sin(4(x-pi/4))=0

{ "query": { "display": "$$\\sin\\left(4\\left(x-\\frac{π}{4}\\right)\\right)=0$$", "symbolab_question": "EQUATION#\\sin(4(x-\\frac{π}{4}))=0" }, "solution": { "level": "PERFORMED", "subject": "Trigonometry", "topic": "Trig Equations", "subTopic": "Trig Equations", "default": "x=\\frac{πn}{2}+\\frac{π}{4},x=\\frac{π+πn}{2}", "degrees": "x=45^{\\circ }+90^{\\circ }n,x=90^{\\circ }+90^{\\circ }n", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\sin\\left(4\\left(x-\\frac{π}{4}\\right)\\right)=0{\\quad:\\quad}x=\\frac{πn}{2}+\\frac{π}{4},\\:x=\\frac{π+πn}{2}$$", "input": "\\sin\\left(4\\left(x-\\frac{π}{4}\\right)\\right)=0", "steps": [ { "type": "interim", "title": "General solutions for $$\\sin\\left(4\\left(x-\\frac{π}{4}\\right)\\right)=0$$", "result": "4\\left(x-\\frac{π}{4}\\right)=0+2πn,\\:4\\left(x-\\frac{π}{4}\\right)=π+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": "4\\left(x-\\frac{π}{4}\\right)=0+2πn,\\:4\\left(x-\\frac{π}{4}\\right)=π+2πn" } ], "meta": { "interimType": "Trig General Solutions sin 1Eq" } }, { "type": "interim", "title": "Solve $$4\\left(x-\\frac{π}{4}\\right)=0+2πn:{\\quad}x=\\frac{πn}{2}+\\frac{π}{4}$$", "input": "4\\left(x-\\frac{π}{4}\\right)=0+2πn", "steps": [ { "type": "step", "primary": "$$0+2πn=2πn$$", "result": "4\\left(x-\\frac{π}{4}\\right)=2πn" }, { "type": "interim", "title": "Divide both sides by $$4$$", "input": "4\\left(x-\\frac{π}{4}\\right)=2πn", "result": "x-\\frac{π}{4}=\\frac{πn}{2}", "steps": [ { "type": "step", "primary": "Divide both sides by $$4$$", "result": "\\frac{4\\left(x-\\frac{π}{4}\\right)}{4}=\\frac{2πn}{4}" }, { "type": "step", "primary": "Simplify", "result": "x-\\frac{π}{4}=\\frac{πn}{2}" } ], "meta": { "interimType": "Divide Both Sides Specific 1Eq", "gptData": "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" } }, { "type": "interim", "title": "Move $$\\frac{π}{4}\\:$$to the right side", "input": "x-\\frac{π}{4}=\\frac{πn}{2}", "result": "x=\\frac{πn}{2}+\\frac{π}{4}", "steps": [ { "type": "step", "primary": "Add $$\\frac{π}{4}$$ to both sides", "result": "x-\\frac{π}{4}+\\frac{π}{4}=\\frac{πn}{2}+\\frac{π}{4}" }, { "type": "step", "primary": "Simplify", "result": "x=\\frac{πn}{2}+\\frac{π}{4}" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": "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" } } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve Title 1Eq" } }, { "type": "interim", "title": "Solve $$4\\left(x-\\frac{π}{4}\\right)=π+2πn:{\\quad}x=\\frac{π+πn}{2}$$", "input": "4\\left(x-\\frac{π}{4}\\right)=π+2πn", "steps": [ { "type": "interim", "title": "Divide both sides by $$4$$", "input": "4\\left(x-\\frac{π}{4}\\right)=π+2πn", "result": "x-\\frac{π}{4}=\\frac{π}{4}+\\frac{πn}{2}", "steps": [ { "type": "step", "primary": "Divide both sides by $$4$$", "result": "\\frac{4\\left(x-\\frac{π}{4}\\right)}{4}=\\frac{π}{4}+\\frac{2πn}{4}" }, { "type": "step", "primary": "Simplify", "result": "x-\\frac{π}{4}=\\frac{π}{4}+\\frac{πn}{2}" } ], "meta": { "interimType": "Divide Both Sides Specific 1Eq", "gptData": "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" } }, { "type": "interim", "title": "Move $$\\frac{π}{4}\\:$$to the right side", "input": "x-\\frac{π}{4}=\\frac{π}{4}+\\frac{πn}{2}", "result": "x=\\frac{π+πn}{2}", "steps": [ { "type": "step", "primary": "Add $$\\frac{π}{4}$$ to both sides", "result": "x-\\frac{π}{4}+\\frac{π}{4}=\\frac{π}{4}+\\frac{πn}{2}+\\frac{π}{4}" }, { "type": "interim", "title": "Simplify", "input": "x-\\frac{π}{4}+\\frac{π}{4}=\\frac{π}{4}+\\frac{πn}{2}+\\frac{π}{4}", "result": "x=\\frac{π+πn}{2}", "steps": [ { "type": "interim", "title": "Simplify $$x-\\frac{π}{4}+\\frac{π}{4}:{\\quad}x$$", "input": "x-\\frac{π}{4}+\\frac{π}{4}", "steps": [ { "type": "step", "primary": "Add similar elements: $$-\\frac{π}{4}+\\frac{π}{4}=0$$" }, { "type": "step", "result": "=x" } ], "meta": { "interimType": "Generic Simplify Specific 1Eq" } }, { "type": "interim", "title": "Simplify $$\\frac{π}{4}+\\frac{πn}{2}+\\frac{π}{4}:{\\quad}\\frac{π+πn}{2}$$", "input": "\\frac{π}{4}+\\frac{πn}{2}+\\frac{π}{4}", "steps": [ { "type": "step", "primary": "Group like terms", "result": "=\\frac{π}{4}+\\frac{π}{4}+\\frac{πn}{2}" }, { "type": "interim", "title": "Combine the fractions $$\\frac{π}{4}+\\frac{π}{4}:{\\quad}\\frac{π}{2}$$", "result": "=\\frac{π}{2}+\\frac{πn}{2}", "steps": [ { "type": "step", "primary": "Apply rule $$\\frac{a}{c}\\pm\\frac{b}{c}=\\frac{a\\pm\\:b}{c}$$", "result": "=\\frac{π+π}{4}" }, { "type": "step", "primary": "Add similar elements: $$π+π=2π$$", "result": "=\\frac{2π}{4}" }, { "type": "step", "primary": "Cancel the common factor: $$2$$", "result": "=\\frac{π}{2}" } ], "meta": { "interimType": "LCD Top Title 1Eq" } }, { "type": "step", "primary": "Since the denominators are equal, combine the fractions: $$\\frac{a}{c}\\pm\\frac{b}{c}=\\frac{a\\pm\\:b}{c}$$", "result": "=\\frac{π+πn}{2}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7OJ2UrfHP7HArhS+8hRjY5gltyh474n9QDGHXrSr9d33besk31dEFyg5Juz9K/kvLICf2WQN9mSJxQaQ7cQX4il2BgpogRffZXrJl2fYiHyfEWADrSjp4QZpFbYX2vtqP8LfSxJ+0AgVLpCSnLX0iSpuSlq7+cu05ZlcV/9tnDJdOHF8mq7W2qxw5yzijxw5mbmMfxEW8Z3NfoDSv1XaIG+DWWff852g/EVWUsaGA5IQ=" } }, { "type": "step", "result": "x=\\frac{π+πn}{2}" } ], "meta": { "interimType": "Generic Simplify 0Eq" } } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": "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" } } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve Title 1Eq" } }, { "type": "step", "result": "x=\\frac{πn}{2}+\\frac{π}{4},\\:x=\\frac{π+πn}{2}" } ], "meta": { "solvingClass": "Trig Equations", "practiceLink": "/practice/trigonometry-practice#area=main&subtopic=Trig%20Equations", "practiceTopic": "Trig Equations" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "\\sin(4(x-\\frac{π}{4}))" }, "showViewLarger": true } }, "meta": { "showVerify": true } }