cos(2t)-cos(t)=0.5

{ "query": { "display": "$$\\cos\\left(2t\\right)-\\cos\\left(t\\right)=0.5$$", "symbolab_question": "EQUATION#\\cos(2t)-\\cos(t)=0.5" }, "solution": { "level": "PERFORMED", "subject": "Trigonometry", "topic": "Trig Equations", "subTopic": "Trig Equations", "default": "t=2.28020…+2πn,t=-2.28020…+2πn", "degrees": "t=130.64631…^{\\circ }+360^{\\circ }n,t=-130.64631…^{\\circ }+360^{\\circ }n", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\cos\\left(2t\\right)-\\cos\\left(t\\right)=0.5{\\quad:\\quad}t=2.28020…+2πn,\\:t=-2.28020…+2πn$$", "input": "\\cos\\left(2t\\right)-\\cos\\left(t\\right)=0.5", "steps": [ { "type": "step", "primary": "Subtract $$0.5$$ from both sides", "result": "\\cos\\left(2t\\right)-\\cos\\left(t\\right)-0.5=0" }, { "type": "interim", "title": "Rewrite using trig identities", "input": "-0.5+\\cos\\left(2t\\right)-\\cos\\left(t\\right)", "result": "-1.5-\\cos\\left(t\\right)+2\\cos^{2}\\left(t\\right)=0", "steps": [ { "type": "step", "primary": "Use the Double Angle identity: $$\\cos\\left(2x\\right)=2\\cos^{2}\\left(x\\right)-1$$", "result": "=-0.5+2\\cos^{2}\\left(t\\right)-1-\\cos\\left(t\\right)" }, { "type": "interim", "title": "Simplify $$-0.5+2\\cos^{2}\\left(t\\right)-1-\\cos\\left(t\\right):{\\quad}2\\cos^{2}\\left(t\\right)-\\cos\\left(t\\right)-1.5$$", "input": "-0.5+2\\cos^{2}\\left(t\\right)-1-\\cos\\left(t\\right)", "result": "=2\\cos^{2}\\left(t\\right)-\\cos\\left(t\\right)-1.5", "steps": [ { "type": "step", "primary": "Group like terms", "result": "=2\\cos^{2}\\left(t\\right)-\\cos\\left(t\\right)-0.5-1" }, { "type": "step", "primary": "Subtract the numbers: $$-0.5-1=-1.5$$", "result": "=2\\cos^{2}\\left(t\\right)-\\cos\\left(t\\right)-1.5" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7BqoZ2G4jxp4E6VyWXdlno9j8zfwYjEGCr5E33o5mvjMtOtZYwUjyXhDTsNnn6ElriOB8hTxY3wN8anamyuwHt1vwNn2oBW9s42jy6BdZRon5c/qFgsAjumWdecXSXImfHjb2+5NLFZrsH9fcPWg/TZhli5XxLXqgnZrAO26BLBqgp7Nx+qXcYrKAbQBXv+X7PH5ZD6aFl7Cbfthb/5gVxA==" } } ], "meta": { "interimType": "Trig Rewrite Using Trig identities 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7rrqBN2OyzpnjR5QUtF+XjQB5F+rwllWIX7ue4YyB0LQUOLSbMox6vSWmoo6JD5Rw6oDeFBEXdI8LVv9QsNOFoRacrm3LsJ1aIjJN65AlUxeJDKsu+SuqQAE/BN/8mXKTenAS7uHRcrNuY/YiV2CuSe/i1qBOrdmwxPB+GDbCRqnwt9LEn7QCBUukJKctfSJKk2VQdqPGsmYTaYyesllHTRc4KXs8xgJNiLkJwr0b/5OJqVxX90jlMfh9fKn6dzC4" } }, { "type": "interim", "title": "Solve by substitution", "input": "-1.5-\\cos\\left(t\\right)+2\\cos^{2}\\left(t\\right)=0", "result": "\\cos\\left(t\\right)=\\frac{1+\\sqrt{13}}{4},\\:\\cos\\left(t\\right)=\\frac{1-\\sqrt{13}}{4}", "steps": [ { "type": "step", "primary": "Let: $$\\cos\\left(t\\right)=u$$", "result": "-1.5-u+2u^{2}=0" }, { "type": "interim", "title": "$$-1.5-u+2u^{2}=0{\\quad:\\quad}u=\\frac{1+\\sqrt{13}}{4},\\:u=\\frac{1-\\sqrt{13}}{4}$$", "input": "-1.5-u+2u^{2}=0", "steps": [ { "type": "interim", "title": "Multiply both sides by $$10$$", "input": "-1.5-u+2u^{2}=0", "result": "-15-10u+20u^{2}=0", "steps": [ { "type": "step", "primary": "To eliminate decimal points, multiply by 10 for every digit after the decimal point", "secondary": [ "There is one digit to the right of the decimal point, therefore multiply by $$10$$" ], "result": "-1.5\\cdot\\:10-u\\cdot\\:10+2u^{2}\\cdot\\:10=0\\cdot\\:10" }, { "type": "step", "primary": "Refine", "result": "-15-10u+20u^{2}=0" } ], "meta": { "interimType": "Multiply Both Sides Specific 1Eq", "gptData": "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" } }, { "type": "step", "primary": "Write in the standard form $$ax^{2}+bx+c=0$$", "result": "20u^{2}-10u-15=0" }, { "type": "interim", "title": "Solve with the quadratic formula", "input": "20u^{2}-10u-15=0", "result": "{u}_{1,\\:2}=\\frac{-\\left(-10\\right)\\pm\\:\\sqrt{\\left(-10\\right)^{2}-4\\cdot\\:20\\left(-15\\right)}}{2\\cdot\\:20}", "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 $${\\quad}a=20,\\:b=-10,\\:c=-15$$", "result": "{u}_{1,\\:2}=\\frac{-\\left(-10\\right)\\pm\\:\\sqrt{\\left(-10\\right)^{2}-4\\cdot\\:20\\left(-15\\right)}}{2\\cdot\\:20}" } ], "meta": { "interimType": "Solving The Quadratic Equation With Quadratic Formula Definition 0Eq", "gptData": "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" } }, { "type": "interim", "title": "$$\\sqrt{\\left(-10\\right)^{2}-4\\cdot\\:20\\left(-15\\right)}=10\\sqrt{13}$$", "input": "\\sqrt{\\left(-10\\right)^{2}-4\\cdot\\:20\\left(-15\\right)}", "result": "{u}_{1,\\:2}=\\frac{-\\left(-10\\right)\\pm\\:10\\sqrt{13}}{2\\cdot\\:20}", "steps": [ { "type": "step", "primary": "Apply rule $$-\\left(-a\\right)=a$$", "result": "=\\sqrt{\\left(-10\\right)^{2}+4\\cdot\\:20\\cdot\\:15}" }, { "type": "step", "primary": "Apply exponent rule: $$\\left(-a\\right)^{n}=a^{n},\\:$$if $$n$$ is even", "secondary": [ "$$\\left(-10\\right)^{2}=10^{2}$$" ], "result": "=\\sqrt{10^{2}+4\\cdot\\:20\\cdot\\:15}" }, { "type": "step", "primary": "Multiply the numbers: $$4\\cdot\\:20\\cdot\\:15=1200$$", "result": "=\\sqrt{10^{2}+1200}" }, { "type": "step", "primary": "$$10^{2}=100$$", "result": "=\\sqrt{100+1200}" }, { "type": "step", "primary": "Add the numbers: $$100+1200=1300$$", "result": "=\\sqrt{1300}" }, { "type": "interim", "title": "Prime factorization of $$1300:{\\quad}2^{2}\\cdot\\:5^{2}\\cdot\\:13$$", "input": "1300", "result": "=\\sqrt{2^{2}\\cdot\\:5^{2}\\cdot\\:13}", "steps": [ { "type": "step", "primary": "$$1300\\:$$divides by $$2\\quad\\:1300=650\\cdot\\:2$$", "result": "=2\\cdot\\:650" }, { "type": "step", "primary": "$$650\\:$$divides by $$2\\quad\\:650=325\\cdot\\:2$$", "result": "=2\\cdot\\:2\\cdot\\:325" }, { "type": "step", "primary": "$$325\\:$$divides by $$5\\quad\\:325=65\\cdot\\:5$$", "result": "=2\\cdot\\:2\\cdot\\:5\\cdot\\:65" }, { "type": "step", "primary": "$$65\\:$$divides by $$5\\quad\\:65=13\\cdot\\:5$$", "result": "=2\\cdot\\:2\\cdot\\:5\\cdot\\:5\\cdot\\:13" }, { "type": "step", "primary": "$$2,\\:5,\\:13$$ are all prime numbers, therefore no further factorization is possible", "result": "=2\\cdot\\:2\\cdot\\:5\\cdot\\:5\\cdot\\:13" }, { "type": "step", "result": "=2^{2}\\cdot\\:5^{2}\\cdot\\:13" } ], "meta": { "solvingClass": "Composite Integer", "interimType": "Prime Fac 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xyoU2cWyPLDgE1QLLHeauuc3PHQdChPJ2JhfqHT+ZU0OMrfn8NOj0LUzuzje6xTyxRvTIRluRNPwUULD5JCqpmdzlQpqXZdOp1ODEbM70NuQa+ZE2CuWYXhz/1zRh+DBwSWJYgvXjl1/GaaWGY6UA4hGVE/90DxkmiXfmFDUpmLWr" } }, { "type": "step", "primary": "Apply radical rule: $$\\sqrt[n]{ab}=\\sqrt[n]{a}\\sqrt[n]{b}$$", "result": "=\\sqrt{13}\\sqrt{2^{2}}\\sqrt{5^{2}}", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } }, { "type": "step", "primary": "Apply radical rule: $$\\sqrt[n]{a^n}=a$$", "secondary": [ "$$\\sqrt{2^{2}}=2$$" ], "result": "=2\\sqrt{13}\\sqrt{5^{2}}", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } }, { "type": "step", "primary": "Apply radical rule: $$\\sqrt[n]{a^n}=a$$", "secondary": [ "$$\\sqrt{5^{2}}=5$$" ], "result": "=2\\cdot\\:5\\sqrt{13}", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } }, { "type": "step", "primary": "Refine", "result": "=10\\sqrt{13}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7xYKyhiwThaKsz2Tl55KkNCSDE5ZXiT7i0G6OHwfSBwlucrZDjeKCXHUZNhNytfd/o5FYteSPKwXny4uCMrdsKzbZ4FbVup0SkWLdsUlh7BiGPrKVLAeMmSZA8offj7t/W80sdbB1u2v2qyNh5c5IvLFqBAt5T67Tq8n98acoS6mvdIV+S/e6Y0xIlZs2Mpn1" } }, { "type": "step", "primary": "Separate the solutions", "result": "{u}_{1}=\\frac{-\\left(-10\\right)+10\\sqrt{13}}{2\\cdot\\:20},\\:{u}_{2}=\\frac{-\\left(-10\\right)-10\\sqrt{13}}{2\\cdot\\:20}" }, { "type": "interim", "title": "$$u=\\frac{-\\left(-10\\right)+10\\sqrt{13}}{2\\cdot\\:20}:{\\quad}\\frac{1+\\sqrt{13}}{4}$$", "input": "\\frac{-\\left(-10\\right)+10\\sqrt{13}}{2\\cdot\\:20}", "steps": [ { "type": "step", "primary": "Apply rule $$-\\left(-a\\right)=a$$", "result": "=\\frac{10+10\\sqrt{13}}{2\\cdot\\:20}" }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:20=40$$", "result": "=\\frac{10+10\\sqrt{13}}{40}" }, { "type": "interim", "title": "Factor $$10+10\\sqrt{13}:{\\quad}10\\left(1+\\sqrt{13}\\right)$$", "input": "10+10\\sqrt{13}", "result": "=\\frac{10\\left(1+\\sqrt{13}\\right)}{40}", "steps": [ { "type": "step", "primary": "Rewrite as", "result": "=10\\cdot\\:1+10\\sqrt{13}" }, { "type": "step", "primary": "Factor out common term $$10$$", "result": "=10\\left(1+\\sqrt{13}\\right)", "meta": { "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } } ], "meta": { "interimType": "Algebraic Manipulation Factor Title 1Eq" } }, { "type": "step", "primary": "Cancel the common factor: $$10$$", "result": "=\\frac{1+\\sqrt{13}}{4}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79/ul7VUbFVJ26S67qVBPqYW5HOGna/RaXa/Gk09/Z4PWpsRiJtjAg9hOGwXNhKOvA585Wz2Y8ioMtXlAhbC3eT/T0/f0601DgXfK88BhMN23W9KZmrXNYoAN18UqX8DRQWI3C7Ag9M6KUYO8l4hxQHrSBc9g/Ric4s/oUP5h9w6otGg9ku5UmRJdiMk4VVplDXnKs87BlN0+mj3MLu4hIw==" } }, { "type": "interim", "title": "$$u=\\frac{-\\left(-10\\right)-10\\sqrt{13}}{2\\cdot\\:20}:{\\quad}\\frac{1-\\sqrt{13}}{4}$$", "input": "\\frac{-\\left(-10\\right)-10\\sqrt{13}}{2\\cdot\\:20}", "steps": [ { "type": "step", "primary": "Apply rule $$-\\left(-a\\right)=a$$", "result": "=\\frac{10-10\\sqrt{13}}{2\\cdot\\:20}" }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:20=40$$", "result": "=\\frac{10-10\\sqrt{13}}{40}" }, { "type": "interim", "title": "Factor $$10-10\\sqrt{13}:{\\quad}10\\left(1-\\sqrt{13}\\right)$$", "input": "10-10\\sqrt{13}", "result": "=\\frac{10\\left(1-\\sqrt{13}\\right)}{40}", "steps": [ { "type": "step", "primary": "Rewrite as", "result": "=10\\cdot\\:1-10\\sqrt{13}" }, { "type": "step", "primary": "Factor out common term $$10$$", "result": "=10\\left(1-\\sqrt{13}\\right)", "meta": { "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } } ], "meta": { "interimType": "Algebraic Manipulation Factor Title 1Eq" } }, { "type": "step", "primary": "Cancel the common factor: $$10$$", "result": "=\\frac{1-\\sqrt{13}}{4}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7fFFzGKGpQqH1+arANkx3PoW5HOGna/RaXa/Gk09/Z4PWpsRiJtjAg9hOGwXNhKOvA585Wz2Y8ioMtXlAhbC3eVgQFGmNsbNcgLmy9BJd1ju3W9KZmrXNYoAN18UqX8DRQWI3C7Ag9M6KUYO8l4hxQOz1AwEVb+AUK8qLtzuQnMGotGg9ku5UmRJdiMk4VVplDXnKs87BlN0+mj3MLu4hIw==" } }, { "type": "step", "primary": "The solutions to the quadratic equation are:", "result": "u=\\frac{1+\\sqrt{13}}{4},\\:u=\\frac{1-\\sqrt{13}}{4}" } ], "meta": { "solvingClass": "Equations", "interimType": "Equations" } }, { "type": "step", "primary": "Substitute back $$u=\\cos\\left(t\\right)$$", "result": "\\cos\\left(t\\right)=\\frac{1+\\sqrt{13}}{4},\\:\\cos\\left(t\\right)=\\frac{1-\\sqrt{13}}{4}" } ], "meta": { "interimType": "Substitution Method 0Eq" } }, { "type": "interim", "title": "$$\\cos\\left(t\\right)=\\frac{1+\\sqrt{13}}{4}{\\quad:\\quad}$$No Solution", "input": "\\cos\\left(t\\right)=\\frac{1+\\sqrt{13}}{4}", "steps": [ { "type": "step", "primary": "$$-1\\le\\cos\\left(x\\right)\\le1$$", "result": "\\mathrm{No\\:Solution}" } ], "meta": { "interimType": "N/A" } }, { "type": "interim", "title": "$$\\cos\\left(t\\right)=\\frac{1-\\sqrt{13}}{4}{\\quad:\\quad}t=\\arccos\\left(\\frac{1-\\sqrt{13}}{4}\\right)+2πn,\\:t=-\\arccos\\left(\\frac{1-\\sqrt{13}}{4}\\right)+2πn$$", "input": "\\cos\\left(t\\right)=\\frac{1-\\sqrt{13}}{4}", "steps": [ { "type": "interim", "title": "Apply trig inverse properties", "input": "\\cos\\left(t\\right)=\\frac{1-\\sqrt{13}}{4}", "result": "t=\\arccos\\left(\\frac{1-\\sqrt{13}}{4}\\right)+2πn,\\:t=-\\arccos\\left(\\frac{1-\\sqrt{13}}{4}\\right)+2πn", "steps": [ { "type": "step", "primary": "General solutions for $$\\cos\\left(t\\right)=\\frac{1-\\sqrt{13}}{4}$$", "secondary": [ "$$\\cos\\left(x\\right)=-a\\quad\\Rightarrow\\quad\\:x=\\arccos\\left(-a\\right)+2πn,\\:\\quad\\:x=-\\arccos\\left(-a\\right)+2πn$$" ], "result": "t=\\arccos\\left(\\frac{1-\\sqrt{13}}{4}\\right)+2πn,\\:t=-\\arccos\\left(\\frac{1-\\sqrt{13}}{4}\\right)+2πn" } ], "meta": { "interimType": "Trig Apply Inverse Props 0Eq" } } ], "meta": { "interimType": "N/A" } }, { "type": "step", "primary": "Combine all the solutions", "result": "t=\\arccos\\left(\\frac{1-\\sqrt{13}}{4}\\right)+2πn,\\:t=-\\arccos\\left(\\frac{1-\\sqrt{13}}{4}\\right)+2πn" }, { "type": "step", "primary": "Show solutions in decimal form", "result": "t=2.28020…+2πn,\\:t=-2.28020…+2πn" } ], "meta": { "solvingClass": "Trig Equations", "practiceLink": "/practice/trigonometry-practice#area=main&subtopic=Trig%20Equations", "practiceTopic": "Trig Equations" } }, "plot_output": { "meta": { "plotInfo": { "variable": "t", "plotRequest": "\\cos(2t)-\\cos(t)-0.5" }, "showViewLarger": true } }, "meta": { "showVerify": true } }