cos(x)+3sin(x)=2

{ "query": { "display": "$$\\cos\\left(x\\right)+3\\sin\\left(x\\right)=2$$", "symbolab_question": "EQUATION#\\cos(x)+3\\sin(x)=2" }, "solution": { "level": "PERFORMED", "subject": "Trigonometry", "topic": "Trig Equations", "subTopic": "Trig Equations", "default": "x=0.36296…+2πn,x=π-1.00646…+2πn", "degrees": "x=20.79657…^{\\circ }+360^{\\circ }n,x=122.33353…^{\\circ }+360^{\\circ }n", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\cos\\left(x\\right)+3\\sin\\left(x\\right)=2{\\quad:\\quad}x=0.36296…+2πn,\\:x=π-1.00646…+2πn$$", "input": "\\cos\\left(x\\right)+3\\sin\\left(x\\right)=2", "steps": [ { "type": "step", "primary": "Subtract $$3\\sin\\left(x\\right)$$ from both sides", "result": "\\cos\\left(x\\right)=2-3\\sin\\left(x\\right)" }, { "type": "step", "primary": "Square both sides", "result": "\\cos^{2}\\left(x\\right)=\\left(2-3\\sin\\left(x\\right)\\right)^{2}" }, { "type": "step", "primary": "Subtract $$\\left(2-3\\sin\\left(x\\right)\\right)^{2}$$ from both sides", "result": "\\cos^{2}\\left(x\\right)-4+12\\sin\\left(x\\right)-9\\sin^{2}\\left(x\\right)=0" }, { "type": "interim", "title": "Rewrite using trig identities", "input": "-4+\\cos^{2}\\left(x\\right)+12\\sin\\left(x\\right)-9\\sin^{2}\\left(x\\right)", "result": "-3-10\\sin^{2}\\left(x\\right)+12\\sin\\left(x\\right)=0", "steps": [ { "type": "step", "primary": "Use the Pythagorean identity: $$\\cos^{2}\\left(x\\right)+\\sin^{2}\\left(x\\right)=1$$", "secondary": [ "$$\\cos^{2}\\left(x\\right)=1-\\sin^{2}\\left(x\\right)$$" ], "result": "=-4+1-\\sin^{2}\\left(x\\right)+12\\sin\\left(x\\right)-9\\sin^{2}\\left(x\\right)" }, { "type": "interim", "title": "Simplify $$-4+1-\\sin^{2}\\left(x\\right)+12\\sin\\left(x\\right)-9\\sin^{2}\\left(x\\right):{\\quad}12\\sin\\left(x\\right)-10\\sin^{2}\\left(x\\right)-3$$", "input": "-4+1-\\sin^{2}\\left(x\\right)+12\\sin\\left(x\\right)-9\\sin^{2}\\left(x\\right)", "result": "=12\\sin\\left(x\\right)-10\\sin^{2}\\left(x\\right)-3", "steps": [ { "type": "step", "primary": "Group like terms", "result": "=-\\sin^{2}\\left(x\\right)+12\\sin\\left(x\\right)-9\\sin^{2}\\left(x\\right)-4+1" }, { "type": "step", "primary": "Add similar elements: $$-\\sin^{2}\\left(x\\right)-9\\sin^{2}\\left(x\\right)=-10\\sin^{2}\\left(x\\right)$$", "result": "=-10\\sin^{2}\\left(x\\right)+12\\sin\\left(x\\right)-4+1" }, { "type": "step", "primary": "Add/Subtract the numbers: $$-4+1=-3$$", "result": "=12\\sin\\left(x\\right)-10\\sin^{2}\\left(x\\right)-3" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s73fyUYAINZy7XQMOBRObIC87Bo0eNfl4FFAHgOelCNxG8YmzyCJU6fioVZZtDM4gVICf2WQN9mSJxQaQ7cQX4igyR5TGRXNMYuUjUeC6o1fNv3Lo52MhMckvL/1Vdy+KXKvDND9zWzzCLxDATqtlmDpjcBIL5pmo83UMFZRSzJsWE4rMnaTOFhXKtGdXO59ZPQrPJLLKKdzNMtGgO5cRKwZmEaILIfUrZKd38JQdn+y7iAEmXhYw7WsDRrfT9tRiW" } } ], "meta": { "interimType": "Trig Rewrite Using Trig identities 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7L3hjgZFKTWfrMkuwjPw7axutUEssT1CN2MGTAJRSjdIkLFS4iekNYXK4QDFBG7Y2pPhT6LCbotBL2w28fN/wqjztF0iqmlpTiuphw71TNdGCGFoMnekmy9HkUbEx6Mv2J0ihef+PudalWNFOCpy5sUZVPv3HXuXyV2xRhwXIlGecK0zoKZKlrmMI1Egm4dTF72wZm7kDUxdE6YSmfEbr2gao/79SBz1O81ZHch8Vm4T2a2H7GhgDjJqgadr3zWCmhn7L5gTBUzF4g+U/sLR6Sg==" } }, { "type": "interim", "title": "Solve by substitution", "input": "-3-10\\sin^{2}\\left(x\\right)+12\\sin\\left(x\\right)=0", "result": "\\sin\\left(x\\right)=\\frac{6-\\sqrt{6}}{10},\\:\\sin\\left(x\\right)=\\frac{6+\\sqrt{6}}{10}", "steps": [ { "type": "step", "primary": "Let: $$\\sin\\left(x\\right)=u$$", "result": "-3-10u^{2}+12u=0" }, { "type": "interim", "title": "$$-3-10u^{2}+12u=0{\\quad:\\quad}u=\\frac{6-\\sqrt{6}}{10},\\:u=\\frac{6+\\sqrt{6}}{10}$$", "input": "-3-10u^{2}+12u=0", "steps": [ { "type": "step", "primary": "Write in the standard form $$ax^{2}+bx+c=0$$", "result": "-10u^{2}+12u-3=0" }, { "type": "interim", "title": "Solve with the quadratic formula", "input": "-10u^{2}+12u-3=0", "result": "{u}_{1,\\:2}=\\frac{-12\\pm\\:\\sqrt{12^{2}-4\\left(-10\\right)\\left(-3\\right)}}{2\\left(-10\\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 $${\\quad}a=-10,\\:b=12,\\:c=-3$$", "result": "{u}_{1,\\:2}=\\frac{-12\\pm\\:\\sqrt{12^{2}-4\\left(-10\\right)\\left(-3\\right)}}{2\\left(-10\\right)}" } ], "meta": { "interimType": "Solving The Quadratic Equation With Quadratic Formula Definition 0Eq", "gptData": "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" } }, { "type": "interim", "title": "$$\\sqrt{12^{2}-4\\left(-10\\right)\\left(-3\\right)}=2\\sqrt{6}$$", "input": "\\sqrt{12^{2}-4\\left(-10\\right)\\left(-3\\right)}", "result": "{u}_{1,\\:2}=\\frac{-12\\pm\\:2\\sqrt{6}}{2\\left(-10\\right)}", "steps": [ { "type": "step", "primary": "Apply rule $$-\\left(-a\\right)=a$$", "result": "=\\sqrt{12^{2}-4\\cdot\\:10\\cdot\\:3}" }, { "type": "step", "primary": "Multiply the numbers: $$4\\cdot\\:10\\cdot\\:3=120$$", "result": "=\\sqrt{12^{2}-120}" }, { "type": "step", "primary": "$$12^{2}=144$$", "result": "=\\sqrt{144-120}" }, { "type": "step", "primary": "Subtract the numbers: $$144-120=24$$", "result": "=\\sqrt{24}" }, { "type": "interim", "title": "Prime factorization of $$24:{\\quad}2^{3}\\cdot\\:3$$", "input": "24", "result": "=\\sqrt{2^{3}\\cdot\\:3}", "steps": [ { "type": "step", "primary": "$$24\\:$$divides by $$2\\quad\\:24=12\\cdot\\:2$$", "result": "=2\\cdot\\:12" }, { "type": "step", "primary": "$$12\\:$$divides by $$2\\quad\\:12=6\\cdot\\:2$$", "result": "=2\\cdot\\:2\\cdot\\:6" }, { "type": "step", "primary": "$$6\\:$$divides by $$2\\quad\\:6=3\\cdot\\:2$$", "result": "=2\\cdot\\:2\\cdot\\:2\\cdot\\:3" }, { "type": "step", "primary": "$$2,\\:3$$ are all prime numbers, therefore no further factorization is possible", "result": "=2\\cdot\\:2\\cdot\\:2\\cdot\\:3" }, { "type": "step", "result": "=2^{3}\\cdot\\:3" } ], "meta": { "solvingClass": "Composite Integer", "interimType": "Prime Fac 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xyoU2cWyPLDgE1QLLHeauuc3PHQdChPJ2JhfqHT+ZU0OMrfn8NOj0LUzuzje6xTyxRl8ZboA8wPLg0yhI4RzfjFwi+wm/9WmYGY6WU37AycNdB4gitN/2ICkrV6ivfiR3BLFRzd4QlsM8ugKm4vxBIEBR4dvLeP4r0I8y1/8ztmjm" } }, { "type": "step", "primary": "Apply exponent rule: $$a^{b+c}=a^b\\cdot\\:a^c$$", "result": "=\\sqrt{2^{2}\\cdot\\:2\\cdot\\:3}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Apply radical rule: $$\\sqrt[n]{ab}=\\sqrt[n]{a}\\sqrt[n]{b}$$", "result": "=\\sqrt{2^{2}}\\sqrt{2\\cdot\\:3}", "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{2\\cdot\\:3}", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } }, { "type": "step", "primary": "Refine", "result": "=2\\sqrt{6}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7UIKTqwzY8/jQt4wJ26+eReY7wLvDmu+SipmhluLwf3Tdd47a0hQ8flDbGsI5To1dlcVozG+SjJH275Ljf0glJKvKzRUWvR7H7TI+qhAXIp2BI94i/tU2rQugaePIV+ntJN0/NDjFI739AyV1OlVrK6alIGONMlwCHUFyQocxGyg=" } }, { "type": "step", "primary": "Separate the solutions", "result": "{u}_{1}=\\frac{-12+2\\sqrt{6}}{2\\left(-10\\right)},\\:{u}_{2}=\\frac{-12-2\\sqrt{6}}{2\\left(-10\\right)}" }, { "type": "interim", "title": "$$u=\\frac{-12+2\\sqrt{6}}{2\\left(-10\\right)}:{\\quad}\\frac{6-\\sqrt{6}}{10}$$", "input": "\\frac{-12+2\\sqrt{6}}{2\\left(-10\\right)}", "steps": [ { "type": "step", "primary": "Remove parentheses: $$\\left(-a\\right)=-a$$", "result": "=\\frac{-12+2\\sqrt{6}}{-2\\cdot\\:10}" }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:10=20$$", "result": "=\\frac{-12+2\\sqrt{6}}{-20}" }, { "type": "step", "primary": "Apply the fraction rule: $$\\frac{-a}{-b}=\\frac{a}{b}$$", "secondary": [ "$$-12+2\\sqrt{6}=-\\left(12-2\\sqrt{6}\\right)$$" ], "result": "=\\frac{12-2\\sqrt{6}}{20}" }, { "type": "interim", "title": "Factor $$12-2\\sqrt{6}:{\\quad}2\\left(6-\\sqrt{6}\\right)$$", "input": "12-2\\sqrt{6}", "result": "=\\frac{2\\left(6-\\sqrt{6}\\right)}{20}", "steps": [ { "type": "step", "primary": "Rewrite as", "result": "=2\\cdot\\:6-2\\sqrt{6}" }, { "type": "step", "primary": "Factor out common term $$2$$", "result": "=2\\left(6-\\sqrt{6}\\right)", "meta": { "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } } ], "meta": { "interimType": "Algebraic Manipulation Factor Title 1Eq" } }, { "type": "step", "primary": "Cancel the common factor: $$2$$", "result": "=\\frac{6-\\sqrt{6}}{10}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7MCb583FVSl7g6uMtQ+sAgs8zIYWgbR56gzJacTtbiosAlilG71elit3w1IBbYN0P8rEus7TgCihQBF5omOFkJnXRG+n4+K3E1S0ulQ4el8HN/AGVx9Caek8XbK0x6n2BIvqrXyWWgKKlNLvd9+7RtrCUZydEH5ubSieLW4zAjcl3uYlBTn+7m2IaWa65+YYW" } }, { "type": "interim", "title": "$$u=\\frac{-12-2\\sqrt{6}}{2\\left(-10\\right)}:{\\quad}\\frac{6+\\sqrt{6}}{10}$$", "input": "\\frac{-12-2\\sqrt{6}}{2\\left(-10\\right)}", "steps": [ { "type": "step", "primary": "Remove parentheses: $$\\left(-a\\right)=-a$$", "result": "=\\frac{-12-2\\sqrt{6}}{-2\\cdot\\:10}" }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:10=20$$", "result": "=\\frac{-12-2\\sqrt{6}}{-20}" }, { "type": "step", "primary": "Apply the fraction rule: $$\\frac{-a}{-b}=\\frac{a}{b}$$", "secondary": [ "$$-12-2\\sqrt{6}=-\\left(12+2\\sqrt{6}\\right)$$" ], "result": "=\\frac{12+2\\sqrt{6}}{20}" }, { "type": "interim", "title": "Factor $$12+2\\sqrt{6}:{\\quad}2\\left(6+\\sqrt{6}\\right)$$", "input": "12+2\\sqrt{6}", "result": "=\\frac{2\\left(6+\\sqrt{6}\\right)}{20}", "steps": [ { "type": "step", "primary": "Rewrite as", "result": "=2\\cdot\\:6+2\\sqrt{6}" }, { "type": "step", "primary": "Factor out common term $$2$$", "result": "=2\\left(6+\\sqrt{6}\\right)", "meta": { "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } } ], "meta": { "interimType": "Algebraic Manipulation Factor Title 1Eq" } }, { "type": "step", "primary": "Cancel the common factor: $$2$$", "result": "=\\frac{6+\\sqrt{6}}{10}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7HZWvtZJ1VkiSvJQfLFiyks8zIYWgbR56gzJacTtbiosAlilG71elit3w1IBbYN0P8rEus7TgCihQBF5omOFkJqBdCa+CH142U4hjOmS+BgbN/AGVx9Caek8XbK0x6n2BIvqrXyWWgKKlNLvd9+7RtrFtIe7q8ssAN2oMVaDYwpN3uYlBTn+7m2IaWa65+YYW" } }, { "type": "step", "primary": "The solutions to the quadratic equation are:", "result": "u=\\frac{6-\\sqrt{6}}{10},\\:u=\\frac{6+\\sqrt{6}}{10}" } ], "meta": { "solvingClass": "Equations", "interimType": "Equations" } }, { "type": "step", "primary": "Substitute back $$u=\\sin\\left(x\\right)$$", "result": "\\sin\\left(x\\right)=\\frac{6-\\sqrt{6}}{10},\\:\\sin\\left(x\\right)=\\frac{6+\\sqrt{6}}{10}" } ], "meta": { "interimType": "Substitution Method 0Eq" } }, { "type": "interim", "title": "$$\\sin\\left(x\\right)=\\frac{6-\\sqrt{6}}{10}{\\quad:\\quad}x=\\arcsin\\left(\\frac{6-\\sqrt{6}}{10}\\right)+2πn,\\:x=π-\\arcsin\\left(\\frac{6-\\sqrt{6}}{10}\\right)+2πn$$", "input": "\\sin\\left(x\\right)=\\frac{6-\\sqrt{6}}{10}", "steps": [ { "type": "interim", "title": "Apply trig inverse properties", "input": "\\sin\\left(x\\right)=\\frac{6-\\sqrt{6}}{10}", "result": "x=\\arcsin\\left(\\frac{6-\\sqrt{6}}{10}\\right)+2πn,\\:x=π-\\arcsin\\left(\\frac{6-\\sqrt{6}}{10}\\right)+2πn", "steps": [ { "type": "step", "primary": "General solutions for $$\\sin\\left(x\\right)=\\frac{6-\\sqrt{6}}{10}$$", "secondary": [ "$$\\sin\\left(x\\right)=a\\quad\\Rightarrow\\quad\\:x=\\arcsin\\left(a\\right)+2πn,\\:\\quad\\:x=π-\\arcsin\\left(a\\right)+2πn$$" ], "result": "x=\\arcsin\\left(\\frac{6-\\sqrt{6}}{10}\\right)+2πn,\\:x=π-\\arcsin\\left(\\frac{6-\\sqrt{6}}{10}\\right)+2πn" } ], "meta": { "interimType": "Trig Apply Inverse Props 0Eq" } } ], "meta": { "interimType": "N/A" } }, { "type": "interim", "title": "$$\\sin\\left(x\\right)=\\frac{6+\\sqrt{6}}{10}{\\quad:\\quad}x=\\arcsin\\left(\\frac{6+\\sqrt{6}}{10}\\right)+2πn,\\:x=π-\\arcsin\\left(\\frac{6+\\sqrt{6}}{10}\\right)+2πn$$", "input": "\\sin\\left(x\\right)=\\frac{6+\\sqrt{6}}{10}", "steps": [ { "type": "interim", "title": "Apply trig inverse properties", "input": "\\sin\\left(x\\right)=\\frac{6+\\sqrt{6}}{10}", "result": "x=\\arcsin\\left(\\frac{6+\\sqrt{6}}{10}\\right)+2πn,\\:x=π-\\arcsin\\left(\\frac{6+\\sqrt{6}}{10}\\right)+2πn", "steps": [ { "type": "step", "primary": "General solutions for $$\\sin\\left(x\\right)=\\frac{6+\\sqrt{6}}{10}$$", "secondary": [ "$$\\sin\\left(x\\right)=a\\quad\\Rightarrow\\quad\\:x=\\arcsin\\left(a\\right)+2πn,\\:\\quad\\:x=π-\\arcsin\\left(a\\right)+2πn$$" ], "result": "x=\\arcsin\\left(\\frac{6+\\sqrt{6}}{10}\\right)+2πn,\\:x=π-\\arcsin\\left(\\frac{6+\\sqrt{6}}{10}\\right)+2πn" } ], "meta": { "interimType": "Trig Apply Inverse Props 0Eq" } } ], "meta": { "interimType": "N/A" } }, { "type": "step", "primary": "Combine all the solutions", "result": "x=\\arcsin\\left(\\frac{6-\\sqrt{6}}{10}\\right)+2πn,\\:x=π-\\arcsin\\left(\\frac{6-\\sqrt{6}}{10}\\right)+2πn,\\:x=\\arcsin\\left(\\frac{6+\\sqrt{6}}{10}\\right)+2πn,\\:x=π-\\arcsin\\left(\\frac{6+\\sqrt{6}}{10}\\right)+2πn" }, { "type": "interim", "title": "Verify solutions by plugging them into the original equation", "steps": [ { "type": "step", "primary": "Check the solutions by plugging them into $$\\cos\\left(x\\right)+3\\sin\\left(x\\right)=2$$<br/>Remove the ones that don't agree with the equation." }, { "type": "interim", "title": "Check the solution $$\\arcsin\\left(\\frac{6-\\sqrt{6}}{10}\\right)+2πn:{\\quad}$$True", "input": "\\arcsin\\left(\\frac{6-\\sqrt{6}}{10}\\right)+2πn", "steps": [ { "type": "step", "primary": "Plug in $$n=1$$", "result": "\\arcsin\\left(\\frac{6-\\sqrt{6}}{10}\\right)+2π1" }, { "type": "step", "primary": "For $$\\cos\\left(x\\right)+3\\sin\\left(x\\right)=2{\\quad}$$plug in$${\\quad}x=\\arcsin\\left(\\frac{6-\\sqrt{6}}{10}\\right)+2π1$$", "result": "\\cos\\left(\\arcsin\\left(\\frac{6-\\sqrt{6}}{10}\\right)+2π1\\right)+3\\sin\\left(\\arcsin\\left(\\frac{6-\\sqrt{6}}{10}\\right)+2π1\\right)=2" }, { "type": "step", "primary": "Refine", "result": "2=2" }, { "type": "step", "result": "\\Rightarrow\\:\\mathrm{True}" } ], "meta": { "interimType": "Check One Solution 1Eq" } }, { "type": "interim", "title": "Check the solution $$π-\\arcsin\\left(\\frac{6-\\sqrt{6}}{10}\\right)+2πn:{\\quad}$$False", "input": "π-\\arcsin\\left(\\frac{6-\\sqrt{6}}{10}\\right)+2πn", "steps": [ { "type": "step", "primary": "Plug in $$n=1$$", "result": "π-\\arcsin\\left(\\frac{6-\\sqrt{6}}{10}\\right)+2π1" }, { "type": "step", "primary": "For $$\\cos\\left(x\\right)+3\\sin\\left(x\\right)=2{\\quad}$$plug in$${\\quad}x=π-\\arcsin\\left(\\frac{6-\\sqrt{6}}{10}\\right)+2π1$$", "result": "\\cos\\left(π-\\arcsin\\left(\\frac{6-\\sqrt{6}}{10}\\right)+2π1\\right)+3\\sin\\left(π-\\arcsin\\left(\\frac{6-\\sqrt{6}}{10}\\right)+2π1\\right)=2" }, { "type": "step", "primary": "Refine", "result": "0.13030…=2" }, { "type": "step", "result": "\\Rightarrow\\:\\mathrm{False}" } ], "meta": { "interimType": "Check One Solution 1Eq" } }, { "type": "interim", "title": "Check the solution $$\\arcsin\\left(\\frac{6+\\sqrt{6}}{10}\\right)+2πn:{\\quad}$$False", "input": "\\arcsin\\left(\\frac{6+\\sqrt{6}}{10}\\right)+2πn", "steps": [ { "type": "step", "primary": "Plug in $$n=1$$", "result": "\\arcsin\\left(\\frac{6+\\sqrt{6}}{10}\\right)+2π1" }, { "type": "step", "primary": "For $$\\cos\\left(x\\right)+3\\sin\\left(x\\right)=2{\\quad}$$plug in$${\\quad}x=\\arcsin\\left(\\frac{6+\\sqrt{6}}{10}\\right)+2π1$$", "result": "\\cos\\left(\\arcsin\\left(\\frac{6+\\sqrt{6}}{10}\\right)+2π1\\right)+3\\sin\\left(\\arcsin\\left(\\frac{6+\\sqrt{6}}{10}\\right)+2π1\\right)=2" }, { "type": "step", "primary": "Refine", "result": "3.06969…=2" }, { "type": "step", "result": "\\Rightarrow\\:\\mathrm{False}" } ], "meta": { "interimType": "Check One Solution 1Eq" } }, { "type": "interim", "title": "Check the solution $$π-\\arcsin\\left(\\frac{6+\\sqrt{6}}{10}\\right)+2πn:{\\quad}$$True", "input": "π-\\arcsin\\left(\\frac{6+\\sqrt{6}}{10}\\right)+2πn", "steps": [ { "type": "step", "primary": "Plug in $$n=1$$", "result": "π-\\arcsin\\left(\\frac{6+\\sqrt{6}}{10}\\right)+2π1" }, { "type": "step", "primary": "For $$\\cos\\left(x\\right)+3\\sin\\left(x\\right)=2{\\quad}$$plug in$${\\quad}x=π-\\arcsin\\left(\\frac{6+\\sqrt{6}}{10}\\right)+2π1$$", "result": "\\cos\\left(π-\\arcsin\\left(\\frac{6+\\sqrt{6}}{10}\\right)+2π1\\right)+3\\sin\\left(π-\\arcsin\\left(\\frac{6+\\sqrt{6}}{10}\\right)+2π1\\right)=2" }, { "type": "step", "primary": "Refine", "result": "2=2" }, { "type": "step", "result": "\\Rightarrow\\:\\mathrm{True}" } ], "meta": { "interimType": "Check One Solution 1Eq" } } ], "meta": { "interimType": "Check Solutions Plug Preface 1Eq" } }, { "type": "step", "result": "x=\\arcsin\\left(\\frac{6-\\sqrt{6}}{10}\\right)+2πn,\\:x=π-\\arcsin\\left(\\frac{6+\\sqrt{6}}{10}\\right)+2πn" }, { "type": "step", "primary": "Show solutions in decimal form", "result": "x=0.36296…+2πn,\\:x=π-1.00646…+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(x)+3\\sin(x)-2" }, "showViewLarger": true } }, "meta": { "showVerify": true } }