cos^4(x)=1-8sin^2(x)cos^2(x)

{ "query": { "display": "$$\\cos^{4}\\left(x\\right)=1-8\\sin^{2}\\left(x\\right)\\cos^{2}\\left(x\\right)$$", "symbolab_question": "EQUATION#\\cos^{4}(x)=1-8\\sin^{2}(x)\\cos^{2}(x)" }, "solution": { "level": "PERFORMED", "subject": "Trigonometry", "topic": "Trig Equations", "subTopic": "Trig Equations", "default": "x=1.18319…+2πn,x=2π-1.18319…+2πn,x=1.95839…+2πn,x=-1.95839…+2πn,x=2πn,x=π+2πn", "degrees": "x=67.79234…^{\\circ }+360^{\\circ }n,x=292.20765…^{\\circ }+360^{\\circ }n,x=112.20765…^{\\circ }+360^{\\circ }n,x=-112.20765…^{\\circ }+360^{\\circ }n,x=0^{\\circ }+360^{\\circ }n,x=180^{\\circ }+360^{\\circ }n", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\cos^{4}\\left(x\\right)=1-8\\sin^{2}\\left(x\\right)\\cos^{2}\\left(x\\right){\\quad:\\quad}x=1.18319…+2πn,\\:x=2π-1.18319…+2πn,\\:x=1.95839…+2πn,\\:x=-1.95839…+2πn,\\:x=2πn,\\:x=π+2πn$$", "input": "\\cos^{4}\\left(x\\right)=1-8\\sin^{2}\\left(x\\right)\\cos^{2}\\left(x\\right)", "steps": [ { "type": "step", "primary": "Subtract $$1-8\\sin^{2}\\left(x\\right)\\cos^{2}\\left(x\\right)$$ from both sides", "result": "\\cos^{4}\\left(x\\right)-1+8\\sin^{2}\\left(x\\right)\\cos^{2}\\left(x\\right)=0" }, { "type": "interim", "title": "Rewrite using trig identities", "input": "-1+\\cos^{4}\\left(x\\right)+8\\cos^{2}\\left(x\\right)\\sin^{2}\\left(x\\right)", "result": "-1-7\\cos^{4}\\left(x\\right)+8\\cos^{2}\\left(x\\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": "=-1+\\cos^{4}\\left(x\\right)+8\\cos^{2}\\left(x\\right)\\left(1-\\cos^{2}\\left(x\\right)\\right)" }, { "type": "interim", "title": "Simplify $$-1+\\cos^{4}\\left(x\\right)+8\\cos^{2}\\left(x\\right)\\left(1-\\cos^{2}\\left(x\\right)\\right):{\\quad}-7\\cos^{4}\\left(x\\right)+8\\cos^{2}\\left(x\\right)-1$$", "input": "-1+\\cos^{4}\\left(x\\right)+8\\cos^{2}\\left(x\\right)\\left(1-\\cos^{2}\\left(x\\right)\\right)", "result": "=-7\\cos^{4}\\left(x\\right)+8\\cos^{2}\\left(x\\right)-1", "steps": [ { "type": "interim", "title": "Expand $$8\\cos^{2}\\left(x\\right)\\left(1-\\cos^{2}\\left(x\\right)\\right):{\\quad}8\\cos^{2}\\left(x\\right)-8\\cos^{4}\\left(x\\right)$$", "input": "8\\cos^{2}\\left(x\\right)\\left(1-\\cos^{2}\\left(x\\right)\\right)", "result": "=-1+\\cos^{4}\\left(x\\right)+8\\cos^{2}\\left(x\\right)-8\\cos^{4}\\left(x\\right)", "steps": [ { "type": "step", "primary": "Apply the distributive law: $$a\\left(b-c\\right)=ab-ac$$", "secondary": [ "$$a=8\\cos^{2}\\left(x\\right),\\:b=1,\\:c=\\cos^{2}\\left(x\\right)$$" ], "result": "=8\\cos^{2}\\left(x\\right)\\cdot\\:1-8\\cos^{2}\\left(x\\right)\\cos^{2}\\left(x\\right)", "meta": { "practiceLink": "/practice/expansion-practice", "practiceTopic": "Expand Rules" } }, { "type": "step", "result": "=8\\cdot\\:1\\cdot\\:\\cos^{2}\\left(x\\right)-8\\cos^{2}\\left(x\\right)\\cos^{2}\\left(x\\right)" }, { "type": "interim", "title": "Simplify $$8\\cdot\\:1\\cdot\\:\\cos^{2}\\left(x\\right)-8\\cos^{2}\\left(x\\right)\\cos^{2}\\left(x\\right):{\\quad}8\\cos^{2}\\left(x\\right)-8\\cos^{4}\\left(x\\right)$$", "input": "8\\cdot\\:1\\cdot\\:\\cos^{2}\\left(x\\right)-8\\cos^{2}\\left(x\\right)\\cos^{2}\\left(x\\right)", "result": "=8\\cos^{2}\\left(x\\right)-8\\cos^{4}\\left(x\\right)", "steps": [ { "type": "interim", "title": "$$8\\cdot\\:1\\cdot\\:\\cos^{2}\\left(x\\right)=8\\cos^{2}\\left(x\\right)$$", "input": "8\\cdot\\:1\\cdot\\:\\cos^{2}\\left(x\\right)", "steps": [ { "type": "step", "primary": "Multiply the numbers: $$8\\cdot\\:1=8$$", "result": "=8\\cos^{2}\\left(x\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7Bq686OP/7IAg/ySFBBKOHJa+YzW+tr7DkRknqB3er/BV00rpv8+ZC6TM10tVCSHsY4iiZsmRwy9aHQzhNlMmBzt5/8OORxN5fiIfVmhACJIKSxeyS74QDHegRWj8Z40sAbNA/JCldtsE68mebWGmVZ/hMVoFqzxKecssCAilUeDzEbe6tY97b8HX6Oo5G/x2" } }, { "type": "interim", "title": "$$8\\cos^{2}\\left(x\\right)\\cos^{2}\\left(x\\right)=8\\cos^{4}\\left(x\\right)$$", "input": "8\\cos^{2}\\left(x\\right)\\cos^{2}\\left(x\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\cos^{2}\\left(x\\right)\\cos^{2}\\left(x\\right)=\\:\\cos^{2+2}\\left(x\\right)$$" ], "result": "=8\\cos^{2+2}\\left(x\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$2+2=4$$", "result": "=8\\cos^{4}\\left(x\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7Bt8DCs9UyyXnQOwioeiSX0kyybNiG9VmH85fxHMAalTdd47a0hQ8flDbGsI5To1d+Z4XqDMwDnDPbDk6X/zqNKvlo2sl9eQ7wCsHwSaO+hOLCurFZgd80YYn9ptrZH0frpQEAC2HyLW9DhKvJtWw0ol3/byT2ucO83elyA50rl8=" } }, { "type": "step", "result": "=8\\cos^{2}\\left(x\\right)-8\\cos^{4}\\left(x\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } } ], "meta": { "interimType": "Algebraic Manipulation Expand Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7dLe8948d/vPWnQYN++Axxghut6CvUc61Zk6i+Q6zyAwtOtZYwUjyXhDTsNnn6ElrIx5qQTWVy0HGfjRKVYlv6T2BhI3PgbNjtThe7vDn/g0CCOlp++N9bpmT9cFR1wX9TeQKHeh69S6dnv9vSoUoFJDYL6OzKcZpZAAgQF/Aw9YxDS/430m4zxdLSStrgXh9HTlzACCiqhxcuWntH0mlhQ==" } }, { "type": "interim", "title": "Simplify $$-1+\\cos^{4}\\left(x\\right)+8\\cos^{2}\\left(x\\right)-8\\cos^{4}\\left(x\\right):{\\quad}-7\\cos^{4}\\left(x\\right)+8\\cos^{2}\\left(x\\right)-1$$", "input": "-1+\\cos^{4}\\left(x\\right)+8\\cos^{2}\\left(x\\right)-8\\cos^{4}\\left(x\\right)", "result": "=-7\\cos^{4}\\left(x\\right)+8\\cos^{2}\\left(x\\right)-1", "steps": [ { "type": "step", "primary": "Group like terms", "result": "=\\cos^{4}\\left(x\\right)+8\\cos^{2}\\left(x\\right)-8\\cos^{4}\\left(x\\right)-1" }, { "type": "step", "primary": "Add similar elements: $$\\cos^{4}\\left(x\\right)-8\\cos^{4}\\left(x\\right)=-7\\cos^{4}\\left(x\\right)$$", "result": "=-7\\cos^{4}\\left(x\\right)+8\\cos^{2}\\left(x\\right)-1" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7kYwwUV3fCVVXr+ORPx1wU8sIVLUBQWcFLlepDM6ZwdcIbregr1HOtWZOovkOs8gMLTrWWMFI8l4Q07DZ5+hJa0bQbcJ+CTcgH54y1JpyGkPYeWvRDoCEKqM6y++roA/gQoOJJPj2SsNefeq9NMKVh/C30sSftAIFS6Qkpy19Ikp30Q7hVBU2/JXqgWPYlzFUV1W1G9clO7wx1shEnSshwfOpcxfRWZ0OyzP0tCuyG2wgO+kN/ZPt2Z7TmhrY72Lv" } } ], "meta": { "interimType": "Trig Rewrite Using Trig identities 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7XQoqDkagBdkaGF3BP1M6VOv7ui3rZ/nXWl+MOfUJ9CZDXuFxlB40oTbmcw/dnIjIKc3wH80si0VAhXhVPUrr0Qi9Vnn4imguIoWn3N1eqjdGIkb3QW2w7y4N83RSktbc/kUupyp7xXxB8YmKEQ5HoFYAwrnanqhW5xWw9ebBKxfg0KtgR/fC/DdrhTCABpCZkLJY3LPe2m2/Cbkt64MT90mFP65WPUHCIGTpzT/EAccetg60+ZUlqP5jB+ReldUA5AMRxXWt+t9va0qZyagxyQ==" } }, { "type": "interim", "title": "Solve by substitution", "input": "-1-7\\cos^{4}\\left(x\\right)+8\\cos^{2}\\left(x\\right)=0", "result": "\\cos\\left(x\\right)=\\sqrt{\\frac{1}{7}},\\:\\cos\\left(x\\right)=-\\sqrt{\\frac{1}{7}},\\:\\cos\\left(x\\right)=1,\\:\\cos\\left(x\\right)=-1", "steps": [ { "type": "step", "primary": "Let: $$\\cos\\left(x\\right)=u$$", "result": "-1-7u^{4}+8u^{2}=0" }, { "type": "interim", "title": "$$-1-7u^{4}+8u^{2}=0{\\quad:\\quad}u=\\sqrt{\\frac{1}{7}},\\:u=-\\sqrt{\\frac{1}{7}},\\:u=1,\\:u=-1$$", "input": "-1-7u^{4}+8u^{2}=0", "steps": [ { "type": "step", "primary": "Write in the standard form $$a_{n}x^{n}+\\ldots\\:+a_{1}x+a_{0}=0$$", "result": "-7u^{4}+8u^{2}-1=0" }, { "type": "step", "primary": "Rewrite the equation with $$v=u^{2}$$ and $$v^{2}=u^{4}$$", "result": "-7v^{2}+8v-1=0" }, { "type": "interim", "title": "Solve $$-7v^{2}+8v-1=0:{\\quad}v=\\frac{1}{7},\\:v=1$$", "input": "-7v^{2}+8v-1=0", "steps": [ { "type": "interim", "title": "Solve with the quadratic formula", "input": "-7v^{2}+8v-1=0", "result": "{v}_{1,\\:2}=\\frac{-8\\pm\\:\\sqrt{8^{2}-4\\left(-7\\right)\\left(-1\\right)}}{2\\left(-7\\right)}", "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 $${\\quad}a=-7,\\:b=8,\\:c=-1$$", "result": "{v}_{1,\\:2}=\\frac{-8\\pm\\:\\sqrt{8^{2}-4\\left(-7\\right)\\left(-1\\right)}}{2\\left(-7\\right)}" } ], "meta": { "interimType": "Solving The Quadratic Equation With Quadratic Formula Definition 0Eq", "gptData": "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" } }, { "type": "interim", "title": "$$\\sqrt{8^{2}-4\\left(-7\\right)\\left(-1\\right)}=6$$", "input": "\\sqrt{8^{2}-4\\left(-7\\right)\\left(-1\\right)}", "result": "{v}_{1,\\:2}=\\frac{-8\\pm\\:6}{2\\left(-7\\right)}", "steps": [ { "type": "step", "primary": "Apply rule $$-\\left(-a\\right)=a$$", "result": "=\\sqrt{8^{2}-4\\cdot\\:7\\cdot\\:1}" }, { "type": "step", "primary": "Multiply the numbers: $$4\\cdot\\:7\\cdot\\:1=28$$", "result": "=\\sqrt{8^{2}-28}" }, { "type": "step", "primary": "$$8^{2}=64$$", "result": "=\\sqrt{64-28}" }, { "type": "step", "primary": "Subtract the numbers: $$64-28=36$$", "result": "=\\sqrt{36}" }, { "type": "step", "primary": "Factor the number: $$36=6^{2}$$", "result": "=\\sqrt{6^{2}}" }, { "type": "step", "primary": "Apply radical rule: $$\\sqrt[n]{a^n}=a$$", "secondary": [ "$$\\sqrt{6^{2}}=6$$" ], "result": "=6", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7XUwrx7ra00fs8AJJLipg0OsWeJHZgxv0a65S+PfYLEcDnzlbPZjyKgy1eUCFsLd5lsiuj99DBn8VKy1QP0O8ApfdFWg5APU0tVfpuarPdpDOcsORPc0POcS7DvTZCbPwsIjaxJ4DvjTb2fbKjbvtlQ==" } }, { "type": "step", "primary": "Separate the solutions", "result": "{v}_{1}=\\frac{-8+6}{2\\left(-7\\right)},\\:{v}_{2}=\\frac{-8-6}{2\\left(-7\\right)}" }, { "type": "interim", "title": "$$v=\\frac{-8+6}{2\\left(-7\\right)}:{\\quad}\\frac{1}{7}$$", "input": "\\frac{-8+6}{2\\left(-7\\right)}", "steps": [ { "type": "step", "primary": "Remove parentheses: $$\\left(-a\\right)=-a$$", "result": "=\\frac{-8+6}{-2\\cdot\\:7}" }, { "type": "step", "primary": "Add/Subtract the numbers: $$-8+6=-2$$", "result": "=\\frac{-2}{-2\\cdot\\:7}" }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:7=14$$", "result": "=\\frac{-2}{-14}" }, { "type": "step", "primary": "Apply the fraction rule: $$\\frac{-a}{-b}=\\frac{a}{b}$$", "result": "=\\frac{2}{14}" }, { "type": "step", "primary": "Cancel the common factor: $$2$$", "result": "=\\frac{1}{7}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s71ON1pP8C8WiNjVSl7184Jn+h6uF6hbVZ9m5i5es4MbZwkKGJWEPFPk38sdJMsyPIS8bcxMG+aTtpoaeacjKEVddvA0B8kANhaIQeN+eC9yqMrO8TZqgLnveehShu5LqGXnaKrZn5f8vv2EyxI1+wkA==" } }, { "type": "interim", "title": "$$v=\\frac{-8-6}{2\\left(-7\\right)}:{\\quad}1$$", "input": "\\frac{-8-6}{2\\left(-7\\right)}", "steps": [ { "type": "step", "primary": "Remove parentheses: $$\\left(-a\\right)=-a$$", "result": "=\\frac{-8-6}{-2\\cdot\\:7}" }, { "type": "step", "primary": "Subtract the numbers: $$-8-6=-14$$", "result": "=\\frac{-14}{-2\\cdot\\:7}" }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:7=14$$", "result": "=\\frac{-14}{-14}" }, { "type": "step", "primary": "Apply the fraction rule: $$\\frac{-a}{-b}=\\frac{a}{b}$$", "result": "=\\frac{14}{14}" }, { "type": "step", "primary": "Apply rule $$\\frac{a}{a}=1$$", "result": "=1" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7xMXnQwa4a5PDKSUOIbB7QH+h6uF6hbVZ9m5i5es4MbZwkKGJWEPFPk38sdJMsyPIeqXfySbC6vm4UawE43QWXWlj9gus2oPYlXAufhA2c5wuLhtk715Lv2PR7LPx9Xxw" } }, { "type": "step", "primary": "The solutions to the quadratic equation are:", "result": "v=\\frac{1}{7},\\:v=1" } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve Title 1Eq" } }, { "type": "step", "result": "v=\\frac{1}{7},\\:v=1" }, { "type": "step", "primary": "Substitute back $$v=u^{2},\\:$$solve for $$u$$" }, { "type": "interim", "title": "Solve $$u^{2}=\\frac{1}{7}:{\\quad}u=\\sqrt{\\frac{1}{7}},\\:u=-\\sqrt{\\frac{1}{7}}$$", "input": "u^{2}=\\frac{1}{7}", "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}{7}},\\:u=-\\sqrt{\\frac{1}{7}}" } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve Title 1Eq" } }, { "type": "interim", "title": "Solve $$u^{2}=1:{\\quad}u=1,\\:u=-1$$", "input": "u^{2}=1", "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{1},\\:u=-\\sqrt{1}" }, { "type": "interim", "title": "$$\\sqrt{1}=1$$", "input": "\\sqrt{1}", "steps": [ { "type": "step", "primary": "Apply rule $$\\sqrt{1}=1$$", "result": "=1" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7KfzlHGGU7KN8vfEO0eL8NN13jtrSFDx+UNsawjlOjV3ZuCguaNudj5qbY1K8A+fScubCnYZOJ5L8/2gsdymw1PSOscTE6qsKVI9GkIdY/eI=" } }, { "type": "interim", "title": "$$-\\sqrt{1}=-1$$", "input": "-\\sqrt{1}", "steps": [ { "type": "step", "primary": "Apply rule $$\\sqrt{1}=1$$", "result": "=-1" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7kWDE1Jsjy5jGSP2mctwcnCAn9lkDfZkicUGkO3EF+IpIQKToZa7Vmz9RWrIHzooCMHIu6EZfZrJ7HpyNTqg74lPlyk515FWfACaTxs0eUEM=" } }, { "type": "step", "result": "u=1,\\:u=-1" } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve Title 1Eq" } }, { "type": "step", "primary": "The solutions are" }, { "type": "step", "result": "u=\\sqrt{\\frac{1}{7}},\\:u=-\\sqrt{\\frac{1}{7}},\\:u=1,\\:u=-1" } ], "meta": { "solvingClass": "Equations", "interimType": "Equations" } }, { "type": "step", "primary": "Substitute back $$u=\\cos\\left(x\\right)$$", "result": "\\cos\\left(x\\right)=\\sqrt{\\frac{1}{7}},\\:\\cos\\left(x\\right)=-\\sqrt{\\frac{1}{7}},\\:\\cos\\left(x\\right)=1,\\:\\cos\\left(x\\right)=-1" } ], "meta": { "interimType": "Substitution Method 0Eq" } }, { "type": "interim", "title": "$$\\cos\\left(x\\right)=\\sqrt{\\frac{1}{7}}{\\quad:\\quad}x=\\arccos\\left(\\sqrt{\\frac{1}{7}}\\right)+2πn,\\:x=2π-\\arccos\\left(\\sqrt{\\frac{1}{7}}\\right)+2πn$$", "input": "\\cos\\left(x\\right)=\\sqrt{\\frac{1}{7}}", "steps": [ { "type": "interim", "title": "Apply trig inverse properties", "input": "\\cos\\left(x\\right)=\\sqrt{\\frac{1}{7}}", "result": "x=\\arccos\\left(\\sqrt{\\frac{1}{7}}\\right)+2πn,\\:x=2π-\\arccos\\left(\\sqrt{\\frac{1}{7}}\\right)+2πn", "steps": [ { "type": "step", "primary": "General solutions for $$\\cos\\left(x\\right)=\\sqrt{\\frac{1}{7}}$$", "secondary": [ "$$\\cos\\left(x\\right)=a\\quad\\Rightarrow\\quad\\:x=\\arccos\\left(a\\right)+2πn,\\:\\quad\\:x=2π-\\arccos\\left(a\\right)+2πn$$" ], "result": "x=\\arccos\\left(\\sqrt{\\frac{1}{7}}\\right)+2πn,\\:x=2π-\\arccos\\left(\\sqrt{\\frac{1}{7}}\\right)+2πn" } ], "meta": { "interimType": "Trig Apply Inverse Props 0Eq" } } ], "meta": { "interimType": "N/A" } }, { "type": "interim", "title": "$$\\cos\\left(x\\right)=-\\sqrt{\\frac{1}{7}}{\\quad:\\quad}x=\\arccos\\left(-\\sqrt{\\frac{1}{7}}\\right)+2πn,\\:x=-\\arccos\\left(-\\sqrt{\\frac{1}{7}}\\right)+2πn$$", "input": "\\cos\\left(x\\right)=-\\sqrt{\\frac{1}{7}}", "steps": [ { "type": "interim", "title": "Apply trig inverse properties", "input": "\\cos\\left(x\\right)=-\\sqrt{\\frac{1}{7}}", "result": "x=\\arccos\\left(-\\sqrt{\\frac{1}{7}}\\right)+2πn,\\:x=-\\arccos\\left(-\\sqrt{\\frac{1}{7}}\\right)+2πn", "steps": [ { "type": "step", "primary": "General solutions for $$\\cos\\left(x\\right)=-\\sqrt{\\frac{1}{7}}$$", "secondary": [ "$$\\cos\\left(x\\right)=-a\\quad\\Rightarrow\\quad\\:x=\\arccos\\left(-a\\right)+2πn,\\:\\quad\\:x=-\\arccos\\left(-a\\right)+2πn$$" ], "result": "x=\\arccos\\left(-\\sqrt{\\frac{1}{7}}\\right)+2πn,\\:x=-\\arccos\\left(-\\sqrt{\\frac{1}{7}}\\right)+2πn" } ], "meta": { "interimType": "Trig Apply Inverse Props 0Eq" } } ], "meta": { "interimType": "N/A" } }, { "type": "interim", "title": "$$\\cos\\left(x\\right)=1{\\quad:\\quad}x=2πn$$", "input": "\\cos\\left(x\\right)=1", "steps": [ { "type": "interim", "title": "General solutions for $$\\cos\\left(x\\right)=1$$", "result": "x=0+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": "x=0+2πn" } ], "meta": { "interimType": "Trig General Solutions cos 1Eq" } }, { "type": "interim", "title": "Solve $$x=0+2πn:{\\quad}x=2πn$$", "input": "x=0+2πn", "steps": [ { "type": "step", "primary": "$$0+2πn=2πn$$", "result": "x=2πn" } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve Title 1Eq" } }, { "type": "step", "result": "x=2πn" } ], "meta": { "interimType": "N/A" } }, { "type": "interim", "title": "$$\\cos\\left(x\\right)=-1{\\quad:\\quad}x=π+2πn$$", "input": "\\cos\\left(x\\right)=-1", "steps": [ { "type": "interim", "title": "General solutions for $$\\cos\\left(x\\right)=-1$$", "result": "x=π+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": "x=π+2πn" } ], "meta": { "interimType": "Trig General Solutions cos 1Eq" } } ], "meta": { "interimType": "N/A" } }, { "type": "step", "primary": "Combine all the solutions", "result": "x=\\arccos\\left(\\sqrt{\\frac{1}{7}}\\right)+2πn,\\:x=2π-\\arccos\\left(\\sqrt{\\frac{1}{7}}\\right)+2πn,\\:x=\\arccos\\left(-\\sqrt{\\frac{1}{7}}\\right)+2πn,\\:x=-\\arccos\\left(-\\sqrt{\\frac{1}{7}}\\right)+2πn,\\:x=2πn,\\:x=π+2πn" }, { "type": "step", "primary": "Show solutions in decimal form", "result": "x=1.18319…+2πn,\\:x=2π-1.18319…+2πn,\\:x=1.95839…+2πn,\\:x=-1.95839…+2πn,\\:x=2πn,\\:x=π+2πn" } ], "meta": { "solvingClass": "Trig Equations", "practiceLink": "/practice/trigonometry-practice#area=main&subtopic=Trig%20Equations", "practiceTopic": "Trig Equations" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "\\cos^{4}(x)-1+8\\sin^{2}(x)\\cos^{2}(x)" }, "showViewLarger": true } }, "meta": { "showVerify": true } }