sin(2x)-0.8=0

{ "query": { "display": "$$\\sin\\left(2x\\right)-0.8=0$$", "symbolab_question": "EQUATION#\\sin(2x)-0.8=0" }, "solution": { "level": "PERFORMED", "subject": "Trigonometry", "topic": "Trig Equations", "subTopic": "Trig Equations", "default": "x=\\frac{0.92729…}{2}+πn,x=\\frac{π}{2}-\\frac{0.92729…}{2}+πn", "degrees": "x=26.56505…^{\\circ }+180^{\\circ }n,x=63.43494…^{\\circ }+180^{\\circ }n", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\sin\\left(2x\\right)-0.8=0{\\quad:\\quad}x=\\frac{0.92729…}{2}+πn,\\:x=\\frac{π}{2}-\\frac{0.92729…}{2}+πn$$", "input": "\\sin\\left(2x\\right)-0.8=0", "steps": [ { "type": "interim", "title": "Move $$0.8\\:$$to the right side", "input": "\\sin\\left(2x\\right)-0.8=0", "result": "\\sin\\left(2x\\right)=0.8", "steps": [ { "type": "step", "primary": "Add $$0.8$$ to both sides", "result": "\\sin\\left(2x\\right)-0.8+0.8=0+0.8" }, { "type": "step", "primary": "Simplify", "result": "\\sin\\left(2x\\right)=0.8" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": "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" } }, { "type": "interim", "title": "Apply trig inverse properties", "input": "\\sin\\left(2x\\right)=0.8", "result": "2x=\\arcsin\\left(0.8\\right)+2πn,\\:2x=π-\\arcsin\\left(0.8\\right)+2πn", "steps": [ { "type": "step", "primary": "General solutions for $$\\sin\\left(2x\\right)=0.8$$", "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(0.8\\right)+2πn,\\:2x=π-\\arcsin\\left(0.8\\right)+2πn" } ], "meta": { "interimType": "Trig Apply Inverse Props 0Eq" } }, { "type": "interim", "title": "Solve $$2x=\\arcsin\\left(0.8\\right)+2πn:{\\quad}x=\\frac{\\arcsin\\left(0.8\\right)}{2}+πn$$", "input": "2x=\\arcsin\\left(0.8\\right)+2πn", "steps": [ { "type": "interim", "title": "Divide both sides by $$2$$", "input": "2x=\\arcsin\\left(0.8\\right)+2πn", "result": "x=\\frac{\\arcsin\\left(0.8\\right)}{2}+πn", "steps": [ { "type": "step", "primary": "Divide both sides by $$2$$", "result": "\\frac{2x}{2}=\\frac{\\arcsin\\left(0.8\\right)}{2}+\\frac{2πn}{2}" }, { "type": "step", "primary": "Simplify", "result": "x=\\frac{\\arcsin\\left(0.8\\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 $$2x=π-\\arcsin\\left(0.8\\right)+2πn:{\\quad}x=\\frac{π}{2}-\\frac{\\arcsin\\left(0.8\\right)}{2}+πn$$", "input": "2x=π-\\arcsin\\left(0.8\\right)+2πn", "steps": [ { "type": "interim", "title": "Divide both sides by $$2$$", "input": "2x=π-\\arcsin\\left(0.8\\right)+2πn", "result": "x=\\frac{π}{2}-\\frac{\\arcsin\\left(0.8\\right)}{2}+πn", "steps": [ { "type": "step", "primary": "Divide both sides by $$2$$", "result": "\\frac{2x}{2}=\\frac{π}{2}-\\frac{\\arcsin\\left(0.8\\right)}{2}+\\frac{2πn}{2}" }, { "type": "step", "primary": "Simplify", "result": "x=\\frac{π}{2}-\\frac{\\arcsin\\left(0.8\\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(0.8\\right)}{2}+πn,\\:x=\\frac{π}{2}-\\frac{\\arcsin\\left(0.8\\right)}{2}+πn" }, { "type": "step", "primary": "Show solutions in decimal form", "result": "x=\\frac{0.92729…}{2}+πn,\\:x=\\frac{π}{2}-\\frac{0.92729…}{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": "\\sin(2x)-0.8" }, "showViewLarger": true } }, "meta": { "showVerify": true } }