-2sin(t)=-sqrt(2)

{ "query": { "display": "$$-2\\sin\\left(t\\right)=-\\sqrt{2}$$", "symbolab_question": "EQUATION#-2\\sin(t)=-\\sqrt{2}" }, "solution": { "level": "PERFORMED", "subject": "Trigonometry", "topic": "Trig Equations", "subTopic": "Trig Equations", "default": "t=\\frac{π}{4}+2πn,t=\\frac{3π}{4}+2πn", "degrees": "t=45^{\\circ }+360^{\\circ }n,t=135^{\\circ }+360^{\\circ }n", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$-2\\sin\\left(t\\right)=-\\sqrt{2}{\\quad:\\quad}t=\\frac{π}{4}+2πn,\\:t=\\frac{3π}{4}+2πn$$", "input": "-2\\sin\\left(t\\right)=-\\sqrt{2}", "steps": [ { "type": "interim", "title": "Divide both sides by $$-2$$", "input": "-2\\sin\\left(t\\right)=-\\sqrt{2}", "result": "\\sin\\left(t\\right)=\\frac{\\sqrt{2}}{2}", "steps": [ { "type": "step", "primary": "Divide both sides by $$-2$$", "result": "\\frac{-2\\sin\\left(t\\right)}{-2}=\\frac{-\\sqrt{2}}{-2}" }, { "type": "step", "primary": "Simplify", "result": "\\sin\\left(t\\right)=\\frac{\\sqrt{2}}{2}" } ], "meta": { "interimType": "Divide Both Sides Specific 1Eq", "gptData": "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" } }, { "type": "interim", "title": "General solutions for $$\\sin\\left(t\\right)=\\frac{\\sqrt{2}}{2}$$", "result": "t=\\frac{π}{4}+2πn,\\:t=\\frac{3π}{4}+2πn", "steps": [ { "type": "step", "primary": "$$\\sin\\left(x\\right)$$ periodicity table with $$2πn$$ cycle:<br/>$$\\begin{array}{|c|c|c|c|}\\hline x&\\sin(x)&x&\\sin(x)\\\\\\hline 0&0&π&0\\\\\\hline \\frac{π}{6}&\\frac{1}{2}&\\frac{7π}{6}&-\\frac{1}{2}\\\\\\hline \\frac{π}{4}&\\frac{\\sqrt{2}}{2}&\\frac{5π}{4}&-\\frac{\\sqrt{2}}{2}\\\\\\hline \\frac{π}{3}&\\frac{\\sqrt{3}}{2}&\\frac{4π}{3}&-\\frac{\\sqrt{3}}{2}\\\\\\hline \\frac{π}{2}&1&\\frac{3π}{2}&-1\\\\\\hline \\frac{2π}{3}&\\frac{\\sqrt{3}}{2}&\\frac{5π}{3}&-\\frac{\\sqrt{3}}{2}\\\\\\hline \\frac{3π}{4}&\\frac{\\sqrt{2}}{2}&\\frac{7π}{4}&-\\frac{\\sqrt{2}}{2}\\\\\\hline \\frac{5π}{6}&\\frac{1}{2}&\\frac{11π}{6}&-\\frac{1}{2}\\\\\\hline \\end{array}$$" }, { "type": "step", "result": "t=\\frac{π}{4}+2πn,\\:t=\\frac{3π}{4}+2πn" } ], "meta": { "interimType": "Trig General Solutions sin 1Eq" } } ], "meta": { "solvingClass": "Trig Equations", "practiceLink": "/practice/trigonometry-practice#area=main&subtopic=Trig%20Equations", "practiceTopic": "Trig Equations" } }, "plot_output": { "meta": { "plotInfo": { "variable": "t", "plotRequest": "-2\\sin(t)+\\sqrt{2}" }, "showViewLarger": true } }, "meta": { "showVerify": true } }