sin^2(θ)-2sin(θ)+3=0

{ "query": { "display": "$$\\sin^{2}\\left(θ\\right)-2\\sin\\left(θ\\right)+3=0$$", "symbolab_question": "EQUATION#\\sin^{2}(θ)-2\\sin(θ)+3=0" }, "solution": { "level": "PERFORMED", "subject": "Trigonometry", "topic": "Trig Equations", "subTopic": "Trig Equations", "default": "\\mathrm{No\\:Solution\\:for}\\:θ\\in\\mathbb{R}", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\sin^{2}\\left(θ\\right)-2\\sin\\left(θ\\right)+3=0{\\quad:\\quad}$$No Solution for $$θ\\in\\mathbb{R}$$", "input": "\\sin^{2}\\left(θ\\right)-2\\sin\\left(θ\\right)+3=0", "steps": [ { "type": "interim", "title": "Solve by substitution", "input": "\\sin^{2}\\left(θ\\right)-2\\sin\\left(θ\\right)+3=0", "result": "\\sin\\left(θ\\right)=1+\\sqrt{2}i,\\:\\sin\\left(θ\\right)=1-\\sqrt{2}i", "steps": [ { "type": "step", "primary": "Let: $$\\sin\\left(θ\\right)=u$$", "result": "u^{2}-2u+3=0" }, { "type": "interim", "title": "$$u^{2}-2u+3=0{\\quad:\\quad}u=1+\\sqrt{2}i,\\:u=1-\\sqrt{2}i$$", "input": "u^{2}-2u+3=0", "steps": [ { "type": "interim", "title": "Solve with the quadratic formula", "input": "u^{2}-2u+3=0", "result": "{u}_{1,\\:2}=\\frac{-\\left(-2\\right)\\pm\\:\\sqrt{\\left(-2\\right)^{2}-4\\cdot\\:1\\cdot\\:3}}{2\\cdot\\:1}", "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 $${\\quad}a=1,\\:b=-2,\\:c=3$$", "result": "{u}_{1,\\:2}=\\frac{-\\left(-2\\right)\\pm\\:\\sqrt{\\left(-2\\right)^{2}-4\\cdot\\:1\\cdot\\:3}}{2\\cdot\\:1}" } ], "meta": { "interimType": "Solving The Quadratic Equation With Quadratic Formula Definition 0Eq", "gptData": "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" } }, { "type": "interim", "title": "Simplify $$\\sqrt{\\left(-2\\right)^{2}-4\\cdot\\:1\\cdot\\:3}:{\\quad}2\\sqrt{2}i$$", "input": "\\sqrt{\\left(-2\\right)^{2}-4\\cdot\\:1\\cdot\\:3}", "result": "{u}_{1,\\:2}=\\frac{-\\left(-2\\right)\\pm\\:2\\sqrt{2}i}{2\\cdot\\:1}", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$\\left(-a\\right)^{n}=a^{n},\\:$$if $$n$$ is even", "secondary": [ "$$\\left(-2\\right)^{2}=2^{2}$$" ], "result": "=\\sqrt{2^{2}-4\\cdot\\:1\\cdot\\:3}" }, { "type": "step", "primary": "Multiply the numbers: $$4\\cdot\\:1\\cdot\\:3=12$$", "result": "=\\sqrt{2^{2}-12}" }, { "type": "step", "primary": "Apply imaginary number rule: $$\\sqrt{-a}=i\\sqrt{a}$$", "result": "=i\\sqrt{12-2^{2}}", "meta": { "practiceLink": "/practice/complex-numbers-practice", "practiceTopic": "Expand FOIL" } }, { "type": "interim", "title": "$$\\sqrt{-2^{2}+12}=2\\sqrt{2}$$", "input": "\\sqrt{-2^{2}+12}", "steps": [ { "type": "step", "primary": "$$2^{2}=4$$", "result": "=\\sqrt{-4+12}" }, { "type": "step", "primary": "Add/Subtract the numbers: $$-4+12=8$$", "result": "=\\sqrt{8}" }, { "type": "interim", "title": "Prime factorization of $$8:{\\quad}2^{3}$$", "input": "8", "result": "=\\sqrt{2^{3}}", "steps": [ { "type": "step", "primary": "$$8\\:$$divides by $$2\\quad\\:8=4\\cdot\\:2$$", "result": "=2\\cdot\\:4" }, { "type": "step", "primary": "$$4\\:$$divides by $$2\\quad\\:4=2\\cdot\\:2$$", "result": "=2\\cdot\\:2\\cdot\\:2" }, { "type": "step", "primary": "$$2$$ is a prime number, therefore no further factorization is possible", "result": "=2\\cdot\\:2\\cdot\\:2" }, { "type": "step", "result": "=2^{3}" } ], "meta": { "solvingClass": "Composite Integer", "interimType": "Prime Fac 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xyoU2cWyPLDgE1QLLHeauuc3PHQdChPJ2JhfqHT+ZU0OMrfn8NOj0LUzuzje6xTyxRtXnBik8vDPXVw+nKWp28DI/y9DKGIPglJ+qMi9xDu2KE1OovxZAaXg7BtrFPk4UcCzRnGgMN6CYRfod7Mq0dp1RcgsS082tQWmOBW6FvhEw" } }, { "type": "step", "primary": "Apply exponent rule: $$a^{b+c}=a^b\\cdot\\:a^c$$", "result": "=\\sqrt{2^{2}\\cdot\\:2}", "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}\\sqrt{2^{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{2}", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7/icOWPT9SRkgHO9psN1CxMAYOMuP4JW3uWQsjElsZW2jkVi15I8rBefLi4Iyt2wrc26nYQa9oNg8OxdsrL7z/MxkARoM44WPcMutmKWsbuxu2P2z7eK9a6RsVaR6bjpWsaSLbKptYiECV5o+VdjWWQ==" } }, { "type": "step", "result": "=2\\sqrt{2}i" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "primary": "Separate the solutions", "result": "{u}_{1}=\\frac{-\\left(-2\\right)+2\\sqrt{2}i}{2\\cdot\\:1},\\:{u}_{2}=\\frac{-\\left(-2\\right)-2\\sqrt{2}i}{2\\cdot\\:1}" }, { "type": "interim", "title": "$$u=\\frac{-\\left(-2\\right)+2\\sqrt{2}i}{2\\cdot\\:1}:{\\quad}1+\\sqrt{2}i$$", "input": "\\frac{-\\left(-2\\right)+2\\sqrt{2}i}{2\\cdot\\:1}", "steps": [ { "type": "step", "primary": "Apply rule $$-\\left(-a\\right)=a$$", "result": "=\\frac{2+2\\sqrt{2}i}{2\\cdot\\:1}" }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:1=2$$", "result": "=\\frac{2+2\\sqrt{2}i}{2}" }, { "type": "interim", "title": "Factor $$2+2\\sqrt{2}i:{\\quad}2\\left(1+\\sqrt{2}i\\right)$$", "input": "2+2\\sqrt{2}i", "result": "=\\frac{2\\left(1+\\sqrt{2}i\\right)}{2}", "steps": [ { "type": "step", "primary": "Rewrite as", "result": "=2\\cdot\\:1+2\\sqrt{2}i" }, { "type": "step", "primary": "Factor out common term $$2$$", "result": "=2\\left(1+\\sqrt{2}i\\right)", "meta": { "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } } ], "meta": { "interimType": "Algebraic Manipulation Factor Title 1Eq" } }, { "type": "step", "primary": "Divide the numbers: $$\\frac{2}{2}=1$$", "result": "=1+\\sqrt{2}i" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7Fv9gVNWyI/X8TvzzSZV8ny+AaCHYlLeOBdo6gCPWlAhZjwHrXwLiU12lumi8FmRWq47vuWedXv2WUg94ER8Iwf2tOGz1qRxi2vgu1E7GBKUt+fVdm7Nauu6V8feKuVwKcSwlkB3snN59uXX26hhfL6hk0+9FoRyjuZ1aILn4CxY0wnI6LdgQd2AzzCI0KAvH" } }, { "type": "interim", "title": "$$u=\\frac{-\\left(-2\\right)-2\\sqrt{2}i}{2\\cdot\\:1}:{\\quad}1-\\sqrt{2}i$$", "input": "\\frac{-\\left(-2\\right)-2\\sqrt{2}i}{2\\cdot\\:1}", "steps": [ { "type": "step", "primary": "Apply rule $$-\\left(-a\\right)=a$$", "result": "=\\frac{2-2\\sqrt{2}i}{2\\cdot\\:1}" }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:1=2$$", "result": "=\\frac{2-2\\sqrt{2}i}{2}" }, { "type": "interim", "title": "Factor $$2-2\\sqrt{2}i:{\\quad}2\\left(1-\\sqrt{2}i\\right)$$", "input": "2-2\\sqrt{2}i", "result": "=\\frac{2\\left(1-\\sqrt{2}i\\right)}{2}", "steps": [ { "type": "step", "primary": "Rewrite as", "result": "=2\\cdot\\:1-2\\sqrt{2}i" }, { "type": "step", "primary": "Factor out common term $$2$$", "result": "=2\\left(1-\\sqrt{2}i\\right)", "meta": { "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } } ], "meta": { "interimType": "Algebraic Manipulation Factor Title 1Eq" } }, { "type": "step", "primary": "Divide the numbers: $$\\frac{2}{2}=1$$", "result": "=1-\\sqrt{2}i" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7yzZ+Rg5UPtRAM2q6UZuzBi+AaCHYlLeOBdo6gCPWlAhZjwHrXwLiU12lumi8FmRWq47vuWedXv2WUg94ER8IweYbvixi59HHkttSyV9PFikt+fVdm7Nauu6V8feKuVwKFgRDz6/QgY1fnuUkuJ3voqhk0+9FoRyjuZ1aILn4CxY0wnI6LdgQd2AzzCI0KAvH" } }, { "type": "step", "primary": "The solutions to the quadratic equation are:", "result": "u=1+\\sqrt{2}i,\\:u=1-\\sqrt{2}i" } ], "meta": { "solvingClass": "Equations", "interimType": "Equations" } }, { "type": "step", "primary": "Substitute back $$u=\\sin\\left(θ\\right)$$", "result": "\\sin\\left(θ\\right)=1+\\sqrt{2}i,\\:\\sin\\left(θ\\right)=1-\\sqrt{2}i" } ], "meta": { "interimType": "Substitution Method 0Eq" } }, { "type": "interim", "title": "$$\\sin\\left(θ\\right)=1+\\sqrt{2}i{\\quad:\\quad}$$No Solution", "input": "\\sin\\left(θ\\right)=1+\\sqrt{2}i", "steps": [ { "type": "step", "result": "\\mathrm{No\\:Solution}" } ], "meta": { "interimType": "N/A" } }, { "type": "interim", "title": "$$\\sin\\left(θ\\right)=1-\\sqrt{2}i{\\quad:\\quad}$$No Solution", "input": "\\sin\\left(θ\\right)=1-\\sqrt{2}i", "steps": [ { "type": "step", "result": "\\mathrm{No\\:Solution}" } ], "meta": { "interimType": "N/A" } }, { "type": "step", "primary": "Combine all the solutions", "result": "\\mathrm{No\\:Solution\\:for}\\:θ\\in\\mathbb{R}" } ], "meta": { "solvingClass": "Trig Equations", "practiceLink": "/practice/trigonometry-practice#area=main&subtopic=Trig%20Equations", "practiceTopic": "Trig Equations" } }, "plot_output": { "meta": { "plotInfo": { "variable": "θ", "plotRequest": "\\sin^{2}(θ)-2\\sin(θ)+3" }, "showViewLarger": true } }, "meta": { "showVerify": true } }