solvefor y,a*z=5sin(2y)

{ "query": { "display": "solve for $$y,\\:a\\cdot\\:z=5\\sin\\left(2y\\right)$$", "symbolab_question": "SOLVE_FOR#solvefor y,a\\cdot z=5\\sin(2y)" }, "solution": { "level": "PERFORMED", "subject": "Trigonometry", "topic": "Trig Equations", "subTopic": "Trig Equations", "default": "y=\\frac{\\arcsin(\\frac{az}{5})}{2}+πn,y=\\frac{π}{2}+\\frac{\\arcsin(-\\frac{az}{5})}{2}+πn", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$solve\\:for\\:y,\\:az=5\\sin\\left(2y\\right){\\quad:\\quad}y=\\frac{\\arcsin\\left(\\frac{az}{5}\\right)}{2}+πn,\\:y=\\frac{π}{2}+\\frac{\\arcsin\\left(-\\frac{az}{5}\\right)}{2}+πn$$", "input": "az=5\\sin\\left(2y\\right)", "steps": [ { "type": "step", "primary": "Switch sides", "result": "5\\sin\\left(2y\\right)=az" }, { "type": "interim", "title": "Divide both sides by $$5$$", "input": "5\\sin\\left(2y\\right)=az", "result": "\\sin\\left(2y\\right)=\\frac{az}{5}", "steps": [ { "type": "step", "primary": "Divide both sides by $$5$$", "result": "\\frac{5\\sin\\left(2y\\right)}{5}=\\frac{az}{5}" }, { "type": "step", "primary": "Simplify", "result": "\\sin\\left(2y\\right)=\\frac{az}{5}" } ], "meta": { "interimType": "Divide Both Sides Specific 1Eq", "gptData": "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" } }, { "type": "interim", "title": "Apply trig inverse properties", "input": "\\sin\\left(2y\\right)=\\frac{az}{5}", "result": "2y=\\arcsin\\left(\\frac{az}{5}\\right)+2πn,\\:2y=π+\\arcsin\\left(-\\frac{az}{5}\\right)+2πn", "steps": [ { "type": "step", "primary": "General solutions for $$\\sin\\left(2y\\right)=\\frac{az}{5}$$", "secondary": [ "$$\\sin\\left(x\\right)=a\\quad\\Rightarrow\\quad\\:x=\\arcsin\\left(a\\right)+2πn,\\:\\quad\\:x=π+\\arcsin\\left(a\\right)+2πn$$" ], "result": "2y=\\arcsin\\left(\\frac{az}{5}\\right)+2πn,\\:2y=π+\\arcsin\\left(-\\frac{az}{5}\\right)+2πn" } ], "meta": { "interimType": "Trig Apply Inverse Props 0Eq" } }, { "type": "interim", "title": "Solve $$2y=\\arcsin\\left(\\frac{az}{5}\\right)+2πn:{\\quad}y=\\frac{\\arcsin\\left(\\frac{az}{5}\\right)}{2}+πn$$", "input": "2y=\\arcsin\\left(\\frac{az}{5}\\right)+2πn", "steps": [ { "type": "interim", "title": "Divide both sides by $$2$$", "input": "2y=\\arcsin\\left(\\frac{az}{5}\\right)+2πn", "result": "y=\\frac{\\arcsin\\left(\\frac{az}{5}\\right)}{2}+πn", "steps": [ { "type": "step", "primary": "Divide both sides by $$2$$", "result": "\\frac{2y}{2}=\\frac{\\arcsin\\left(\\frac{az}{5}\\right)}{2}+\\frac{2πn}{2}" }, { "type": "step", "primary": "Simplify", "result": "y=\\frac{\\arcsin\\left(\\frac{az}{5}\\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 $$2y=π+\\arcsin\\left(-\\frac{az}{5}\\right)+2πn:{\\quad}y=\\frac{π}{2}+\\frac{\\arcsin\\left(-\\frac{az}{5}\\right)}{2}+πn$$", "input": "2y=π+\\arcsin\\left(-\\frac{az}{5}\\right)+2πn", "steps": [ { "type": "interim", "title": "Divide both sides by $$2$$", "input": "2y=π+\\arcsin\\left(-\\frac{az}{5}\\right)+2πn", "result": "y=\\frac{π}{2}+\\frac{\\arcsin\\left(-\\frac{az}{5}\\right)}{2}+πn", "steps": [ { "type": "step", "primary": "Divide both sides by $$2$$", "result": "\\frac{2y}{2}=\\frac{π}{2}+\\frac{\\arcsin\\left(-\\frac{az}{5}\\right)}{2}+\\frac{2πn}{2}" }, { "type": "step", "primary": "Simplify", "result": "y=\\frac{π}{2}+\\frac{\\arcsin\\left(-\\frac{az}{5}\\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": "y=\\frac{\\arcsin\\left(\\frac{az}{5}\\right)}{2}+πn,\\:y=\\frac{π}{2}+\\frac{\\arcsin\\left(-\\frac{az}{5}\\right)}{2}+πn" } ], "meta": { "solvingClass": "Trig Equations", "practiceLink": "/practice/trigonometry-practice#area=main&subtopic=Trig%20Equations", "practiceTopic": "Trig Equations" } }, "plot_output": { "meta": { "plotInfo": { "variable": "y", "plotRequest": "a\\cdot z=5\\sin(2y)" }, "showViewLarger": true } }, "meta": { "showVerify": true } }