solvefor y,ln(x^2+10)+csc(y)=c

{ "query": { "display": "solve for $$y,\\:\\ln\\left(x^{2}+10\\right)+\\csc\\left(y\\right)=c$$", "symbolab_question": "SOLVE_FOR#solvefor y,\\ln(x^{2}+10)+\\csc(y)=c" }, "solution": { "level": "PERFORMED", "subject": "Trigonometry", "topic": "Trig Equations", "subTopic": "Trig Equations", "default": "y=\\arccsc(c-\\ln(x^{2}+10))+2πn,y=π+\\arccsc(-c+\\ln(x^{2}+10))+2πn", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$solve\\:for\\:y,\\:\\ln\\left(x^{2}+10\\right)+\\csc\\left(y\\right)=c{\\quad:\\quad}y=\\arccsc\\left(c-\\ln\\left(x^{2}+10\\right)\\right)+2πn,\\:y=π+\\arccsc\\left(-c+\\ln\\left(x^{2}+10\\right)\\right)+2πn$$", "input": "\\ln\\left(x^{2}+10\\right)+\\csc\\left(y\\right)=c", "steps": [ { "type": "interim", "title": "Move $$\\ln\\left(x^{2}+10\\right)\\:$$to the right side", "input": "\\ln\\left(x^{2}+10\\right)+\\csc\\left(y\\right)=c", "result": "\\csc\\left(y\\right)=c-\\ln\\left(x^{2}+10\\right)", "steps": [ { "type": "step", "primary": "Subtract $$\\ln\\left(x^{2}+10\\right)$$ from both sides", "result": "\\ln\\left(x^{2}+10\\right)+\\csc\\left(y\\right)-\\ln\\left(x^{2}+10\\right)=c-\\ln\\left(x^{2}+10\\right)" }, { "type": "step", "primary": "Simplify", "result": "\\csc\\left(y\\right)=c-\\ln\\left(x^{2}+10\\right)" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s72g5TZdkJwY/HFcjVCNUrn+wM+PfR+J0W/5SvfjqUAHw9G83iCe7WBZdMTMHRKFB8lebWmmhSIMEsjsOQRKarS3P8s+EM1Ne5lcxEywd3PPBEq/CkKwyb90gNYp2dJQm03MBmKeMiEzo8rM+eebUyBqStd8hIOfYSxp3nFALNiC5YBrYhzeKxE6dKEtxJqF9X2P9G1NgmTAdEyme8zwDr9QpcRLepjOMMzCaiiOME7MFf7P+wNS80rBqjBnsQ0EJ6JLhZDNt0lrzRBnGK85tP1rPf2wcqILJpKwQth0CAM2DkU5TkMMWES51yFsTkJT6lflB44+cAhbl05D7N1o4XnYUDxBuztWdWdEmOXBe3LLpWPNqzEhkP5qD6ioNU4s9fi9c/nHGhIqvPRpttdSi2lpJj/5xKKdB5FHwO4G/fHn8mxF5pxAejgMBdXf8NHl+mk3dK9hN6pPkzVNfz8kjct5mbarPQr4AzHT/tAXtM1FKK53zizIgYqLsqdv3wHnhFf1iKGIZH+SGJ+Pb2nr6myg==" } }, { "type": "interim", "title": "Apply trig inverse properties", "input": "\\csc\\left(y\\right)=c-\\ln\\left(x^{2}+10\\right)", "result": "y=\\arccsc\\left(c-\\ln\\left(x^{2}+10\\right)\\right)+2πn,\\:y=π+\\arccsc\\left(-c+\\ln\\left(x^{2}+10\\right)\\right)+2πn", "steps": [ { "type": "step", "primary": "General solutions for $$\\csc\\left(y\\right)=c-\\ln\\left(x^{2}+10\\right)$$", "secondary": [ "$$\\csc\\left(x\\right)=a\\quad\\Rightarrow\\quad\\:x=\\arccsc\\left(a\\right)+2πn,\\:\\quad\\:x=π+\\arccsc\\left(a\\right)+2πn$$" ], "result": "y=\\arccsc\\left(c-\\ln\\left(x^{2}+10\\right)\\right)+2πn,\\:y=π+\\arccsc\\left(-c+\\ln\\left(x^{2}+10\\right)\\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": "y", "plotRequest": "\\ln(x^{2}+10)+\\csc(y)=c" }, "showViewLarger": true } }, "meta": { "showVerify": true } }