2cos(θ)=3cos(3θ)

{ "query": { "display": "$$2\\cos\\left(θ\\right)=3\\cos\\left(3θ\\right)$$", "symbolab_question": "EQUATION#2\\cos(θ)=3\\cos(3θ)" }, "solution": { "level": "PERFORMED", "subject": "Trigonometry", "topic": "Trig Equations", "subTopic": "Trig Equations", "default": "θ=\\frac{π}{2}+2πn,θ=\\frac{3π}{2}+2πn,θ=2.84874…+2πn,θ=-2.84874…+2πn,θ=0.29284…+2πn,θ=2π-0.29284…+2πn", "degrees": "θ=90^{\\circ }+360^{\\circ }n,θ=270^{\\circ }+360^{\\circ }n,θ=163.22134…^{\\circ }+360^{\\circ }n,θ=-163.22134…^{\\circ }+360^{\\circ }n,θ=16.77865…^{\\circ }+360^{\\circ }n,θ=343.22134…^{\\circ }+360^{\\circ }n", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$2\\cos\\left(θ\\right)=3\\cos\\left(3θ\\right){\\quad:\\quad}θ=\\frac{π}{2}+2πn,\\:θ=\\frac{3π}{2}+2πn,\\:θ=2.84874…+2πn,\\:θ=-2.84874…+2πn,\\:θ=0.29284…+2πn,\\:θ=2π-0.29284…+2πn$$", "input": "2\\cos\\left(θ\\right)=3\\cos\\left(3θ\\right)", "steps": [ { "type": "step", "primary": "Subtract $$3\\cos\\left(3θ\\right)$$ from both sides", "result": "2\\cos\\left(θ\\right)-3\\cos\\left(3θ\\right)=0" }, { "type": "interim", "title": "Rewrite using trig identities", "input": "2\\cos\\left(θ\\right)-3\\cos\\left(3θ\\right)", "result": "11\\cos\\left(θ\\right)-12\\cos^{3}\\left(θ\\right)=0", "steps": [ { "type": "interim", "title": "$$\\cos\\left(3θ\\right)=4\\cos^{3}\\left(θ\\right)-3\\cos\\left(θ\\right)$$", "input": "\\cos\\left(3θ\\right)", "steps": [ { "type": "interim", "title": "Rewrite using trig identities", "input": "\\cos\\left(3θ\\right)", "result": "=\\cos\\left(θ\\right)\\cos\\left(2θ\\right)-2\\sin^{2}\\left(θ\\right)\\cos\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Rewrite as", "result": "=\\cos\\left(2θ+θ\\right)" }, { "type": "step", "primary": "Use the Angle Sum identity: $$\\cos\\left(s+t\\right)=\\cos\\left(s\\right)\\cos\\left(t\\right)-\\sin\\left(s\\right)\\sin\\left(t\\right)$$", "result": "=\\cos\\left(2θ\\right)\\cos\\left(θ\\right)-\\sin\\left(2θ\\right)\\sin\\left(θ\\right)" }, { "type": "step", "primary": "Use the Double Angle identity: $$\\sin\\left(2θ\\right)=2\\sin\\left(θ\\right)\\cos\\left(θ\\right)$$", "result": "=\\cos\\left(2θ\\right)\\cos\\left(θ\\right)-2\\sin\\left(θ\\right)\\cos\\left(θ\\right)\\sin\\left(θ\\right)" }, { "type": "interim", "title": "Simplify $$\\cos\\left(2θ\\right)\\cos\\left(θ\\right)-2\\sin\\left(θ\\right)\\cos\\left(θ\\right)\\sin\\left(θ\\right):{\\quad}\\cos\\left(θ\\right)\\cos\\left(2θ\\right)-2\\sin^{2}\\left(θ\\right)\\cos\\left(θ\\right)$$", "input": "\\cos\\left(2θ\\right)\\cos\\left(θ\\right)-2\\sin\\left(θ\\right)\\cos\\left(θ\\right)\\sin\\left(θ\\right)", "result": "=\\cos\\left(θ\\right)\\cos\\left(2θ\\right)-2\\sin^{2}\\left(θ\\right)\\cos\\left(θ\\right)", "steps": [ { "type": "interim", "title": "$$2\\sin\\left(θ\\right)\\cos\\left(θ\\right)\\sin\\left(θ\\right)=2\\sin^{2}\\left(θ\\right)\\cos\\left(θ\\right)$$", "input": "2\\sin\\left(θ\\right)\\cos\\left(θ\\right)\\sin\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\sin\\left(θ\\right)\\sin\\left(θ\\right)=\\:\\sin^{1+1}\\left(θ\\right)$$" ], "result": "=2\\cos\\left(θ\\right)\\sin^{1+1}\\left(θ\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$1+1=2$$", "result": "=2\\cos\\left(θ\\right)\\sin^{2}\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7MK8yICad9aR5K9D9XMMpG/ltR1M3NKZf3jmoZOL1F7ZV00rpv8+ZC6TM10tVCSHse4qBpIkVy2vvJRCXZt6ryFDf5kvQDZqJRhu9yOQDSxmuqvW3hAHH635V/vEEYIatFNUT6leA6S7YtNQKJo5vb9jsbgDd/rDfleqC5sUrnqjiSkEDZ0xH9T2+Da75TQ1UQEffeV5IX3igOUFqeHqssA==" } }, { "type": "step", "result": "=\\cos\\left(θ\\right)\\cos\\left(2θ\\right)-2\\sin^{2}\\left(θ\\right)\\cos\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } } ], "meta": { "interimType": "Trig Rewrite Using Trig identities 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7lRPp6QndqgNT+L6IZutAbP8oSWOnU5wiIU2FkfnbZ5ISwtk/DRWb0wPqX3+QHVz3V/ZUTeJpWKv8yNHYMRFxnT73Z4YtLN5Wfgye6xIk3qkujmThGQoCfX5tOxzG+J9yZ0FgGdO/fOiO3O0j7zqZbTBJKWzSyIlbs0M5YmS71wujeh7+jKEzLb7VNCEMF3Z/AUodDpEZVCOJY6rkfXcVkqaOQILKf053R9DIJaaWk7Mc6tftTNY2KsUtvVbis5zN" } }, { "type": "step", "primary": "Use the Double Angle identity: $$\\cos\\left(2θ\\right)=2\\cos^{2}\\left(θ\\right)-1$$", "result": "=\\left(2\\cos^{2}\\left(θ\\right)-1\\right)\\cos\\left(θ\\right)-2\\sin^{2}\\left(θ\\right)\\cos\\left(θ\\right)" }, { "type": "step", "primary": "Use the Pythagorean identity: $$\\cos^{2}\\left(θ\\right)+\\sin^{2}\\left(θ\\right)=1$$", "secondary": [ "$$\\sin^{2}\\left(θ\\right)=1-\\cos^{2}\\left(θ\\right)$$" ], "result": "=\\left(2\\cos^{2}\\left(θ\\right)-1\\right)\\cos\\left(θ\\right)-2\\left(1-\\cos^{2}\\left(θ\\right)\\right)\\cos\\left(θ\\right)" }, { "type": "interim", "title": "Expand $$\\left(2\\cos^{2}\\left(θ\\right)-1\\right)\\cos\\left(θ\\right)-2\\left(1-\\cos^{2}\\left(θ\\right)\\right)\\cos\\left(θ\\right):{\\quad}4\\cos^{3}\\left(θ\\right)-3\\cos\\left(θ\\right)$$", "input": "\\left(2\\cos^{2}\\left(θ\\right)-1\\right)\\cos\\left(θ\\right)-2\\left(1-\\cos^{2}\\left(θ\\right)\\right)\\cos\\left(θ\\right)", "result": "=4\\cos^{3}\\left(θ\\right)-3\\cos\\left(θ\\right)", "steps": [ { "type": "step", "result": "=\\cos\\left(θ\\right)\\left(2\\cos^{2}\\left(θ\\right)-1\\right)-2\\cos\\left(θ\\right)\\left(1-\\cos^{2}\\left(θ\\right)\\right)" }, { "type": "interim", "title": "Expand $$\\cos\\left(θ\\right)\\left(2\\cos^{2}\\left(θ\\right)-1\\right):{\\quad}2\\cos^{3}\\left(θ\\right)-\\cos\\left(θ\\right)$$", "input": "\\cos\\left(θ\\right)\\left(2\\cos^{2}\\left(θ\\right)-1\\right)", "result": "=2\\cos^{3}\\left(θ\\right)-\\cos\\left(θ\\right)-2\\left(1-\\cos^{2}\\left(θ\\right)\\right)\\cos\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Apply the distributive law: $$a\\left(b-c\\right)=ab-ac$$", "secondary": [ "$$a=\\cos\\left(θ\\right),\\:b=2\\cos^{2}\\left(θ\\right),\\:c=1$$" ], "result": "=\\cos\\left(θ\\right)2\\cos^{2}\\left(θ\\right)-\\cos\\left(θ\\right)1", "meta": { "practiceLink": "/practice/expansion-practice", "practiceTopic": "Expand Rules" } }, { "type": "step", "result": "=2\\cos^{2}\\left(θ\\right)\\cos\\left(θ\\right)-1\\cos\\left(θ\\right)" }, { "type": "interim", "title": "Simplify $$2\\cos^{2}\\left(θ\\right)\\cos\\left(θ\\right)-1\\cdot\\:\\cos\\left(θ\\right):{\\quad}2\\cos^{3}\\left(θ\\right)-\\cos\\left(θ\\right)$$", "input": "2\\cos^{2}\\left(θ\\right)\\cos\\left(θ\\right)-1\\cos\\left(θ\\right)", "result": "=2\\cos^{3}\\left(θ\\right)-\\cos\\left(θ\\right)", "steps": [ { "type": "interim", "title": "$$2\\cos^{2}\\left(θ\\right)\\cos\\left(θ\\right)=2\\cos^{3}\\left(θ\\right)$$", "input": "2\\cos^{2}\\left(θ\\right)\\cos\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\cos^{2}\\left(θ\\right)\\cos\\left(θ\\right)=\\:\\cos^{2+1}\\left(θ\\right)$$" ], "result": "=2\\cos^{2+1}\\left(θ\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$2+1=3$$", "result": "=2\\cos^{3}\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7cajM9UviOJJxp+0LMAC84U4pcWUl32yFd8oXXkas/3MDnzlbPZjyKgy1eUCFsLd5FERgidIIuxED1Sr+ZtOL8Gx9uc3EyK04O37j7Cp6dyVBBf2okYUW/qlIM0erhCXeStw/5zqZ8xmPIpRjOa2e2hgjsGJrWClN24OM/cBg/lw=" } }, { "type": "interim", "title": "$$1\\cdot\\:\\cos\\left(θ\\right)=\\cos\\left(θ\\right)$$", "input": "1\\cos\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Multiply: $$1\\cdot\\:\\cos\\left(θ\\right)=\\cos\\left(θ\\right)$$", "result": "=\\cos\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7exYCpNgwPj/Kq1zDts5QbSAn9lkDfZkicUGkO3EF+IoFvrKTMZyMas6ny/YtZVXXrqr1t4QBx+t+Vf7xBGCGrWxMpficWwb+Vl4/Dl5Ghkgx6ZKSJsBHHhtdkGeEDCnM" } }, { "type": "step", "result": "=2\\cos^{3}\\left(θ\\right)-\\cos\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } } ], "meta": { "interimType": "Algebraic Manipulation Expand Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7wABr6nTd26EF3k9yvcQdkD68fw39Lv3eS42fhUxGMSzehkKrn0era9rz8TlL+x/vK681lBK6gAQJrscFq11Ga3966c4seBxesEijhgCMvXquqvW3hAHH635V/vEEYIatgnLy9o0s8NVZUXBl+JGuqgib5sgP1/HaSflC0kqFIsrNCn+y1gccR+lGtmWXqcNJJLd1ohke2Wgml78++2zI0g==" } }, { "type": "interim", "title": "Expand $$-2\\cos\\left(θ\\right)\\left(1-\\cos^{2}\\left(θ\\right)\\right):{\\quad}-2\\cos\\left(θ\\right)+2\\cos^{3}\\left(θ\\right)$$", "input": "-2\\cos\\left(θ\\right)\\left(1-\\cos^{2}\\left(θ\\right)\\right)", "result": "=2\\cos^{3}\\left(θ\\right)-\\cos\\left(θ\\right)-2\\cos\\left(θ\\right)+2\\cos^{3}\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Apply the distributive law: $$a\\left(b-c\\right)=ab-ac$$", "secondary": [ "$$a=-2\\cos\\left(θ\\right),\\:b=1,\\:c=\\cos^{2}\\left(θ\\right)$$" ], "result": "=-2\\cos\\left(θ\\right)1-\\left(-2\\cos\\left(θ\\right)\\right)\\cos^{2}\\left(θ\\right)", "meta": { "practiceLink": "/practice/expansion-practice", "practiceTopic": "Expand Rules" } }, { "type": "step", "primary": "Apply minus-plus rules", "secondary": [ "$$-\\left(-a\\right)=a$$" ], "result": "=-2\\cdot\\:1\\cos\\left(θ\\right)+2\\cos^{2}\\left(θ\\right)\\cos\\left(θ\\right)" }, { "type": "interim", "title": "Simplify $$-2\\cdot\\:1\\cdot\\:\\cos\\left(θ\\right)+2\\cos^{2}\\left(θ\\right)\\cos\\left(θ\\right):{\\quad}-2\\cos\\left(θ\\right)+2\\cos^{3}\\left(θ\\right)$$", "input": "-2\\cdot\\:1\\cos\\left(θ\\right)+2\\cos^{2}\\left(θ\\right)\\cos\\left(θ\\right)", "result": "=-2\\cos\\left(θ\\right)+2\\cos^{3}\\left(θ\\right)", "steps": [ { "type": "interim", "title": "$$2\\cdot\\:1\\cdot\\:\\cos\\left(θ\\right)=2\\cos\\left(θ\\right)$$", "input": "2\\cdot\\:1\\cos\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:1=2$$", "result": "=2\\cos\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7hA94Ei8IRXRSFT5crEz4sDHt7fABx6LEy6vdX9TQoXDMwViaLUXkeD+JukROhWdjB+p8J68KnmLoRZO+z7HW4dUr2P3iRs7xJpVu4aDxEw7fgZXmlYOfV+AT+8iqmLZXrN0V/bLyMAbQTi3lcqeOSL8yD3hLQ33B7/8/LpbPE3o=" } }, { "type": "interim", "title": "$$2\\cos^{2}\\left(θ\\right)\\cos\\left(θ\\right)=2\\cos^{3}\\left(θ\\right)$$", "input": "2\\cos^{2}\\left(θ\\right)\\cos\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\cos^{2}\\left(θ\\right)\\cos\\left(θ\\right)=\\:\\cos^{2+1}\\left(θ\\right)$$" ], "result": "=2\\cos^{2+1}\\left(θ\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$2+1=3$$", "result": "=2\\cos^{3}\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7cajM9UviOJJxp+0LMAC84U4pcWUl32yFd8oXXkas/3MDnzlbPZjyKgy1eUCFsLd5FERgidIIuxED1Sr+ZtOL8Gx9uc3EyK04O37j7Cp6dyVBBf2okYUW/qlIM0erhCXeStw/5zqZ8xmPIpRjOa2e2hgjsGJrWClN24OM/cBg/lw=" } }, { "type": "step", "result": "=-2\\cos\\left(θ\\right)+2\\cos^{3}\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } } ], "meta": { "interimType": "Algebraic Manipulation Expand Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7mF3X3x+oGi1JiNt3VUuyxwx+68Ns1gJL5ctjkKZoS/tV00rpv8+ZC6TM10tVCSHslHTZA2JjKbs3sGXpKbpArQhtFpFWLro3uOe0n6mc50v5XtLqQXOvrM0ZezvwvFqKEnJ/zNrC2ARoMQpaswsTNj+0L5GI0hha18B//0HcX7Ldr3Yucrbc2Bj0TAmgkmJJtmRi0E3eC0He/xESpULhVg==" } }, { "type": "interim", "title": "Simplify $$2\\cos^{3}\\left(θ\\right)-\\cos\\left(θ\\right)-2\\cos\\left(θ\\right)+2\\cos^{3}\\left(θ\\right):{\\quad}4\\cos^{3}\\left(θ\\right)-3\\cos\\left(θ\\right)$$", "input": "2\\cos^{3}\\left(θ\\right)-\\cos\\left(θ\\right)-2\\cos\\left(θ\\right)+2\\cos^{3}\\left(θ\\right)", "result": "=4\\cos^{3}\\left(θ\\right)-3\\cos\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Group like terms", "result": "=2\\cos^{3}\\left(θ\\right)+2\\cos^{3}\\left(θ\\right)-\\cos\\left(θ\\right)-2\\cos\\left(θ\\right)" }, { "type": "step", "primary": "Add similar elements: $$2\\cos^{3}\\left(θ\\right)+2\\cos^{3}\\left(θ\\right)=4\\cos^{3}\\left(θ\\right)$$", "result": "=4\\cos^{3}\\left(θ\\right)-\\cos\\left(θ\\right)-2\\cos\\left(θ\\right)" }, { "type": "step", "primary": "Add similar elements: $$-\\cos\\left(θ\\right)-2\\cos\\left(θ\\right)=-3\\cos\\left(θ\\right)$$", "result": "=4\\cos^{3}\\left(θ\\right)-3\\cos\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Expand Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7szb5nQPKa/tfNp2vTdmJG0ns053UO1DFWxkuOlA/Del1rU8AGCNCj4N4I8fsj0GdTilxZSXfbIV3yhdeRqz/cwOfOVs9mPIqDLV5QIWwt3kj2NDR1/PQrZr09feip8OkfeXv2w7yB2pDDB1RpaWWNWRLd2VwIqlBNByF6663syRU6H0nS++8kDqP632fuVHPLSQ5g0+jHJsCa6uZug5hRlYnYcuL2xyC5d3fyqnwu9xC2UnuMHyD2Dm2LPwiDuewXKXGPDxA7c7YUtD9/bBQDA==" } } ], "meta": { "interimType": "N/A" } }, { "type": "step", "result": "=2\\cos\\left(θ\\right)-3\\left(4\\cos^{3}\\left(θ\\right)-3\\cos\\left(θ\\right)\\right)" }, { "type": "interim", "title": "Simplify $$2\\cos\\left(θ\\right)-3\\left(4\\cos^{3}\\left(θ\\right)-3\\cos\\left(θ\\right)\\right):{\\quad}11\\cos\\left(θ\\right)-12\\cos^{3}\\left(θ\\right)$$", "input": "2\\cos\\left(θ\\right)-3\\left(4\\cos^{3}\\left(θ\\right)-3\\cos\\left(θ\\right)\\right)", "result": "=11\\cos\\left(θ\\right)-12\\cos^{3}\\left(θ\\right)", "steps": [ { "type": "interim", "title": "Expand $$-3\\left(4\\cos^{3}\\left(θ\\right)-3\\cos\\left(θ\\right)\\right):{\\quad}-12\\cos^{3}\\left(θ\\right)+9\\cos\\left(θ\\right)$$", "input": "-3\\left(4\\cos^{3}\\left(θ\\right)-3\\cos\\left(θ\\right)\\right)", "result": "=2\\cos\\left(θ\\right)-12\\cos^{3}\\left(θ\\right)+9\\cos\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Apply the distributive law: $$a\\left(b-c\\right)=ab-ac$$", "secondary": [ "$$a=-3,\\:b=4\\cos^{3}\\left(θ\\right),\\:c=3\\cos\\left(θ\\right)$$" ], "result": "=-3\\cdot\\:4\\cos^{3}\\left(θ\\right)-\\left(-3\\right)\\cdot\\:3\\cos\\left(θ\\right)", "meta": { "practiceLink": "/practice/expansion-practice", "practiceTopic": "Expand Rules" } }, { "type": "step", "primary": "Apply minus-plus rules", "secondary": [ "$$-\\left(-a\\right)=a$$" ], "result": "=-3\\cdot\\:4\\cos^{3}\\left(θ\\right)+3\\cdot\\:3\\cos\\left(θ\\right)" }, { "type": "interim", "title": "Simplify $$-3\\cdot\\:4\\cos^{3}\\left(θ\\right)+3\\cdot\\:3\\cos\\left(θ\\right):{\\quad}-12\\cos^{3}\\left(θ\\right)+9\\cos\\left(θ\\right)$$", "input": "-3\\cdot\\:4\\cos^{3}\\left(θ\\right)+3\\cdot\\:3\\cos\\left(θ\\right)", "result": "=-12\\cos^{3}\\left(θ\\right)+9\\cos\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Multiply the numbers: $$3\\cdot\\:4=12$$", "result": "=-12\\cos^{3}\\left(θ\\right)+3\\cdot\\:3\\cos\\left(θ\\right)" }, { "type": "step", "primary": "Multiply the numbers: $$3\\cdot\\:3=9$$", "result": "=-12\\cos^{3}\\left(θ\\right)+9\\cos\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } } ], "meta": { "interimType": "Algebraic Manipulation Expand Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79nmRK7JAZhq0Clr1TWcmHLfr/TmRyGwVLqMhEd7jO1EtOtZYwUjyXhDTsNnn6Elr9MY8Kv0ZrSPuIt8GfKvpd4xlpEl3IsWuzzdQQ6QpsTkdfS1tBe3kCsSJdPlROpk1TeQKHeh69S6dnv9vSoUoFLMIhVgaUyyEaI6NEDsKYsv2E4RgEOiKjEtw7FgmCZ/nSIUZGk+g6gqVwxD/XktSUw==" } }, { "type": "step", "primary": "Add similar elements: $$2\\cos\\left(θ\\right)+9\\cos\\left(θ\\right)=11\\cos\\left(θ\\right)$$", "result": "=11\\cos\\left(θ\\right)-12\\cos^{3}\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7A5t2+ubl0k14qzYV6HZGVpaP3F+l+CM8MWVpmsei+epJmCtbqgdghayyaLQrrD83A585Wz2Y8ioMtXlAhbC3eXIVQIpI48w95TiBtyJ58LfKXrQLqgLl6We1dRTVgLCj7lvKtANBUJdQPS8f9+853HKF3u2OIb4bFA3EO8aRlSWKTbnjTbFgHUk64X8k1KXbC1ZycoghuAetv5iyKyfawS1fzewKDfghR/mdzaTzYE0=" } } ], "meta": { "interimType": "Trig Rewrite Using Trig identities 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s791AXYceO9Pc4CRAMY99YBULCadmY8XFWR95p/rdpdYAaM37KF/jK0q16Pg+4JBw35yFOVAzu8w4itJwsMopep5nX/LTRLwEhE6Awmk/6zhUIAqZBkW8uMb72Rzx6Zt+trLQFXXCtKKX+NEiYuJQ1MhxPYGZvRD7tun+y/Hlzt4jwt9LEn7QCBUukJKctfSJKk2VQdqPGsmYTaYyesllHTRc4KXs8xgJNiLkJwr0b/5OJqVxX90jlMfh9fKn6dzC4" } }, { "type": "interim", "title": "Solve by substitution", "input": "11\\cos\\left(θ\\right)-12\\cos^{3}\\left(θ\\right)=0", "result": "\\cos\\left(θ\\right)=0,\\:\\cos\\left(θ\\right)=-\\frac{\\sqrt{33}}{6},\\:\\cos\\left(θ\\right)=\\frac{\\sqrt{33}}{6}", "steps": [ { "type": "step", "primary": "Let: $$\\cos\\left(θ\\right)=u$$", "result": "11u-12u^{3}=0" }, { "type": "interim", "title": "$$11u-12u^{3}=0{\\quad:\\quad}u=0,\\:u=-\\frac{\\sqrt{33}}{6},\\:u=\\frac{\\sqrt{33}}{6}$$", "input": "11u-12u^{3}=0", "steps": [ { "type": "interim", "title": "Factor $$11u-12u^{3}:{\\quad}-u\\left(2\\sqrt{3}u+\\sqrt{11}\\right)\\left(2\\sqrt{3}u-\\sqrt{11}\\right)$$", "input": "11u-12u^{3}", "steps": [ { "type": "interim", "title": "Factor out common term $$-u:{\\quad}-u\\left(12u^{2}-11\\right)$$", "input": "-12u^{3}+11u", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^{b+c}=a^{b}a^{c}$$", "secondary": [ "$$u^{3}=u^{2}u$$" ], "result": "=-12u^{2}u+11u", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Factor out common term $$-u$$", "result": "=-u\\left(12u^{2}-11\\right)" } ], "meta": { "interimType": "Factor Take Out Common Term 1Eq", "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } }, { "type": "step", "result": "=-u\\left(12u^{2}-11\\right)" }, { "type": "interim", "title": "Factor $$12u^{2}-11:{\\quad}\\left(\\sqrt{12}u+\\sqrt{11}\\right)\\left(\\sqrt{12}u-\\sqrt{11}\\right)$$", "input": "12u^{2}-11", "steps": [ { "type": "interim", "title": "Rewrite $$12u^{2}-11$$ as $$\\left(\\sqrt{12}u\\right)^{2}-\\left(\\sqrt{11}\\right)^{2}$$", "input": "12u^{2}-11", "result": "=\\left(\\sqrt{12}u\\right)^{2}-\\left(\\sqrt{11}\\right)^{2}", "steps": [ { "type": "step", "primary": "Apply radical rule: $$a=\\left(\\sqrt{a}\\right)^{2}$$", "secondary": [ "$$12=\\left(\\sqrt{12}\\right)^{2}$$" ], "result": "=\\left(\\sqrt{12}\\right)^{2}u^{2}-11", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } }, { "type": "step", "primary": "Apply radical rule: $$a=\\left(\\sqrt{a}\\right)^{2}$$", "secondary": [ "$$11=\\left(\\sqrt{11}\\right)^{2}$$" ], "result": "=\\left(\\sqrt{12}\\right)^{2}u^{2}-\\left(\\sqrt{11}\\right)^{2}", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } }, { "type": "step", "primary": "Apply exponent rule: $$a^{m}b^{m}=\\left(ab\\right)^{m}$$", "secondary": [ "$$\\left(\\sqrt{12}\\right)^{2}u^{2}=\\left(\\sqrt{12}u\\right)^{2}$$" ], "result": "=\\left(\\sqrt{12}u\\right)^{2}-\\left(\\sqrt{11}\\right)^{2}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } } ], "meta": { "interimType": "Generic Rewrite As Specific 2Eq" } }, { "type": "step", "primary": "Apply Difference of Two Squares Formula: $$x^{2}-y^{2}=\\left(x+y\\right)\\left(x-y\\right)$$", "secondary": [ "$$\\left(\\sqrt{12}u\\right)^{2}-\\left(\\sqrt{11}\\right)^{2}=\\left(\\sqrt{12}u+\\sqrt{11}\\right)\\left(\\sqrt{12}u-\\sqrt{11}\\right)$$" ], "result": "=\\left(\\sqrt{12}u+\\sqrt{11}\\right)\\left(\\sqrt{12}u-\\sqrt{11}\\right)", "meta": { "practiceLink": "/practice/factoring-practice#area=main&subtopic=Difference%20of%20Two%20Squares", "practiceTopic": "Factor Difference of Squares" } } ], "meta": { "interimType": "Algebraic Manipulation Factor Title 1Eq" } }, { "type": "step", "result": "=-u\\left(\\sqrt{12}u+\\sqrt{11}\\right)\\left(\\sqrt{12}u-\\sqrt{11}\\right)" }, { "type": "step", "primary": "Refine", "result": "=-u\\left(2\\sqrt{3}u+\\sqrt{11}\\right)\\left(2\\sqrt{3}u-\\sqrt{11}\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Factor Specific 1Eq" } }, { "type": "step", "result": "-u\\left(2\\sqrt{3}u+\\sqrt{11}\\right)\\left(2\\sqrt{3}u-\\sqrt{11}\\right)=0" }, { "type": "step", "primary": "Using the Zero Factor Principle:$$\\quad$$ If $$ab=0\\:$$then $$a=0\\:$$or $$b=0$$", "result": "u=0\\lor\\:2\\sqrt{3}u+\\sqrt{11}=0\\lor\\:2\\sqrt{3}u-\\sqrt{11}=0" }, { "type": "interim", "title": "Solve $$2\\sqrt{3}u+\\sqrt{11}=0:{\\quad}u=-\\frac{\\sqrt{33}}{6}$$", "input": "2\\sqrt{3}u+\\sqrt{11}=0", "steps": [ { "type": "interim", "title": "Move $$\\sqrt{11}\\:$$to the right side", "input": "2\\sqrt{3}u+\\sqrt{11}=0", "result": "2\\sqrt{3}u=-\\sqrt{11}", "steps": [ { "type": "step", "primary": "Subtract $$\\sqrt{11}$$ from both sides", "result": "2\\sqrt{3}u+\\sqrt{11}-\\sqrt{11}=0-\\sqrt{11}" }, { "type": "step", "primary": "Simplify", "result": "2\\sqrt{3}u=-\\sqrt{11}" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": "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" } }, { "type": "interim", "title": "Divide both sides by $$2\\sqrt{3}$$", "input": "2\\sqrt{3}u=-\\sqrt{11}", "result": "u=-\\frac{\\sqrt{33}}{6}", "steps": [ { "type": "step", "primary": "Divide both sides by $$2\\sqrt{3}$$", "result": "\\frac{2\\sqrt{3}u}{2\\sqrt{3}}=\\frac{-\\sqrt{11}}{2\\sqrt{3}}" }, { "type": "interim", "title": "Simplify", "input": "\\frac{2\\sqrt{3}u}{2\\sqrt{3}}=\\frac{-\\sqrt{11}}{2\\sqrt{3}}", "result": "u=-\\frac{\\sqrt{33}}{6}", "steps": [ { "type": "interim", "title": "Simplify $$\\frac{2\\sqrt{3}u}{2\\sqrt{3}}:{\\quad}u$$", "input": "\\frac{2\\sqrt{3}u}{2\\sqrt{3}}", "steps": [ { "type": "step", "primary": "Divide the numbers: $$\\frac{2}{2}=1$$", "result": "=\\frac{\\sqrt{3}u}{\\sqrt{3}}" }, { "type": "step", "primary": "Cancel the common factor: $$\\sqrt{3}$$", "result": "=u" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7iA1jhOJsYg26KMh1oIvzcr+SrK6v43hdihJsRmAb11Z8kR7hsO/rTOTBE0w4+r1RYppoGJ3t+EWMwmq0/EQlQmRLd2VwIqlBNByF6663syR2SpdpleAJc7YgKUwBYoM9rKSn6S8yGX//uYoRDIG/ir+SrK6v43hdihJsRmAb11aJqVxX90jlMfh9fKn6dzC4" } }, { "type": "interim", "title": "Simplify $$\\frac{-\\sqrt{11}}{2\\sqrt{3}}:{\\quad}-\\frac{\\sqrt{33}}{6}$$", "input": "\\frac{-\\sqrt{11}}{2\\sqrt{3}}", "steps": [ { "type": "step", "primary": "Apply the fraction rule: $$\\frac{-a}{b}=-\\frac{a}{b}$$", "result": "=-\\frac{\\sqrt{11}}{2\\sqrt{3}}" }, { "type": "interim", "title": "Rationalize $$-\\frac{\\sqrt{11}}{2\\sqrt{3}}:{\\quad}-\\frac{\\sqrt{33}}{6}$$", "input": "-\\frac{\\sqrt{11}}{2\\sqrt{3}}", "result": "=-\\frac{\\sqrt{33}}{6}", "steps": [ { "type": "step", "primary": "Multiply by the conjugate $$\\frac{\\sqrt{3}}{\\sqrt{3}}$$", "result": "=-\\frac{\\sqrt{11}\\sqrt{3}}{2\\sqrt{3}\\sqrt{3}}", "meta": { "title": { "extension": "To rationalize the denominator, multiply numerator and denominator by the conjugate of the radical $$\\sqrt{3}$$" } } }, { "type": "interim", "title": "$$\\sqrt{11}\\sqrt{3}=\\sqrt{33}$$", "input": "\\sqrt{11}\\sqrt{3}", "steps": [ { "type": "step", "primary": "Apply radical rule: $$\\sqrt{a}\\sqrt{b}=\\sqrt{a\\cdot{b}}$$", "secondary": [ "$$\\sqrt{11}\\sqrt{3}=\\sqrt{11\\cdot\\:3}$$" ], "result": "=\\sqrt{11\\cdot\\:3}", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } }, { "type": "step", "primary": "Multiply the numbers: $$11\\cdot\\:3=33$$", "result": "=\\sqrt{33}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7OOrYgUTRfuJ+rEJC8tzMYCj7guf/B8PEHny4jV/tPelwkKGJWEPFPk38sdJMsyPIT6SAm4l5zkHCOVWGqGW3bYY+spUsB4yZJkDyh9+Pu39a7YlKPs+0J9nd5MP8YUNaHuV7PWraVhR7fPhyDwKS0w==" } }, { "type": "interim", "title": "$$2\\sqrt{3}\\sqrt{3}=6$$", "input": "2\\sqrt{3}\\sqrt{3}", "result": "=-\\frac{\\sqrt{33}}{6}", "steps": [ { "type": "step", "primary": "Apply radical rule: $$\\sqrt{a}\\sqrt{a}=a$$", "secondary": [ "$$\\sqrt{3}\\sqrt{3}=3$$" ], "result": "=2\\cdot\\:3", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:3=6$$", "result": "=6" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7aTdZOU+fmdGxwfasVfdUfSj7guf/B8PEHny4jV/tPelwkKGJWEPFPk38sdJMsyPIOzRnKkVJMpyFFl92V2YjIEDScMkehj8ySs0hZkq/+teFKt1ZlkEUZE0U4/d5bZ1Y" } } ], "meta": { "interimType": "Rationalize Title 1Eq" } } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s70c9DTe1ubFlIcViNwxWNeVOsTGjGRtCMbscLsZVkpd58kR7hsO/rTOTBE0w4+r1RwxWDXidEV9CzsGPnUu41zEoC0Y9e8wrAeY1jx2P5VETQ7wabUGwYq+xsGfhpVfEvmNwEgvmmajzdQwVlFLMmxQOcAqy5kkCcnKs8aluR+24cg6CjzvYrbPodKzddbAQ+AS2KM520E/UQzQVt7k90kw==" } }, { "type": "step", "result": "u=-\\frac{\\sqrt{33}}{6}" } ], "meta": { "interimType": "Generic Simplify 0Eq" } } ], "meta": { "interimType": "Divide Both Sides Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7DejwV0bQN8/X7sW77OKBGOmFRfDC8eq4u98zQUOejTMOUg+KKUwy9kFBDzSx/bFX002SygbK5y4m+OEMFPvH+Cph9FNQkMoDWkmI8PfkOw/D2bCoz9MlEApa6cb6Dr5OpBYZtxaOD0wdVKSxV9MZ5sxuajNO/Clb3mUDmBKNGjyJzb4WJ2gItQG093k/9yqQn0DOUCKhNs+Rsl0Db7zAutv+C/gq4Or8Vp9qc4pKyQJ73JgXZDmpK/eAOUtCgEWOsIU6aaC5t5rGPz5/S19A8WZfqkaAoWwZzpVnzOGh6n02yaU49pEDc6Kv1qZQMlCoZp8KsmzEHn7YoHAtLg8Iaff63LdY40XnL0dUCK/3tYrfsSYC61BeyJuSoO+F6/0KD7pTaolgiN3kM+nQIfYulnVxoTNJoAt9cYYLWbfFUT0l9cyIf9ruxjEJoGIj/SsgInKBDGjlVaZ0zObB85VdeiTlMOO4Flv7vaqtDBlNwnXEFUyF8R257qqxiHuVL/hCwyFI6awgPlyKbcX2/xGzjgeAhEvZxsQ9kubrfhw5IjsbVisicFPt6Ju3hxiG8fAPNFAkR2ckSg80HygcZzT0U7yV64UUEyma1dEKidaHgBn73tRZCxmMu6XzR1u4HvD1rUjrJZ+WwWW3dvO1wqNupzpK9nhncQ6rzGxs9NSg3yjiH73l+44HWYZf+Vv64EMK" } } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve Title 1Eq" } }, { "type": "interim", "title": "Solve $$2\\sqrt{3}u-\\sqrt{11}=0:{\\quad}u=\\frac{\\sqrt{33}}{6}$$", "input": "2\\sqrt{3}u-\\sqrt{11}=0", "steps": [ { "type": "interim", "title": "Move $$\\sqrt{11}\\:$$to the right side", "input": "2\\sqrt{3}u-\\sqrt{11}=0", "result": "2\\sqrt{3}u=\\sqrt{11}", "steps": [ { "type": "step", "primary": "Add $$\\sqrt{11}$$ to both sides", "result": "2\\sqrt{3}u-\\sqrt{11}+\\sqrt{11}=0+\\sqrt{11}" }, { "type": "step", "primary": "Simplify", "result": "2\\sqrt{3}u=\\sqrt{11}" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7DejwV0bQN8/X7sW77OKBGKsIqpzDtSyn/SZIcJK+CsGvTjrEmD8o5mgacYosz9AGLe3ZZClxkCnFHP8TgmpWxYEy3pnZKbrxMKqksGcX2kiNHj5WqM/+D8y9VW3GmUmPyZIYiwcvDWlICwufraU8Hb5OEd3UGuDJRNcpCd+6z3SsHuCWujTcybm8y4lstc3AiiuTvHZMNMidREaJCrkWGcKYbdT6lJjVyUzTxq3e7tjCPNlUYj52tc4Y8B80I30BN1GUh5y/ArLUxZfIpo/yflK8vfC0n6Vf+b+jniaHbWt9TUqbZZFMlcv4c2DLWdwp5+BJD/K81r5iyb8yZXAreGk7sANVFNEZYaG+0BeCEq5NN5nGAWYciH3ut0cCYhXl7qpcs7N6I0H9bbdzYpIwSLfAhmbmQgG+lamDJtNoIIJXFb9/0oEug13geZKIC2HFzH3ZUtGHirnT7E0T0WD8mlBVm8vy0PbbAkF01+U5oPzkU5TkMMWES51yFsTkJT6lflB44+cAhbl05D7N1o4XnYUDxBuztWdWdEmOXBe3LLpWPNqzEhkP5qD6ioNU4s9fi9c/nHGhIqvPRpttdSi2lpJj/5xKKdB5FHwO4G/fHn8mxF5pxAejgMBdXf8NHl+mk3dK9hN6pPkzVNfz8kjct+cy2oW1fENZ7tPIWHFEzP+cqeFNvkW4u+eykK6QpWbUbqgCeJqTEq1trtv68VFB6w==" } }, { "type": "interim", "title": "Divide both sides by $$2\\sqrt{3}$$", "input": "2\\sqrt{3}u=\\sqrt{11}", "result": "u=\\frac{\\sqrt{33}}{6}", "steps": [ { "type": "step", "primary": "Divide both sides by $$2\\sqrt{3}$$", "result": "\\frac{2\\sqrt{3}u}{2\\sqrt{3}}=\\frac{\\sqrt{11}}{2\\sqrt{3}}" }, { "type": "interim", "title": "Simplify", "input": "\\frac{2\\sqrt{3}u}{2\\sqrt{3}}=\\frac{\\sqrt{11}}{2\\sqrt{3}}", "result": "u=\\frac{\\sqrt{33}}{6}", "steps": [ { "type": "interim", "title": "Simplify $$\\frac{2\\sqrt{3}u}{2\\sqrt{3}}:{\\quad}u$$", "input": "\\frac{2\\sqrt{3}u}{2\\sqrt{3}}", "steps": [ { "type": "step", "primary": "Divide the numbers: $$\\frac{2}{2}=1$$", "result": "=\\frac{\\sqrt{3}u}{\\sqrt{3}}" }, { "type": "step", "primary": "Cancel the common factor: $$\\sqrt{3}$$", "result": "=u" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7iA1jhOJsYg26KMh1oIvzcr+SrK6v43hdihJsRmAb11Z8kR7hsO/rTOTBE0w4+r1RYppoGJ3t+EWMwmq0/EQlQmRLd2VwIqlBNByF6663syR2SpdpleAJc7YgKUwBYoM9rKSn6S8yGX//uYoRDIG/ir+SrK6v43hdihJsRmAb11aJqVxX90jlMfh9fKn6dzC4" } }, { "type": "interim", "title": "Simplify $$\\frac{\\sqrt{11}}{2\\sqrt{3}}:{\\quad}\\frac{\\sqrt{33}}{6}$$", "input": "\\frac{\\sqrt{11}}{2\\sqrt{3}}", "steps": [ { "type": "step", "primary": "Multiply by the conjugate $$\\frac{\\sqrt{3}}{\\sqrt{3}}$$", "result": "=\\frac{\\sqrt{11}\\sqrt{3}}{2\\sqrt{3}\\sqrt{3}}", "meta": { "title": { "extension": "To rationalize the denominator, multiply numerator and denominator by the conjugate of the radical $$\\sqrt{3}$$" } } }, { "type": "interim", "title": "$$\\sqrt{11}\\sqrt{3}=\\sqrt{33}$$", "input": "\\sqrt{11}\\sqrt{3}", "steps": [ { "type": "step", "primary": "Apply radical rule: $$\\sqrt{a}\\sqrt{b}=\\sqrt{a\\cdot{b}}$$", "secondary": [ "$$\\sqrt{11}\\sqrt{3}=\\sqrt{11\\cdot\\:3}$$" ], "result": "=\\sqrt{11\\cdot\\:3}", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } }, { "type": "step", "primary": "Multiply the numbers: $$11\\cdot\\:3=33$$", "result": "=\\sqrt{33}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7OOrYgUTRfuJ+rEJC8tzMYCj7guf/B8PEHny4jV/tPelwkKGJWEPFPk38sdJMsyPIT6SAm4l5zkHCOVWGqGW3bYY+spUsB4yZJkDyh9+Pu39a7YlKPs+0J9nd5MP8YUNaHuV7PWraVhR7fPhyDwKS0w==" } }, { "type": "interim", "title": "$$2\\sqrt{3}\\sqrt{3}=6$$", "input": "2\\sqrt{3}\\sqrt{3}", "result": "=\\frac{\\sqrt{33}}{6}", "steps": [ { "type": "step", "primary": "Apply radical rule: $$\\sqrt{a}\\sqrt{a}=a$$", "secondary": [ "$$\\sqrt{3}\\sqrt{3}=3$$" ], "result": "=2\\cdot\\:3", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:3=6$$", "result": "=6" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7aTdZOU+fmdGxwfasVfdUfSj7guf/B8PEHny4jV/tPelwkKGJWEPFPk38sdJMsyPIOzRnKkVJMpyFFl92V2YjIEDScMkehj8ySs0hZkq/+teFKt1ZlkEUZE0U4/d5bZ1Y" } } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s74DwjiBDVBIb8+yh6fkkMMQiFZqUy6HPc0qktejQC2IcAlilG71elit3w1IBbYN0P8rEus7TgCihQBF5omOFkJrfAhmbmQgG+lamDJtNoIIKrys0VFr0ex+0yPqoQFyKdHimBRYRqHSWeJkuUPhfTC/j+KYYgjJt/QT9RFe9c9dPlSx8iuBJi65MK9MlcMqJUy/qAQLPGKf3UiV0iQ3ubxQ==" } }, { "type": "step", "result": "u=\\frac{\\sqrt{33}}{6}" } ], "meta": { "interimType": "Generic Simplify 0Eq" } } ], "meta": { "interimType": "Divide Both Sides Specific 1Eq", "gptData": "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" } } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve Title 1Eq" } }, { "type": "step", "primary": "The solutions are", "result": "u=0,\\:u=-\\frac{\\sqrt{33}}{6},\\:u=\\frac{\\sqrt{33}}{6}" } ], "meta": { "solvingClass": "Equations", "interimType": "Equations" } }, { "type": "step", "primary": "Substitute back $$u=\\cos\\left(θ\\right)$$", "result": "\\cos\\left(θ\\right)=0,\\:\\cos\\left(θ\\right)=-\\frac{\\sqrt{33}}{6},\\:\\cos\\left(θ\\right)=\\frac{\\sqrt{33}}{6}" } ], "meta": { "interimType": "Substitution Method 0Eq" } }, { "type": "interim", "title": "$$\\cos\\left(θ\\right)=0{\\quad:\\quad}θ=\\frac{π}{2}+2πn,\\:θ=\\frac{3π}{2}+2πn$$", "input": "\\cos\\left(θ\\right)=0", "steps": [ { "type": "interim", "title": "General solutions for $$\\cos\\left(θ\\right)=0$$", "result": "θ=\\frac{π}{2}+2πn,\\:θ=\\frac{3π}{2}+2πn", "steps": [ { "type": "step", "primary": "$$\\cos\\left(x\\right)$$ periodicity table with $$2πn$$ cycle:<br/>$$\\begin{array}{|c|c|c|c|}\\hline x&\\cos(x)&x&\\cos(x)\\\\\\hline 0&1&π&-1\\\\\\hline \\frac{π}{6}&\\frac{\\sqrt{3}}{2}&\\frac{7π}{6}&-\\frac{\\sqrt{3}}{2}\\\\\\hline \\frac{π}{4}&\\frac{\\sqrt{2}}{2}&\\frac{5π}{4}&-\\frac{\\sqrt{2}}{2}\\\\\\hline \\frac{π}{3}&\\frac{1}{2}&\\frac{4π}{3}&-\\frac{1}{2}\\\\\\hline \\frac{π}{2}&0&\\frac{3π}{2}&0\\\\\\hline \\frac{2π}{3}&-\\frac{1}{2}&\\frac{5π}{3}&\\frac{1}{2}\\\\\\hline \\frac{3π}{4}&-\\frac{\\sqrt{2}}{2}&\\frac{7π}{4}&\\frac{\\sqrt{2}}{2}\\\\\\hline \\frac{5π}{6}&-\\frac{\\sqrt{3}}{2}&\\frac{11π}{6}&\\frac{\\sqrt{3}}{2}\\\\\\hline \\end{array}$$" }, { "type": "step", "result": "θ=\\frac{π}{2}+2πn,\\:θ=\\frac{3π}{2}+2πn" } ], "meta": { "interimType": "Trig General Solutions cos 1Eq" } } ], "meta": { "interimType": "N/A" } }, { "type": "interim", "title": "$$\\cos\\left(θ\\right)=-\\frac{\\sqrt{33}}{6}{\\quad:\\quad}θ=\\arccos\\left(-\\frac{\\sqrt{33}}{6}\\right)+2πn,\\:θ=-\\arccos\\left(-\\frac{\\sqrt{33}}{6}\\right)+2πn$$", "input": "\\cos\\left(θ\\right)=-\\frac{\\sqrt{33}}{6}", "steps": [ { "type": "interim", "title": "Apply trig inverse properties", "input": "\\cos\\left(θ\\right)=-\\frac{\\sqrt{33}}{6}", "result": "θ=\\arccos\\left(-\\frac{\\sqrt{33}}{6}\\right)+2πn,\\:θ=-\\arccos\\left(-\\frac{\\sqrt{33}}{6}\\right)+2πn", "steps": [ { "type": "step", "primary": "General solutions for $$\\cos\\left(θ\\right)=-\\frac{\\sqrt{33}}{6}$$", "secondary": [ "$$\\cos\\left(x\\right)=-a\\quad\\Rightarrow\\quad\\:x=\\arccos\\left(-a\\right)+2πn,\\:\\quad\\:x=-\\arccos\\left(-a\\right)+2πn$$" ], "result": "θ=\\arccos\\left(-\\frac{\\sqrt{33}}{6}\\right)+2πn,\\:θ=-\\arccos\\left(-\\frac{\\sqrt{33}}{6}\\right)+2πn" } ], "meta": { "interimType": "Trig Apply Inverse Props 0Eq" } } ], "meta": { "interimType": "N/A" } }, { "type": "interim", "title": "$$\\cos\\left(θ\\right)=\\frac{\\sqrt{33}}{6}{\\quad:\\quad}θ=\\arccos\\left(\\frac{\\sqrt{33}}{6}\\right)+2πn,\\:θ=2π-\\arccos\\left(\\frac{\\sqrt{33}}{6}\\right)+2πn$$", "input": "\\cos\\left(θ\\right)=\\frac{\\sqrt{33}}{6}", "steps": [ { "type": "interim", "title": "Apply trig inverse properties", "input": "\\cos\\left(θ\\right)=\\frac{\\sqrt{33}}{6}", "result": "θ=\\arccos\\left(\\frac{\\sqrt{33}}{6}\\right)+2πn,\\:θ=2π-\\arccos\\left(\\frac{\\sqrt{33}}{6}\\right)+2πn", "steps": [ { "type": "step", "primary": "General solutions for $$\\cos\\left(θ\\right)=\\frac{\\sqrt{33}}{6}$$", "secondary": [ "$$\\cos\\left(x\\right)=a\\quad\\Rightarrow\\quad\\:x=\\arccos\\left(a\\right)+2πn,\\:\\quad\\:x=2π-\\arccos\\left(a\\right)+2πn$$" ], "result": "θ=\\arccos\\left(\\frac{\\sqrt{33}}{6}\\right)+2πn,\\:θ=2π-\\arccos\\left(\\frac{\\sqrt{33}}{6}\\right)+2πn" } ], "meta": { "interimType": "Trig Apply Inverse Props 0Eq" } } ], "meta": { "interimType": "N/A" } }, { "type": "step", "primary": "Combine all the solutions", "result": "θ=\\frac{π}{2}+2πn,\\:θ=\\frac{3π}{2}+2πn,\\:θ=\\arccos\\left(-\\frac{\\sqrt{33}}{6}\\right)+2πn,\\:θ=-\\arccos\\left(-\\frac{\\sqrt{33}}{6}\\right)+2πn,\\:θ=\\arccos\\left(\\frac{\\sqrt{33}}{6}\\right)+2πn,\\:θ=2π-\\arccos\\left(\\frac{\\sqrt{33}}{6}\\right)+2πn" }, { "type": "step", "primary": "Show solutions in decimal form", "result": "θ=\\frac{π}{2}+2πn,\\:θ=\\frac{3π}{2}+2πn,\\:θ=2.84874…+2πn,\\:θ=-2.84874…+2πn,\\:θ=0.29284…+2πn,\\:θ=2π-0.29284…+2πn" } ], "meta": { "solvingClass": "Trig Equations", "practiceLink": "/practice/trigonometry-practice#area=main&subtopic=Trig%20Equations", "practiceTopic": "Trig Equations" } }, "plot_output": { "meta": { "plotInfo": { "variable": "θ", "plotRequest": "2\\cos(θ)-3\\cos(3θ)" }, "showViewLarger": true } }, "meta": { "showVerify": true } }