sin^2(2x)+3sin(2x)+2=0,0<= x<= 10pi

{ "query": { "display": "$$\\sin^{2}\\left(2x\\right)+3\\sin\\left(2x\\right)+2=0,\\:0\\le\\:x\\le\\:10π$$", "symbolab_question": "EQUATION#\\sin^{2}(2x)+3\\sin(2x)+2=0,0\\le x\\le 10π" }, "solution": { "level": "PERFORMED", "subject": "Trigonometry", "topic": "Trig Equations", "subTopic": "Trig Equations", "default": "x=\\frac{3π}{4},x=\\frac{7π}{4},x=\\frac{11π}{4},x=\\frac{15π}{4},x=\\frac{19π}{4}", "degrees": "x=135^{\\circ },x=315^{\\circ },x=495^{\\circ },x=675^{\\circ },x=855^{\\circ }", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\sin^{2}\\left(2x\\right)+3\\sin\\left(2x\\right)+2=0,\\:0\\le\\:x\\le\\:10π{\\quad:\\quad}x=\\frac{3π}{4},\\:x=\\frac{7π}{4},\\:x=\\frac{11π}{4},\\:x=\\frac{15π}{4},\\:x=\\frac{19π}{4}$$", "input": "\\sin^{2}\\left(2x\\right)+3\\sin\\left(2x\\right)+2=0,\\:0\\le\\:x\\le\\:10π", "steps": [ { "type": "interim", "title": "Solve by substitution", "input": "\\sin^{2}\\left(2x\\right)+3\\sin\\left(2x\\right)+2=0", "result": "\\sin\\left(2x\\right)=-1,\\:\\sin\\left(2x\\right)=-2", "steps": [ { "type": "step", "primary": "Let: $$\\sin\\left(2x\\right)=u$$", "result": "u^{2}+3u+2=0" }, { "type": "interim", "title": "$$u^{2}+3u+2=0{\\quad:\\quad}u=-1,\\:u=-2$$", "input": "u^{2}+3u+2=0", "steps": [ { "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\\cdot\\:1\\cdot\\:2}}{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 $${\\quad}a=1,\\:b=3,\\:c=2$$", "result": "{u}_{1,\\:2}=\\frac{-3\\pm\\:\\sqrt{3^{2}-4\\cdot\\:1\\cdot\\:2}}{2\\cdot\\:1}" } ], "meta": { "interimType": "Solving The Quadratic Equation With Quadratic Formula Definition 0Eq", "gptData": "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" } }, { "type": "interim", "title": "$$\\sqrt{3^{2}-4\\cdot\\:1\\cdot\\:2}=1$$", "input": "\\sqrt{3^{2}-4\\cdot\\:1\\cdot\\:2}", "result": "{u}_{1,\\:2}=\\frac{-3\\pm\\:1}{2\\cdot\\:1}", "steps": [ { "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": "Subtract the numbers: $$9-8=1$$", "result": "=\\sqrt{1}" }, { "type": "step", "primary": "Apply rule $$\\sqrt{1}=1$$", "result": "=1" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7I8nbFv5ohMEv/aq9F0opf7fOk4ny0fHNijTdWwbnlih8kR7hsO/rTOTBE0w4+r1RRECsuRgN/jNlFWO6hVsmqWRLd2VwIqlBNByF6663syTeuKe78HD0WnFH5j2hDe+Nd9qvoKMjf5PrCBYD3AuHR7CI2sSeA74029n2yo277ZU=" } }, { "type": "step", "primary": "Separate the solutions", "result": "{u}_{1}=\\frac{-3+1}{2\\cdot\\:1},\\:{u}_{2}=\\frac{-3-1}{2\\cdot\\:1}" }, { "type": "interim", "title": "$$u=\\frac{-3+1}{2\\cdot\\:1}:{\\quad}-1$$", "input": "\\frac{-3+1}{2\\cdot\\:1}", "steps": [ { "type": "step", "primary": "Add/Subtract the numbers: $$-3+1=-2$$", "result": "=\\frac{-2}{2\\cdot\\:1}" }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:1=2$$", "result": "=\\frac{-2}{2}" }, { "type": "step", "primary": "Apply the fraction rule: $$\\frac{-a}{b}=-\\frac{a}{b}$$", "result": "=-\\frac{2}{2}" }, { "type": "step", "primary": "Apply rule $$\\frac{a}{a}=1$$", "result": "=-1" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7/sAmUTlhMBxDI2Dy6MlmsiJzNSds22HF6C2iiwwKPXMDnzlbPZjyKgy1eUCFsLd5pmhpfHqzxU6UVoC7gLWdYz/AeYEIX4I11Ba3Yz03hSCJhRCKIb6MHlk1DV35c4X8ialcV/dI5TH4fXyp+ncwuA==" } }, { "type": "interim", "title": "$$u=\\frac{-3-1}{2\\cdot\\:1}:{\\quad}-2$$", "input": "\\frac{-3-1}{2\\cdot\\:1}", "steps": [ { "type": "step", "primary": "Subtract the numbers: $$-3-1=-4$$", "result": "=\\frac{-4}{2\\cdot\\:1}" }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:1=2$$", "result": "=\\frac{-4}{2}" }, { "type": "step", "primary": "Apply the fraction rule: $$\\frac{-a}{b}=-\\frac{a}{b}$$", "result": "=-\\frac{4}{2}" }, { "type": "step", "primary": "Divide the numbers: $$\\frac{4}{2}=2$$", "result": "=-2" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s726nwabKyGUk7dWW1VAq7XSJzNSds22HF6C2iiwwKPXMDnzlbPZjyKgy1eUCFsLd5RDtWG5guT8Xmwx3nth5Sgj/AeYEIX4I11Ba3Yz03hSCbAtAIoDTQz7J1tMaxV8J0ialcV/dI5TH4fXyp+ncwuA==" } }, { "type": "step", "primary": "The solutions to the quadratic equation are:", "result": "u=-1,\\:u=-2" } ], "meta": { "solvingClass": "Equations", "interimType": "Equations" } }, { "type": "step", "primary": "Substitute back $$u=\\sin\\left(2x\\right)$$", "result": "\\sin\\left(2x\\right)=-1,\\:\\sin\\left(2x\\right)=-2" } ], "meta": { "interimType": "Substitution Method 0Eq" } }, { "type": "interim", "title": "$$\\sin\\left(2x\\right)=-1,\\:0\\le\\:x\\le\\:10π{\\quad:\\quad}x=\\frac{3π}{4},\\:x=\\frac{7π}{4},\\:x=\\frac{11π}{4},\\:x=\\frac{15π}{4},\\:x=\\frac{19π}{4}$$", "input": "\\sin\\left(2x\\right)=-1,\\:0\\le\\:x\\le\\:10π", "steps": [ { "type": "interim", "title": "General solutions for $$\\sin\\left(2x\\right)=-1$$", "result": "2x=\\frac{3π}{2}+2πn", "steps": [ { "type": "step", "primary": "$$\\sin\\left(x\\right)$$ periodicity table with $$2πn$$ cycle:<br/>$$\\begin{array}{|c|c|c|c|}\\hline x&\\sin(x)&x&\\sin(x)\\\\\\hline 0&0&π&0\\\\\\hline \\frac{π}{6}&\\frac{1}{2}&\\frac{7π}{6}&-\\frac{1}{2}\\\\\\hline \\frac{π}{4}&\\frac{\\sqrt{2}}{2}&\\frac{5π}{4}&-\\frac{\\sqrt{2}}{2}\\\\\\hline \\frac{π}{3}&\\frac{\\sqrt{3}}{2}&\\frac{4π}{3}&-\\frac{\\sqrt{3}}{2}\\\\\\hline \\frac{π}{2}&1&\\frac{3π}{2}&-1\\\\\\hline \\frac{2π}{3}&\\frac{\\sqrt{3}}{2}&\\frac{5π}{3}&-\\frac{\\sqrt{3}}{2}\\\\\\hline \\frac{3π}{4}&\\frac{\\sqrt{2}}{2}&\\frac{7π}{4}&-\\frac{\\sqrt{2}}{2}\\\\\\hline \\frac{5π}{6}&\\frac{1}{2}&\\frac{11π}{6}&-\\frac{1}{2}\\\\\\hline \\end{array}$$" }, { "type": "step", "result": "2x=\\frac{3π}{2}+2πn" } ], "meta": { "interimType": "Trig General Solutions sin 1Eq" } }, { "type": "interim", "title": "Solve $$2x=\\frac{3π}{2}+2πn:{\\quad}x=\\frac{3π}{4}+πn$$", "input": "2x=\\frac{3π}{2}+2πn", "steps": [ { "type": "interim", "title": "Divide both sides by $$2$$", "input": "2x=\\frac{3π}{2}+2πn", "result": "x=\\frac{3π}{4}+πn", "steps": [ { "type": "step", "primary": "Divide both sides by $$2$$", "result": "\\frac{2x}{2}=\\frac{\\frac{3π}{2}}{2}+\\frac{2πn}{2}" }, { "type": "interim", "title": "Simplify", "input": "\\frac{2x}{2}=\\frac{\\frac{3π}{2}}{2}+\\frac{2πn}{2}", "result": "x=\\frac{3π}{4}+πn", "steps": [ { "type": "interim", "title": "Simplify $$\\frac{2x}{2}:{\\quad}x$$", "input": "\\frac{2x}{2}", "steps": [ { "type": "step", "primary": "Divide the numbers: $$\\frac{2}{2}=1$$", "result": "=x" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7wyZUoCmW9j1Yiq04nFm+Ei061ljBSPJeENOw2efoSWt2cZ6LAXMcKee6PSyjLEFrRSpN33oxZMojoqvYhvSJACLkpiVA7S8UT8ieezZKu6rogJeWDMKCUBljFmQOLy/H" } }, { "type": "interim", "title": "Simplify $$\\frac{\\frac{3π}{2}}{2}+\\frac{2πn}{2}:{\\quad}\\frac{3π}{4}+πn$$", "input": "\\frac{\\frac{3π}{2}}{2}+\\frac{2πn}{2}", "steps": [ { "type": "interim", "title": "$$\\frac{\\frac{3π}{2}}{2}=\\frac{3π}{4}$$", "input": "\\frac{\\frac{3π}{2}}{2}", "steps": [ { "type": "step", "primary": "Apply the fraction rule: $$\\frac{\\frac{b}{c}}{a}=\\frac{b}{c\\:\\cdot\\:a}$$", "result": "=\\frac{3π}{2\\cdot\\:2}" }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:2=4$$", "result": "=\\frac{3π}{4}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7tiTmKkmeEqHWILbBb88ajZJTpMqJaq9bfSnb9kPhskfdd47a0hQ8flDbGsI5To1drfMRPynLSfP3VYe1OQldREkIgLCzuVxsE8bZJAi+zLMWtOub2dAxgaWacwKjDMkBc6krFpzteeOSStKZ2st+3LSXRhuEYkItk0ZmqvQzatokt3WiGR7ZaCaXvz77bMjS" } }, { "type": "interim", "title": "$$\\frac{2πn}{2}=πn$$", "input": "\\frac{2πn}{2}", "steps": [ { "type": "step", "primary": "Divide the numbers: $$\\frac{2}{2}=1$$", "result": "=πn" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7YVyjdFRW0jf8Bxzp61JzqHyRHuGw7+tM5METTDj6vVECnBmppnUV+797UqKeeyBQmGP7cF4dEaUuqswlNsH4qzvxhqyDNGf7EUmy3SORhvXPUvVtHPbCNTQhn1k5UbDm" } }, { "type": "step", "result": "=\\frac{3π}{4}+πn" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7tiTmKkmeEqHWILbBb88ajZ5yKgO88C1F1xZtWMQHGNW2z3vSzoDYIK8LwhuQjcbO3XeO2tIUPH5Q2xrCOU6NXa3zET8py0nz91WHtTkJXUTV9uDRRJh/E3msxuJMsId28LfSxJ+0AgVLpCSnLX0iSpuSlq7+cu05ZlcV/9tnDJcIdcrEX3CFtz7dlmgYCS4trsGj7x6oDpco7rkJ2wqU5oL+85zi6ibrzOxBmpP0LLM=" } }, { "type": "step", "result": "x=\\frac{3π}{4}+πn" } ], "meta": { "interimType": "Generic Simplify 0Eq" } } ], "meta": { "interimType": "Divide Both Sides Specific 1Eq", "gptData": "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" } } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve Title 1Eq" } }, { "type": "step", "result": "x=\\frac{3π}{4}+πn" }, { "type": "step", "primary": "Solutions for the range $$0\\le\\:x\\le\\:10π$$", "result": "x=\\frac{3π}{4},\\:x=\\frac{7π}{4},\\:x=\\frac{11π}{4},\\:x=\\frac{15π}{4},\\:x=\\frac{19π}{4}" } ], "meta": { "interimType": "N/A" } }, { "type": "interim", "title": "$$\\sin\\left(2x\\right)=-2,\\:0\\le\\:x\\le\\:10π{\\quad:\\quad}$$No Solution", "input": "\\sin\\left(2x\\right)=-2,\\:0\\le\\:x\\le\\:10π", "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=\\frac{3π}{4},\\:x=\\frac{7π}{4},\\:x=\\frac{11π}{4},\\:x=\\frac{15π}{4},\\:x=\\frac{19π}{4}" } ], "meta": { "solvingClass": "Trig Equations", "practiceLink": "/practice/trigonometry-practice#area=main&subtopic=Trig%20Equations", "practiceTopic": "Trig Equations" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "\\sin^{2}(2x)+3\\sin(2x)+2,0\\le x\\le 10π" }, "showViewLarger": true } }, "meta": { "showVerify": true } }