solvefor x,sin(x)=e^y

{ "query": { "display": "solve for $$x,\\:\\sin\\left(x\\right)=e^{y}$$", "symbolab_question": "SOLVE_FOR#solvefor x,\\sin(x)=e^{y}" }, "solution": { "level": "PERFORMED", "subject": "Trigonometry", "topic": "Trig Equations", "subTopic": "Trig Equations", "default": "x=\\arcsin(e^{y})+2πn,x=π+\\arcsin(-e^{y})+2πn", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$solve\\:for\\:x,\\:\\sin\\left(x\\right)=e^{y}{\\quad:\\quad}x=\\arcsin\\left(e^{y}\\right)+2πn,\\:x=π+\\arcsin\\left(-e^{y}\\right)+2πn$$", "input": "\\sin\\left(x\\right)=e^{y}", "steps": [ { "type": "interim", "title": "Apply trig inverse properties", "input": "\\sin\\left(x\\right)=e^{y}", "result": "x=\\arcsin\\left(e^{y}\\right)+2πn,\\:x=π+\\arcsin\\left(-e^{y}\\right)+2πn", "steps": [ { "type": "step", "primary": "General solutions for $$\\sin\\left(x\\right)=e^{y}$$", "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(e^{y}\\right)+2πn,\\:x=π+\\arcsin\\left(-e^{y}\\right)+2πn" } ], "meta": { "interimType": "Trig Apply Inverse Props 0Eq" } } ], "meta": { "solvingClass": "Trig Equations", "practiceLink": "/practice/trigonometry-practice#area=main&subtopic=Trig%20Equations", "practiceTopic": "Trig Equations" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "\\sin(x)-e^{y}" }, "showViewLarger": true } }, "meta": { "showVerify": true } }