solvefor x,r^2=sin(2x)

{ "query": { "display": "solve for $$x,\\:r^{2}=\\sin\\left(2x\\right)$$", "symbolab_question": "SOLVE_FOR#solvefor x,r^{2}=\\sin(2x)" }, "solution": { "level": "PERFORMED", "subject": "Trigonometry", "topic": "Trig Equations", "subTopic": "Trig Equations", "default": "x=\\frac{\\arcsin(r^{2})}{2}+πn,x=\\frac{π}{2}+\\frac{\\arcsin(-r^{2})}{2}+πn", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$solve\\:for\\:x,\\:r^{2}=\\sin\\left(2x\\right){\\quad:\\quad}x=\\frac{\\arcsin\\left(r^{2}\\right)}{2}+πn,\\:x=\\frac{π}{2}+\\frac{\\arcsin\\left(-r^{2}\\right)}{2}+πn$$", "input": "r^{2}=\\sin\\left(2x\\right)", "steps": [ { "type": "step", "primary": "Switch sides", "result": "\\sin\\left(2x\\right)=r^{2}" }, { "type": "interim", "title": "Apply trig inverse properties", "input": "\\sin\\left(2x\\right)=r^{2}", "result": "2x=\\arcsin\\left(r^{2}\\right)+2πn,\\:2x=π+\\arcsin\\left(-r^{2}\\right)+2πn", "steps": [ { "type": "step", "primary": "General solutions for $$\\sin\\left(2x\\right)=r^{2}$$", "secondary": [ "$$\\sin\\left(x\\right)=a\\quad\\Rightarrow\\quad\\:x=\\arcsin\\left(a\\right)+2πn,\\:\\quad\\:x=π+\\arcsin\\left(a\\right)+2πn$$" ], "result": "2x=\\arcsin\\left(r^{2}\\right)+2πn,\\:2x=π+\\arcsin\\left(-r^{2}\\right)+2πn" } ], "meta": { "interimType": "Trig Apply Inverse Props 0Eq" } }, { "type": "interim", "title": "Solve $$2x=\\arcsin\\left(r^{2}\\right)+2πn:{\\quad}x=\\frac{\\arcsin\\left(r^{2}\\right)}{2}+πn$$", "input": "2x=\\arcsin\\left(r^{2}\\right)+2πn", "steps": [ { "type": "interim", "title": "Divide both sides by $$2$$", "input": "2x=\\arcsin\\left(r^{2}\\right)+2πn", "result": "x=\\frac{\\arcsin\\left(r^{2}\\right)}{2}+πn", "steps": [ { "type": "step", "primary": "Divide both sides by $$2$$", "result": "\\frac{2x}{2}=\\frac{\\arcsin\\left(r^{2}\\right)}{2}+\\frac{2πn}{2}" }, { "type": "step", "primary": "Simplify", "result": "x=\\frac{\\arcsin\\left(r^{2}\\right)}{2}+πn" } ], "meta": { "interimType": "Divide Both Sides Specific 1Eq", "gptData": "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" } } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve Title 1Eq" } }, { "type": "interim", "title": "Solve $$2x=π+\\arcsin\\left(-r^{2}\\right)+2πn:{\\quad}x=\\frac{π}{2}+\\frac{\\arcsin\\left(-r^{2}\\right)}{2}+πn$$", "input": "2x=π+\\arcsin\\left(-r^{2}\\right)+2πn", "steps": [ { "type": "interim", "title": "Divide both sides by $$2$$", "input": "2x=π+\\arcsin\\left(-r^{2}\\right)+2πn", "result": "x=\\frac{π}{2}+\\frac{\\arcsin\\left(-r^{2}\\right)}{2}+πn", "steps": [ { "type": "step", "primary": "Divide both sides by $$2$$", "result": "\\frac{2x}{2}=\\frac{π}{2}+\\frac{\\arcsin\\left(-r^{2}\\right)}{2}+\\frac{2πn}{2}" }, { "type": "step", "primary": "Simplify", "result": "x=\\frac{π}{2}+\\frac{\\arcsin\\left(-r^{2}\\right)}{2}+πn" } ], "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{\\arcsin\\left(r^{2}\\right)}{2}+πn,\\:x=\\frac{π}{2}+\\frac{\\arcsin\\left(-r^{2}\\right)}{2}+πn" } ], "meta": { "solvingClass": "Trig Equations", "practiceLink": "/practice/trigonometry-practice#area=main&subtopic=Trig%20Equations", "practiceTopic": "Trig Equations" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "r^{2}-\\sin(2x)" }, "showViewLarger": true } }, "meta": { "showVerify": true } }