cos^3(x)+cos^2(x)-cos(x)-1=0

{ "query": { "display": "$$\\cos^{3}\\left(x\\right)+\\cos^{2}\\left(x\\right)-\\cos\\left(x\\right)-1=0$$", "symbolab_question": "EQUATION#\\cos^{3}(x)+\\cos^{2}(x)-\\cos(x)-1=0" }, "solution": { "level": "PERFORMED", "subject": "Trigonometry", "topic": "Trig Equations", "subTopic": "Trig Equations", "default": "x=π+2πn,x=2πn", "degrees": "x=180^{\\circ }+360^{\\circ }n,x=0^{\\circ }+360^{\\circ }n", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\cos^{3}\\left(x\\right)+\\cos^{2}\\left(x\\right)-\\cos\\left(x\\right)-1=0{\\quad:\\quad}x=π+2πn,\\:x=2πn$$", "input": "\\cos^{3}\\left(x\\right)+\\cos^{2}\\left(x\\right)-\\cos\\left(x\\right)-1=0", "steps": [ { "type": "interim", "title": "Solve by substitution", "input": "\\cos^{3}\\left(x\\right)+\\cos^{2}\\left(x\\right)-\\cos\\left(x\\right)-1=0", "result": "\\cos\\left(x\\right)=-1,\\:\\cos\\left(x\\right)=1", "steps": [ { "type": "step", "primary": "Let: $$\\cos\\left(x\\right)=u$$", "result": "u^{3}+u^{2}-u-1=0" }, { "type": "interim", "title": "$$u^{3}+u^{2}-u-1=0{\\quad:\\quad}u=-1,\\:u=1$$", "input": "u^{3}+u^{2}-u-1=0", "steps": [ { "type": "interim", "title": "Factor $$u^{3}+u^{2}-u-1:{\\quad}\\left(u+1\\right)^{2}\\left(u-1\\right)$$", "input": "u^{3}+u^{2}-u-1", "steps": [ { "type": "step", "result": "=\\left(u^{3}+u^{2}\\right)+\\left(-u-1\\right)" }, { "type": "interim", "title": "Factor out $$-1\\:$$from $$-u-1:\\quad\\:-\\left(u+1\\right)$$", "input": "-u-1", "steps": [ { "type": "step", "primary": "Factor out common term $$-1$$", "result": "=-\\left(u+1\\right)", "meta": { "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } } ], "meta": { "interimType": "Factor Out Specific 3Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7nTAMxQPuEJ2N/MP6L/aFUMevWYjE5TzvOKvzjFpIz0IByGqvpVrcovBtOEfrP9G1txRRXbKKq8tv3GwNCH6E9aN6Hv6MoTMtvtU0IQwXdn/LaG2/wGnl6dNw3H9m17XsThnUlbyBEfXJCWlCtkCLAZWuFgPzapB5Djc2fCE3REg=" } }, { "type": "interim", "title": "Factor out $$u^{2}\\:$$from $$u^{3}+u^{2}:\\quad\\:u^{2}\\left(u+1\\right)$$", "input": "u^{3}+u^{2}", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^{b+c}=a^{b}a^{c}$$", "secondary": [ "$$u^{3}=uu^{2}$$" ], "result": "=uu^{2}+u^{2}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Factor out common term $$u^{2}$$", "result": "=u^{2}\\left(u+1\\right)", "meta": { "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } } ], "meta": { "interimType": "Factor Out 3Eq" } }, { "type": "step", "result": "=-\\left(u+1\\right)+u^{2}\\left(u+1\\right)" }, { "type": "step", "primary": "Factor out common term $$u+1$$", "result": "=\\left(u+1\\right)\\left(u^{2}-1\\right)", "meta": { "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } }, { "type": "interim", "title": "Factor $$u^{2}-1:{\\quad}\\left(u+1\\right)\\left(u-1\\right)$$", "input": "u^{2}-1", "steps": [ { "type": "step", "primary": "Rewrite $$1$$ as $$1^{2}$$", "result": "=u^{2}-1^{2}" }, { "type": "step", "primary": "Apply Difference of Two Squares Formula: $$x^{2}-y^{2}=\\left(x+y\\right)\\left(x-y\\right)$$", "secondary": [ "$$u^{2}-1^{2}=\\left(u+1\\right)\\left(u-1\\right)$$" ], "result": "=\\left(u+1\\right)\\left(u-1\\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": "=\\left(u+1\\right)\\left(u+1\\right)\\left(u-1\\right)" }, { "type": "step", "primary": "Refine", "result": "=\\left(u+1\\right)^{2}\\left(u-1\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Factor Specific 1Eq" } }, { "type": "step", "result": "\\left(u+1\\right)^{2}\\left(u-1\\right)=0" }, { "type": "step", "primary": "Using the Zero Factor Principle:$$\\quad$$ If $$ab=0\\:$$then $$a=0\\:$$or $$b=0$$", "result": "u+1=0\\lor\\:u-1=0" }, { "type": "interim", "title": "Solve $$u+1=0:{\\quad}u=-1$$", "input": "u+1=0", "steps": [ { "type": "interim", "title": "Move $$1\\:$$to the right side", "input": "u+1=0", "result": "u=-1", "steps": [ { "type": "step", "primary": "Subtract $$1$$ from both sides", "result": "u+1-1=0-1" }, { "type": "step", "primary": "Simplify", "result": "u=-1" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": "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" } } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve Title 1Eq" } }, { "type": "interim", "title": "Solve $$u-1=0:{\\quad}u=1$$", "input": "u-1=0", "steps": [ { "type": "interim", "title": "Move $$1\\:$$to the right side", "input": "u-1=0", "result": "u=1", "steps": [ { "type": "step", "primary": "Add $$1$$ to both sides", "result": "u-1+1=0+1" }, { "type": "step", "primary": "Simplify", "result": "u=1" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": "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" } } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve Title 1Eq" } }, { "type": "step", "primary": "The solutions are", "result": "u=-1,\\:u=1" } ], "meta": { "solvingClass": "Equations", "interimType": "Equations" } }, { "type": "step", "primary": "Substitute back $$u=\\cos\\left(x\\right)$$", "result": "\\cos\\left(x\\right)=-1,\\:\\cos\\left(x\\right)=1" } ], "meta": { "interimType": "Substitution Method 0Eq" } }, { "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": "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": "step", "primary": "Combine all the solutions", "result": "x=π+2πn,\\:x=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^{3}(x)+\\cos^{2}(x)-\\cos(x)-1" }, "showViewLarger": true } }, "meta": { "showVerify": true } }