdot\\:2πn=6πn$$", "input": "3\\cdot\\:2πn", "steps": [ { "type": "step", "primary": "Multiply the numbers: $$3\\cdot\\:2=6$$", "result": "=6πn" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7YTBxxpumBqXGn+KTPhB1CFXTSum/z5kLpMzXS1UJIezrYzddS4OvRs60Tm4Kkwg7ZgbKfKW0HJe3lrhv5BsQrVFTnRcAzswxSGd6nlYqYFWwiNrEngO+NNvZ9sqNu+2V" } }, { "type": "step", "result": "=2π+6πn" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7CzXS+MvaOUdKL6IkXGMFA01gK/+AkA+N+PG0/W3M3Wzj+O38RAvKhZolrKec34mYcJChiVhDxT5N/LHSTLMjyI/Z1jAf4GLy+dub8fBLPXTwt9LEn7QCBUukJKctfSJK6j6Qb3+Pt4A9UGhgwM3TRIb3yYIQ1HpxvkXFEOxrLc7JhXbwIMTZF2XkNmsW4ZyAsIjaxJ4DvjTb2fbKjbvtlQ==" } }, { "type": "step", "result": "x=2π+6πn" } ], "meta": { "interimType": "Generic Simplify 0Eq" } } ], "meta": { "interimType": "Multiply Both Sides Specific 1Eq", "gptData": "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" } } ], "meta": { "solvingClass": "Equations", "interimType": "Equations" } }, { "type": "interim", "title": "$$\\frac{x}{3}=\\frac{4π}{3}+2πn{\\quad:\\quad}x=4π+6πn$$", "input": "\\frac{x}{3}=\\frac{4π}{3}+2πn", "steps": [ { "type": "interim", "title": "Multiply both sides by $$3$$", "input": "\\frac{x}{3}=\\frac{4π}{3}+2πn", "result": "x=4π+6πn", "steps": [ { "type": "step", "primary": "Multiply both sides by $$3$$", "result": "\\frac{3x}{3}=3\\cdot\\:\\frac{4π}{3}+3\\cdot\\:2πn" }, { "type": "interim", "title": "Simplify", "input": "\\frac{3x}{3}=3\\cdot\\:\\frac{4π}{3}+3\\cdot\\:2πn", "result": "x=4π+6πn", "steps": [ { "type": "interim", "title": "Simplify $$\\frac{3x}{3}:{\\quad}x$$", "input": "\\frac{3x}{3}", "steps": [ { "type": "step", "primary": "Divide the numbers: $$\\frac{3}{3}=1$$", "result": "=x" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7QJV6FEydUSv+hKOzRRB7mi061ljBSPJeENOw2efoSWt2cZ6LAXMcKee6PSyjLEFrRSpN33oxZMojoqvYhvSJACLkpiVA7S8UT8ieezZKu6ot8gw9eOFDGZzXCUlLc4C0" } }, { "type": "interim", "title": "Simplify $$3\\cdot\\:\\frac{4π}{3}+3\\cdot\\:2πn:{\\quad}4π+6πn$$", "input": "3\\cdot\\:\\frac{4π}{3}+3\\cdot\\:2πn", "steps": [ { "type": "interim", "title": "$$3\\cdot\\:\\frac{4π}{3}=4π$$", "input": "3\\cdot\\:\\frac{4π}{3}", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{4π3}{3}" }, { "type": "step", "primary": "Cancel the common factor: $$3$$", "result": "=4π" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7CzXS+MvaOUdKL6IkXGMFA9T42mGBAyxdciARvTs4JcN1g99dC9fj9sg0EHzBIRDRCiRPgHS1PbHUjLZEAJLpHLrgkyCMPB58MhCQWH3BBXQqwKUfHQLOiR6xSwPvlYxAG6Jn6ojVoz2IDd0XmrUm3Q==" } }, { "type": "interim", "title": "$$3\\cdot\\:2πn=6πn$$", "input": "3\\cdot\\:2πn", "steps": [ { "type": "step", "primary": "Multiply the numbers: $$3\\cdot\\:2=6$$", "result": "=6πn" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7YTBxxpumBqXGn+KTPhB1CFXTSum/z5kLpMzXS1UJIezrYzddS4OvRs60Tm4Kkwg7ZgbKfKW0HJe3lrhv5BsQrVFTnRcAzswxSGd6nlYqYFWwiNrEngO+NNvZ9sqNu+2V" } }, { "type": "step", "result": "=4π+6πn" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7CzXS+MvaOUdKL6IkXGMFA0+KO8SuYHcoo+ctrxXKtOTj+O38RAvKhZolrKec34mYcJChiVhDxT5N/LHSTLMjyKyByN+IEXh8bZZd364qf3Pwt9LEn7QCBUukJKctfSJK6j6Qb3+Pt4A9UGhgwM3TRDSBwpCdWguD4W6vGq1dO0jJhXbwIMTZF2XkNmsW4ZyAsIjaxJ4DvjTb2fbKjbvtlQ==" } }, { "type": "step", "result": "x=4π+6πn" } ], "meta": { "interimType": "Generic Simplify 0Eq" } } ], "meta": { "interimType": "Multiply Both Sides Specific 1Eq", "gptData": 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"x=\\frac{π}{2}+6πn,\\:x=\\frac{11π}{2}+6πn,\\:x=\\frac{5π}{2}+6πn,\\:x=\\frac{7π}{2}+6πn,\\:x=π+6πn,\\:x=5π+6πn,\\:x=2π+6πn,\\:x=4π+6πn" } ], "meta": { "solvingClass": "Trig Equations", "practiceLink": "/practice/trigonometry-practice#area=main&subtopic=Trig%20Equations", "practiceTopic": "Trig Equations" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "\\sin^{4}(\\frac{x}{3})+\\cos^{4}(\\frac{x}{3})-\\frac{5}{8}" }, "showViewLarger": true } }, "meta": { "showVerify": true } }