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

{ "query": { "display": "$$\\cos^{2}\\left(x\\right)+3\\sin\\left(x\\right)+1=0$$", "symbolab_question": "EQUATION#\\cos^{2}(x)+3\\sin(x)+1=0" }, "solution": { "level": "PERFORMED", "subject": "Trigonometry", "topic": "Trig Equations", "subTopic": "Trig Equations", "default": "x=-0.59626…+2πn,x=π+0.59626…+2πn", "degrees": "x=-34.16325…^{\\circ }+360^{\\circ }n,x=214.16325…^{\\circ }+360^{\\circ }n", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\cos^{2}\\left(x\\right)+3\\sin\\left(x\\right)+1=0{\\quad:\\quad}x=-0.59626…+2πn,\\:x=π+0.59626…+2πn$$", "input": "\\cos^{2}\\left(x\\right)+3\\sin\\left(x\\right)+1=0", "steps": [ { "type": "interim", "title": "Rewrite using trig identities", "input": "1+\\cos^{2}\\left(x\\right)+3\\sin\\left(x\\right)", "result": "2-\\sin^{2}\\left(x\\right)+3\\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": "=1+1-\\sin^{2}\\left(x\\right)+3\\sin\\left(x\\right)" }, { "type": "step", "primary": "Simplify", "result": "=3\\sin\\left(x\\right)-\\sin^{2}\\left(x\\right)+2", "meta": { "solvingClass": "Solver" } } ], "meta": { "interimType": "Trig Rewrite Using Trig identities 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7AnIq3EpDA43ydxBwz5FMdbsTlJmooTaEa9H1hWbFOfKk+FPosJui0EvbDbx83/CqPO0XSKqaWlOK6mHDvVM10YIYWgyd6SbL0eRRsTHoy/YnSKF5/4+51qVY0U4KnLmxUG0LvRGJBgcmuCidFBfZyG0bxiuengOZrT/BOd6Ox87WwPs1+Gw97t4MeuaNjSYTRvemj3GBE2iIDcXU+cR6iI+gxrQ1tCXeKlYIJ1n6NLewiNrEngO+NNvZ9sqNu+2V" } }, { "type": "interim", "title": "Solve by substitution", "input": "2-\\sin^{2}\\left(x\\right)+3\\sin\\left(x\\right)=0", "result": "\\sin\\left(x\\right)=-\\frac{-3+\\sqrt{17}}{2},\\:\\sin\\left(x\\right)=\\frac{3+\\sqrt{17}}{2}", "steps": [ { "type": "step", "primary": "Let: $$\\sin\\left(x\\right)=u$$", "result": "2-u^{2}+3u=0" }, { "type": "interim", "title": "$$2-u^{2}+3u=0{\\quad:\\quad}u=-\\frac{-3+\\sqrt{17}}{2},\\:u=\\frac{3+\\sqrt{17}}{2}$$", "input": "2-u^{2}+3u=0", "steps": [ { "type": "step", "primary": "Write in the standard form $$ax^{2}+bx+c=0$$", "result": "-u^{2}+3u+2=0" }, { "type": "interim", "title": "Solve with the quadratic formula", "input": "-u^{2}+3u+2=0", "result": "{u}_{1,\\:2}=\\frac{-3\\pm\\:\\sqrt{3^{2}-4\\left(-1\\right)\\cdot\\:2}}{2\\left(-1\\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 $${\\quad}a=-1,\\:b=3,\\:c=2$$", "result": "{u}_{1,\\:2}=\\frac{-3\\pm\\:\\sqrt{3^{2}-4\\left(-1\\right)\\cdot\\:2}}{2\\left(-1\\right)}" } ], "meta": { "interimType": "Solving The Quadratic Equation With Quadratic Formula Definition 0Eq", "gptData": "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" } }, { "type": "interim", "title": "$$\\sqrt{3^{2}-4\\left(-1\\right)\\cdot\\:2}=\\sqrt{17}$$", "input": "\\sqrt{3^{2}-4\\left(-1\\right)\\cdot\\:2}", "result": "{u}_{1,\\:2}=\\frac{-3\\pm\\:\\sqrt{17}}{2\\left(-1\\right)}", "steps": [ { "type": "step", "primary": "Apply rule $$-\\left(-a\\right)=a$$", "result": "=\\sqrt{3^{2}+4\\cdot\\:1\\cdot\\:2}" }, { "type": "step", "primary": "Multiply the numbers: $$4\\cdot\\:1\\cdot\\:2=8$$", "result": "=\\sqrt{3^{2}+8}" }, { "type": "step", "primary": "$$3^{2}=9$$", "result": "=\\sqrt{9+8}" }, { "type": "step", "primary": "Add the numbers: $$9+8=17$$", "result": "=\\sqrt{17}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7I8nbFv5ohMEv/aq9F0opfwZjD379sBMLj6lKXhIJqQnehkKrn0era9rz8TlL+x/vBfhxL0S14glIZCZ6ZA0AY+pH1EPJiqvhY2DBd6gfXrlhseSvZYNUjHXQn3NuifuOHbM44GBfEq1M+6+2CvVwbnCXgawyHrHq8Jx4vqQgXTo=" } }, { "type": "step", "primary": "Separate the solutions", "result": "{u}_{1}=\\frac{-3+\\sqrt{17}}{2\\left(-1\\right)},\\:{u}_{2}=\\frac{-3-\\sqrt{17}}{2\\left(-1\\right)}" }, { "type": "interim", "title": "$$u=\\frac{-3+\\sqrt{17}}{2\\left(-1\\right)}:{\\quad}-\\frac{-3+\\sqrt{17}}{2}$$", "input": "\\frac{-3+\\sqrt{17}}{2\\left(-1\\right)}", "steps": [ { "type": "step", "primary": "Remove parentheses: $$\\left(-a\\right)=-a$$", "result": "=\\frac{-3+\\sqrt{17}}{-2\\cdot\\:1}" }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:1=2$$", "result": "=\\frac{-3+\\sqrt{17}}{-2}" }, { "type": "step", "primary": "Apply the fraction rule: $$\\frac{a}{-b}=-\\frac{a}{b}$$", "result": "=-\\frac{-3+\\sqrt{17}}{2}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7S06bDB+MJbrOisU4Z6oFiGVQA2GZGrPBBUsy+VC63ypV00rpv8+ZC6TM10tVCSHshgFkMSyGhkcJufVqsMeq9GvTGWRLPMCxGdDkwniEUxcVFP5aPp8ShpnPl75aN0Hi3PHe8v6jMkM73jlXEVP+lZ+Trb7gjdvHnX+QLNtqP7cwCc6V2IFbtstep8w9RyPK" } }, { "type": "interim", "title": "$$u=\\frac{-3-\\sqrt{17}}{2\\left(-1\\right)}:{\\quad}\\frac{3+\\sqrt{17}}{2}$$", "input": "\\frac{-3-\\sqrt{17}}{2\\left(-1\\right)}", "steps": [ { "type": "step", "primary": "Remove parentheses: $$\\left(-a\\right)=-a$$", "result": "=\\frac{-3-\\sqrt{17}}{-2\\cdot\\:1}" }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:1=2$$", "result": "=\\frac{-3-\\sqrt{17}}{-2}" }, { "type": "step", "primary": "Apply the fraction rule: $$\\frac{-a}{-b}=\\frac{a}{b}$$", "secondary": [ "$$-3-\\sqrt{17}=-\\left(3+\\sqrt{17}\\right)$$" ], "result": "=\\frac{3+\\sqrt{17}}{2}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7QCrI66+7EgSAvN8Xk/GTa2VQA2GZGrPBBUsy+VC63ypV00rpv8+ZC6TM10tVCSHs0xDS+Y5aj0hl+F6LvDaAlqAnSYkzwNN50TWzH6uFsTARztKE552b3ssyeALQtWRjPSySEEAy/hNMHrX7AoCmQbF1BlPtUtrQxnFOfmnqKx6wiNrEngO+NNvZ9sqNu+2V" } }, { "type": "step", "primary": "The solutions to the quadratic equation are:", "result": "u=-\\frac{-3+\\sqrt{17}}{2},\\:u=\\frac{3+\\sqrt{17}}{2}" } ], "meta": { "solvingClass": "Equations", "interimType": "Equations" } }, { "type": "step", "primary": "Substitute back $$u=\\sin\\left(x\\right)$$", "result": "\\sin\\left(x\\right)=-\\frac{-3+\\sqrt{17}}{2},\\:\\sin\\left(x\\right)=\\frac{3+\\sqrt{17}}{2}" } ], "meta": { "interimType": "Substitution Method 0Eq" } }, { "type": "interim", "title": "$$\\sin\\left(x\\right)=-\\frac{-3+\\sqrt{17}}{2}{\\quad:\\quad}x=\\arcsin\\left(-\\frac{-3+\\sqrt{17}}{2}\\right)+2πn,\\:x=π+\\arcsin\\left(\\frac{-3+\\sqrt{17}}{2}\\right)+2πn$$", "input": "\\sin\\left(x\\right)=-\\frac{-3+\\sqrt{17}}{2}", "steps": [ { "type": "interim", "title": "Apply trig inverse properties", "input": "\\sin\\left(x\\right)=-\\frac{-3+\\sqrt{17}}{2}", "result": "x=\\arcsin\\left(-\\frac{-3+\\sqrt{17}}{2}\\right)+2πn,\\:x=π+\\arcsin\\left(\\frac{-3+\\sqrt{17}}{2}\\right)+2πn", "steps": [ { "type": "step", "primary": "General solutions for $$\\sin\\left(x\\right)=-\\frac{-3+\\sqrt{17}}{2}$$", "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{-3+\\sqrt{17}}{2}\\right)+2πn,\\:x=π+\\arcsin\\left(\\frac{-3+\\sqrt{17}}{2}\\right)+2πn" } ], "meta": { "interimType": "Trig Apply Inverse Props 0Eq" } } ], "meta": { "interimType": "N/A" } }, { "type": "interim", "title": "$$\\sin\\left(x\\right)=\\frac{3+\\sqrt{17}}{2}{\\quad:\\quad}$$No Solution", "input": "\\sin\\left(x\\right)=\\frac{3+\\sqrt{17}}{2}", "steps": [ { "type": "step", "primary": "$$-1\\le\\sin\\left(x\\right)\\le1$$", "result": "\\mathrm{No\\:Solution}" } ], "meta": { "interimType": "N/A" } }, { "type": "step", "primary": "Combine all the solutions", "result": "x=\\arcsin\\left(-\\frac{-3+\\sqrt{17}}{2}\\right)+2πn,\\:x=π+\\arcsin\\left(\\frac{-3+\\sqrt{17}}{2}\\right)+2πn" }, { "type": "step", "primary": "Show solutions in decimal form", "result": "x=-0.59626…+2πn,\\:x=π+0.59626…+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^{2}(x)+3\\sin(x)+1" }, "showViewLarger": true } }, "meta": { "showVerify": true } }