solvefor x,sin^2(x)=((1-cos(y)))/2

{ "query": { "display": "solve for $$x,\\:\\sin^{2}\\left(x\\right)=\\frac{\\left(1-\\cos\\left(y\\right)\\right)}{2}$$", "symbolab_question": "SOLVE_FOR#solvefor x,\\sin^{2}(x)=\\frac{(1-\\cos(y))}{2}" }, "solution": { "level": "PERFORMED", "subject": "Trigonometry", "topic": "Trig Equations", "subTopic": "Trig Equations", "default": "x=\\arcsin(\\sqrt{\\frac{1-\\cos(y)}{2}})+2πn,x=π+\\arcsin(-\\sqrt{\\frac{1-\\cos(y)}{2}})+2πn,x=\\arcsin(-\\sqrt{\\frac{1-\\cos(y)}{2}})+2πn,x=π+\\arcsin(\\sqrt{\\frac{1-\\cos(y)}{2}})+2πn", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$solve\\:for\\:x,\\:\\sin^{2}\\left(x\\right)=\\frac{\\left(1-\\cos\\left(y\\right)\\right)}{2}{\\quad:\\quad}x=\\arcsin\\left(\\sqrt{\\frac{1-\\cos\\left(y\\right)}{2}}\\right)+2πn,\\:x=π+\\arcsin\\left(-\\sqrt{\\frac{1-\\cos\\left(y\\right)}{2}}\\right)+2πn,\\:x=\\arcsin\\left(-\\sqrt{\\frac{1-\\cos\\left(y\\right)}{2}}\\right)+2πn,\\:x=π+\\arcsin\\left(\\sqrt{\\frac{1-\\cos\\left(y\\right)}{2}}\\right)+2πn$$", "input": "\\sin^{2}\\left(x\\right)=\\frac{\\left(1-\\cos\\left(y\\right)\\right)}{2}", "steps": [ { "type": "interim", "title": "Solve by substitution", "input": "\\sin^{2}\\left(x\\right)=\\frac{1-\\cos\\left(y\\right)}{2}", "result": "\\sin\\left(x\\right)=\\sqrt{\\frac{1-\\cos\\left(y\\right)}{2}},\\:\\sin\\left(x\\right)=-\\sqrt{\\frac{1-\\cos\\left(y\\right)}{2}}", "steps": [ { "type": "step", "primary": "Let: $$\\sin\\left(x\\right)=u$$", "result": "u^{2}=\\frac{1-\\cos\\left(y\\right)}{2}" }, { "type": "interim", "title": "$$u^{2}=\\frac{1-\\cos\\left(y\\right)}{2}{\\quad:\\quad}u=\\sqrt{\\frac{1-\\cos\\left(y\\right)}{2}},\\:u=-\\sqrt{\\frac{1-\\cos\\left(y\\right)}{2}}$$", "input": "u^{2}=\\frac{1-\\cos\\left(y\\right)}{2}", "steps": [ { "type": "step", "primary": "For $$x^{2}=f\\left(a\\right)$$ the solutions are $$x=\\sqrt{f\\left(a\\right)},\\:\\:-\\sqrt{f\\left(a\\right)}$$" }, { "type": "step", "result": "u=\\sqrt{\\frac{1-\\cos\\left(y\\right)}{2}},\\:u=-\\sqrt{\\frac{1-\\cos\\left(y\\right)}{2}}" } ], "meta": { "solvingClass": "Equations", "interimType": "Equations" } }, { "type": "step", "primary": "Substitute back $$u=\\sin\\left(x\\right)$$", "result": "\\sin\\left(x\\right)=\\sqrt{\\frac{1-\\cos\\left(y\\right)}{2}},\\:\\sin\\left(x\\right)=-\\sqrt{\\frac{1-\\cos\\left(y\\right)}{2}}" } ], "meta": { "interimType": "Substitution Method 0Eq" } }, { "type": "interim", "title": "$$\\sin\\left(x\\right)=\\sqrt{\\frac{1-\\cos\\left(y\\right)}{2}}{\\quad:\\quad}x=\\arcsin\\left(\\sqrt{\\frac{1-\\cos\\left(y\\right)}{2}}\\right)+2πn,\\:x=π+\\arcsin\\left(-\\sqrt{\\frac{1-\\cos\\left(y\\right)}{2}}\\right)+2πn$$", "input": "\\sin\\left(x\\right)=\\sqrt{\\frac{1-\\cos\\left(y\\right)}{2}}", "steps": [ { "type": "interim", "title": "Apply trig inverse properties", "input": "\\sin\\left(x\\right)=\\sqrt{\\frac{1-\\cos\\left(y\\right)}{2}}", "result": "x=\\arcsin\\left(\\sqrt{\\frac{1-\\cos\\left(y\\right)}{2}}\\right)+2πn,\\:x=π+\\arcsin\\left(-\\sqrt{\\frac{1-\\cos\\left(y\\right)}{2}}\\right)+2πn", "steps": [ { "type": "step", "primary": "General solutions for $$\\sin\\left(x\\right)=\\sqrt{\\frac{1-\\cos\\left(y\\right)}{2}}$$", "secondary": [ "$$\\sin\\left(x\\right)=a\\quad\\Rightarrow\\quad\\:x=\\arcsin\\left(a\\right)+2πn,\\:\\quad\\:x=π+\\arcsin\\left(a\\right)+2πn$$" ], "result": "x=\\arcsin\\left(\\sqrt{\\frac{1-\\cos\\left(y\\right)}{2}}\\right)+2πn,\\:x=π+\\arcsin\\left(-\\sqrt{\\frac{1-\\cos\\left(y\\right)}{2}}\\right)+2πn" } ], "meta": { "interimType": "Trig Apply Inverse Props 0Eq" } } ], "meta": { "interimType": "N/A" } }, { "type": "interim", "title": "$$\\sin\\left(x\\right)=-\\sqrt{\\frac{1-\\cos\\left(y\\right)}{2}}{\\quad:\\quad}x=\\arcsin\\left(-\\sqrt{\\frac{1-\\cos\\left(y\\right)}{2}}\\right)+2πn,\\:x=π+\\arcsin\\left(\\sqrt{\\frac{1-\\cos\\left(y\\right)}{2}}\\right)+2πn$$", "input": "\\sin\\left(x\\right)=-\\sqrt{\\frac{1-\\cos\\left(y\\right)}{2}}", "steps": [ { "type": "interim", "title": "Apply trig inverse properties", "input": "\\sin\\left(x\\right)=-\\sqrt{\\frac{1-\\cos\\left(y\\right)}{2}}", "result": "x=\\arcsin\\left(-\\sqrt{\\frac{1-\\cos\\left(y\\right)}{2}}\\right)+2πn,\\:x=π+\\arcsin\\left(\\sqrt{\\frac{1-\\cos\\left(y\\right)}{2}}\\right)+2πn", "steps": [ { "type": "step", "primary": "General solutions for $$\\sin\\left(x\\right)=-\\sqrt{\\frac{1-\\cos\\left(y\\right)}{2}}$$", "secondary": [ "$$\\sin\\left(x\\right)=a\\quad\\Rightarrow\\quad\\:x=\\arcsin\\left(a\\right)+2πn,\\:\\quad\\:x=π+\\arcsin\\left(a\\right)+2πn$$" ], "result": "x=\\arcsin\\left(-\\sqrt{\\frac{1-\\cos\\left(y\\right)}{2}}\\right)+2πn,\\:x=π+\\arcsin\\left(\\sqrt{\\frac{1-\\cos\\left(y\\right)}{2}}\\right)+2πn" } ], "meta": { "interimType": "Trig Apply Inverse Props 0Eq" } } ], "meta": { "interimType": "N/A" } }, { "type": "step", "primary": "Combine all the solutions", "result": "x=\\arcsin\\left(\\sqrt{\\frac{1-\\cos\\left(y\\right)}{2}}\\right)+2πn,\\:x=π+\\arcsin\\left(-\\sqrt{\\frac{1-\\cos\\left(y\\right)}{2}}\\right)+2πn,\\:x=\\arcsin\\left(-\\sqrt{\\frac{1-\\cos\\left(y\\right)}{2}}\\right)+2πn,\\:x=π+\\arcsin\\left(\\sqrt{\\frac{1-\\cos\\left(y\\right)}{2}}\\right)+2πn" } ], "meta": { "solvingClass": "Trig Equations", "practiceLink": "/practice/trigonometry-practice#area=main&subtopic=Trig%20Equations", "practiceTopic": "Trig Equations" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "\\sin^{2}(x)-\\frac{(1-\\cos(y))}{2}" }, "showViewLarger": true } }, "meta": { "showVerify": true } }