5sin(θ)+12cos(θ)=13

{ "query": { "display": "$$5\\sin\\left(θ\\right)+12\\cos\\left(θ\\right)=13$$", "symbolab_question": "EQUATION#5\\sin(θ)+12\\cos(θ)=13" }, "solution": { "level": "PERFORMED", "subject": "Trigonometry", "topic": "Trig Equations", "subTopic": "Trig Equations", "default": "θ=0.39479…+2πn", "degrees": "θ=22.61986…^{\\circ }+360^{\\circ }n", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$5\\sin\\left(θ\\right)+12\\cos\\left(θ\\right)=13{\\quad:\\quad}θ=0.39479…+2πn$$", "input": "5\\sin\\left(θ\\right)+12\\cos\\left(θ\\right)=13", "steps": [ { "type": "step", "primary": "Subtract $$12\\cos\\left(θ\\right)$$ from both sides", "result": "5\\sin\\left(θ\\right)=13-12\\cos\\left(θ\\right)" }, { "type": "step", "primary": "Square both sides", "result": "\\left(5\\sin\\left(θ\\right)\\right)^{2}=\\left(13-12\\cos\\left(θ\\right)\\right)^{2}" }, { "type": "step", "primary": "Subtract $$\\left(13-12\\cos\\left(θ\\right)\\right)^{2}$$ from both sides", "result": "25\\sin^{2}\\left(θ\\right)-169+312\\cos\\left(θ\\right)-144\\cos^{2}\\left(θ\\right)=0" }, { "type": "interim", "title": "Rewrite using trig identities", "input": "-169-144\\cos^{2}\\left(θ\\right)+25\\sin^{2}\\left(θ\\right)+312\\cos\\left(θ\\right)", "result": "-144-169\\cos^{2}\\left(θ\\right)+312\\cos\\left(θ\\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": "=-169-144\\cos^{2}\\left(θ\\right)+25\\left(1-\\cos^{2}\\left(θ\\right)\\right)+312\\cos\\left(θ\\right)" }, { "type": "interim", "title": "Simplify $$-169-144\\cos^{2}\\left(θ\\right)+25\\left(1-\\cos^{2}\\left(θ\\right)\\right)+312\\cos\\left(θ\\right):{\\quad}312\\cos\\left(θ\\right)-169\\cos^{2}\\left(θ\\right)-144$$", "input": "-169-144\\cos^{2}\\left(θ\\right)+25\\left(1-\\cos^{2}\\left(θ\\right)\\right)+312\\cos\\left(θ\\right)", "result": "=312\\cos\\left(θ\\right)-169\\cos^{2}\\left(θ\\right)-144", "steps": [ { "type": "interim", "title": "Expand $$25\\left(1-\\cos^{2}\\left(θ\\right)\\right):{\\quad}25-25\\cos^{2}\\left(θ\\right)$$", "input": "25\\left(1-\\cos^{2}\\left(θ\\right)\\right)", "result": "=-169-144\\cos^{2}\\left(θ\\right)+25-25\\cos^{2}\\left(θ\\right)+312\\cos\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Apply the distributive law: $$a\\left(b-c\\right)=ab-ac$$", "secondary": [ "$$a=25,\\:b=1,\\:c=\\cos^{2}\\left(θ\\right)$$" ], "result": "=25\\cdot\\:1-25\\cos^{2}\\left(θ\\right)", "meta": { "practiceLink": "/practice/expansion-practice", "practiceTopic": "Expand Rules" } }, { "type": "step", "primary": "Multiply the numbers: $$25\\cdot\\:1=25$$", "result": "=25-25\\cos^{2}\\left(θ\\right)" } ], "meta": { "interimType": "Algebraic Manipulation Expand Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s73c+TMXgaIs0ltp+BImYhu61/TqqEqZbGfmhpsXTelQ9wkKGJWEPFPk38sdJMsyPIrYcE0IaddWwBJ3EtJAoHMGx9uc3EyK04O37j7Cp6dyUezFilETfuNygjs0XPkV2h32qGWSxzRgSNXXpoIEVE+L0Vkt9H4DYA1dCAqr7GzQk=" } }, { "type": "interim", "title": "Simplify $$-169-144\\cos^{2}\\left(θ\\right)+25-25\\cos^{2}\\left(θ\\right)+312\\cos\\left(θ\\right):{\\quad}312\\cos\\left(θ\\right)-169\\cos^{2}\\left(θ\\right)-144$$", "input": "-169-144\\cos^{2}\\left(θ\\right)+25-25\\cos^{2}\\left(θ\\right)+312\\cos\\left(θ\\right)", "result": "=312\\cos\\left(θ\\right)-169\\cos^{2}\\left(θ\\right)-144", "steps": [ { "type": "step", "primary": "Group like terms", "result": "=-144\\cos^{2}\\left(θ\\right)-25\\cos^{2}\\left(θ\\right)+312\\cos\\left(θ\\right)-169+25" }, { "type": "step", "primary": "Add similar elements: $$-144\\cos^{2}\\left(θ\\right)-25\\cos^{2}\\left(θ\\right)=-169\\cos^{2}\\left(θ\\right)$$", "result": "=-169\\cos^{2}\\left(θ\\right)+312\\cos\\left(θ\\right)-169+25" }, { "type": "step", "primary": "Add/Subtract the numbers: $$-169+25=-144$$", "result": "=312\\cos\\left(θ\\right)-169\\cos^{2}\\left(θ\\right)-144" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7tSVjZSOOFx/utTLD6LGV1zRJdZ9jd6D+eThJaoT+7fz1PmjgYFc+7L+eiOxIcASX0VHDR+wRdQQ+t6+tuHArRHWD310L1+P2yDQQfMEhENFbHIKXaFk+809xU22SyLo0kciFM448mJ5hy3Vrjn2DzDYxPG0xjEl2f9oNMgucl2keNvb7k0sVmuwf19w9aD9N2SlilplBANs9KHHObM4ikAkhSLFreFnh2Asyu2U0+ejJJObialLka6D1jrm016OMmecaNEB+/WSjJzpY5JjRg78yD3hLQ33B7/8/LpbPE3o=" } } ], "meta": { "interimType": "Trig Rewrite Using Trig identities 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7+S1gEdH8asjuVjdpaZWDDTTb/eVPikdmbACQuVBBwymZXMZWY91oNmgFNH/8ZMnKFnAWrjhtQ/yQqnlBeB4LDLK5FycWCeHwCKOFg2OGrJFvJljJfoQYwBwAyCM4AOo8tqMjE/Y6uOQEIQJB3xrrDnLY6ufL8xy+C10hFcHTJmpCzDjm98w1RCxdZU5v2lY+QHmMBhCNjsgpLbeK7dF68e9sGZu5A1MXROmEpnxG69oGqP+/Ugc9TvNWR3IfFZuE9mth+xoYA4yaoGna981gpoZ+y+YEwVMxeIPlP7C0eko=" } }, { "type": "interim", "title": "Solve by substitution", "input": "-144-169\\cos^{2}\\left(θ\\right)+312\\cos\\left(θ\\right)=0", "result": "\\cos\\left(θ\\right)=\\frac{12}{13}", "steps": [ { "type": "step", "primary": "Let: $$\\cos\\left(θ\\right)=u$$", "result": "-144-169u^{2}+312u=0" }, { "type": "interim", "title": "$$-144-169u^{2}+312u=0{\\quad:\\quad}u=\\frac{12}{13}$$", "input": "-144-169u^{2}+312u=0", "steps": [ { "type": "step", "primary": "Write in the standard form $$ax^{2}+bx+c=0$$", "result": "-169u^{2}+312u-144=0" }, { "type": "interim", "title": "Solve with the quadratic formula", "input": "-169u^{2}+312u-144=0", "result": "{u}_{1,\\:2}=\\frac{-312\\pm\\:\\sqrt{312^{2}-4\\left(-169\\right)\\left(-144\\right)}}{2\\left(-169\\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 $${\\quad}a=-169,\\:b=312,\\:c=-144$$", "result": "{u}_{1,\\:2}=\\frac{-312\\pm\\:\\sqrt{312^{2}-4\\left(-169\\right)\\left(-144\\right)}}{2\\left(-169\\right)}" } ], "meta": { "interimType": "Solving The Quadratic Equation With Quadratic Formula Definition 0Eq", "gptData": "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" } }, { "type": "interim", "title": "$$312^{2}-4\\left(-169\\right)\\left(-144\\right)=0$$", "input": "312^{2}-4\\left(-169\\right)\\left(-144\\right)", "result": "{u}_{1,\\:2}=\\frac{-312\\pm\\:\\sqrt{0}}{2\\left(-169\\right)}", "steps": [ { "type": "step", "primary": "Apply rule $$-\\left(-a\\right)=a$$", "result": "=312^{2}-4\\cdot\\:169\\cdot\\:144" }, { "type": "step", "primary": "Multiply the numbers: $$4\\cdot\\:169\\cdot\\:144=97344$$", "result": "=312^{2}-97344" }, { "type": "step", "primary": "$$312^{2}=97344$$", "result": "=97344-97344" }, { "type": "step", "primary": "Subtract the numbers: $$97344-97344=0$$", "result": "=0" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7wRHNl6VEr4ID6ae65paWmbj+DIBNS9TT64/+BMB38lcJQJZuTAY5js+oqjdT8kslKXPrgUnq5rRq9Cvw1ceDciFI6byC3hxvd7sztUWEp9tqWHfdJ9CilznI6hKuEnI2" } }, { "type": "step", "result": "u=\\frac{-312}{2\\left(-169\\right)}" }, { "type": "interim", "title": "$$\\frac{-312}{2\\left(-169\\right)}=\\frac{12}{13}$$", "input": "\\frac{-312}{2\\left(-169\\right)}", "result": "u=\\frac{12}{13}", "steps": [ { "type": "step", "primary": "Remove parentheses: $$\\left(-a\\right)=-a$$", "result": "=\\frac{-312}{-2\\cdot\\:169}" }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:169=338$$", "result": "=\\frac{-312}{-338}" }, { "type": "step", "primary": "Apply the fraction rule: $$\\frac{-a}{-b}=\\frac{a}{b}$$", "result": "=\\frac{312}{338}" }, { "type": "step", "primary": "Cancel the common factor: $$26$$", "result": "=\\frac{12}{13}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7OE5GCGOmjaTOhVbqmuvCjXpGG1J7tpm9a/Rdo7Pr8ooJQJZuTAY5js+oqjdT8ksl2w6ZSyVg12UlLV0jIw/TvxSYeD6JDEaUbEpYNvzKYnPRQp9EukjOao0Z6D9sik8HlQucewLuGgU7Wsz9WMOLImo+jo5z8dGLgARDcj1HOno=" } }, { "type": "step", "primary": "The solution to the quadratic equation is:", "result": "u=\\frac{12}{13}" } ], "meta": { "solvingClass": "Equations", "interimType": "Equations" } }, { "type": "step", "primary": "Substitute back $$u=\\cos\\left(θ\\right)$$", "result": "\\cos\\left(θ\\right)=\\frac{12}{13}" } ], "meta": { "interimType": "Substitution Method 0Eq" } }, { "type": "interim", "title": "$$\\cos\\left(θ\\right)=\\frac{12}{13}{\\quad:\\quad}θ=\\arccos\\left(\\frac{12}{13}\\right)+2πn,\\:θ=2π-\\arccos\\left(\\frac{12}{13}\\right)+2πn$$", "input": "\\cos\\left(θ\\right)=\\frac{12}{13}", "steps": [ { "type": "interim", "title": "Apply trig inverse properties", "input": "\\cos\\left(θ\\right)=\\frac{12}{13}", "result": "θ=\\arccos\\left(\\frac{12}{13}\\right)+2πn,\\:θ=2π-\\arccos\\left(\\frac{12}{13}\\right)+2πn", "steps": [ { "type": "step", "primary": "General solutions for $$\\cos\\left(θ\\right)=\\frac{12}{13}$$", "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{12}{13}\\right)+2πn,\\:θ=2π-\\arccos\\left(\\frac{12}{13}\\right)+2πn" } ], "meta": { "interimType": "Trig Apply Inverse Props 0Eq" } } ], "meta": { "interimType": "N/A" } }, { "type": "step", "primary": "Combine all the solutions", "result": "θ=\\arccos\\left(\\frac{12}{13}\\right)+2πn,\\:θ=2π-\\arccos\\left(\\frac{12}{13}\\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 $$5\\sin\\left(θ\\right)+12\\cos\\left(θ\\right)=13$$<br/>Remove the ones that don't agree with the equation." }, { "type": "interim", "title": "Check the solution $$\\arccos\\left(\\frac{12}{13}\\right)+2πn:{\\quad}$$True", "input": "\\arccos\\left(\\frac{12}{13}\\right)+2πn", "steps": [ { "type": "step", "primary": "Plug in $$n=1$$", "result": "\\arccos\\left(\\frac{12}{13}\\right)+2π1" }, { "type": "step", "primary": "For $$5\\sin\\left(θ\\right)+12\\cos\\left(θ\\right)=13{\\quad}$$plug in$${\\quad}θ=\\arccos\\left(\\frac{12}{13}\\right)+2π1$$", "result": "5\\sin\\left(\\arccos\\left(\\frac{12}{13}\\right)+2π1\\right)+12\\cos\\left(\\arccos\\left(\\frac{12}{13}\\right)+2π1\\right)=13" }, { "type": "step", "primary": "Refine", "result": "13=13" }, { "type": "step", "result": "\\Rightarrow\\:\\mathrm{True}" } ], "meta": { "interimType": "Check One Solution 1Eq" } }, { "type": "interim", "title": "Check the solution $$2π-\\arccos\\left(\\frac{12}{13}\\right)+2πn:{\\quad}$$False", "input": "2π-\\arccos\\left(\\frac{12}{13}\\right)+2πn", "steps": [ { "type": "step", "primary": "Plug in $$n=1$$", "result": "2π-\\arccos\\left(\\frac{12}{13}\\right)+2π1" }, { "type": "step", "primary": "For $$5\\sin\\left(θ\\right)+12\\cos\\left(θ\\right)=13{\\quad}$$plug in$${\\quad}θ=2π-\\arccos\\left(\\frac{12}{13}\\right)+2π1$$", "result": "5\\sin\\left(2π-\\arccos\\left(\\frac{12}{13}\\right)+2π1\\right)+12\\cos\\left(2π-\\arccos\\left(\\frac{12}{13}\\right)+2π1\\right)=13" }, { "type": "step", "primary": "Refine", "result": "9.15384…=13" }, { "type": "step", "result": "\\Rightarrow\\:\\mathrm{False}" } ], "meta": { "interimType": "Check One Solution 1Eq" } } ], "meta": { "interimType": "Check Solutions Plug Preface 1Eq" } }, { "type": "step", "result": "θ=\\arccos\\left(\\frac{12}{13}\\right)+2πn" }, { "type": "step", "primary": "Show solutions in decimal form", "result": "θ=0.39479…+2πn" } ], "meta": { "solvingClass": "Trig Equations", "practiceLink": "/practice/trigonometry-practice#area=main&subtopic=Trig%20Equations", "practiceTopic": "Trig Equations" } }, "plot_output": { "meta": { "plotInfo": { "variable": "θ", "plotRequest": "5\\sin(θ)+12\\cos(θ)-13" }, "showViewLarger": true } }, "meta": { "showVerify": true } }