sin(2x)=sin(x)

{ "query": { "display": "$$\\sin\\left(2x\\right)=\\sin\\left(x\\right)$$", "symbolab_question": "EQUATION#\\sin(2x)=\\sin(x)" }, "solution": { "level": "PERFORMED", "subject": "Trigonometry", "topic": "Trig Equations", "subTopic": "Trig Equations", "default": "x=2πn,x=π+2πn,x=\\frac{π}{3}+2πn,x=\\frac{5π}{3}+2πn", "degrees": "x=0^{\\circ }+360^{\\circ }n,x=180^{\\circ }+360^{\\circ }n,x=60^{\\circ }+360^{\\circ }n,x=300^{\\circ }+360^{\\circ }n", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\sin\\left(2x\\right)=\\sin\\left(x\\right){\\quad:\\quad}x=2πn,\\:x=π+2πn,\\:x=\\frac{π}{3}+2πn,\\:x=\\frac{5π}{3}+2πn$$", "input": "\\sin\\left(2x\\right)=\\sin\\left(x\\right)", "steps": [ { "type": "step", "primary": "Subtract $$\\sin\\left(x\\right)$$ from both sides", "result": "\\sin\\left(2x\\right)-\\sin\\left(x\\right)=0" }, { "type": "interim", "title": "Rewrite using trig identities", "input": "\\sin\\left(2x\\right)-\\sin\\left(x\\right)", "result": "-\\sin\\left(x\\right)+2\\cos\\left(x\\right)\\sin\\left(x\\right)=0", "steps": [ { "type": "step", "primary": "Use the Double Angle identity: $$\\sin\\left(2x\\right)=2\\sin\\left(x\\right)\\cos\\left(x\\right)$$", "result": "=2\\sin\\left(x\\right)\\cos\\left(x\\right)-\\sin\\left(x\\right)" } ], "meta": { "interimType": "Trig Rewrite Using Trig identities 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7Awm8OHaUUE5X73KqyTFjFCwx6U47IBIVDH0o0oUI9guyuRcnFgnh8AijhYNjhqyRbyZYyX6EGMAcAMgjOADqPLajIxP2OrjkBCECQd8a6w5LG+giPnP5gBF4jt66ktiEztrFxfa7HutDYNeratXgsqdpa3M4SuhKB9tuevFDkRirve3E7cDlwD8G9VYfu6duTZGH8SqHtUWuYa2dpw0bulhsd89TDzuC3BUug+Pj7eQ=" } }, { "type": "interim", "title": "Factor $$-\\sin\\left(x\\right)+2\\cos\\left(x\\right)\\sin\\left(x\\right):{\\quad}\\sin\\left(x\\right)\\left(2\\cos\\left(x\\right)-1\\right)$$", "input": "-\\sin\\left(x\\right)+2\\cos\\left(x\\right)\\sin\\left(x\\right)", "steps": [ { "type": "step", "primary": "Factor out common term $$\\sin\\left(x\\right)$$", "result": "=\\sin\\left(x\\right)\\left(-1+2\\cos\\left(x\\right)\\right)", "meta": { "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Factor Title 1Eq" } }, { "type": "step", "result": "\\sin\\left(x\\right)\\left(2\\cos\\left(x\\right)-1\\right)=0" }, { "type": "step", "primary": "Solving each part separately", "result": "\\sin\\left(x\\right)=0\\lor\\:2\\cos\\left(x\\right)-1=0" }, { "type": "interim", "title": "$$\\sin\\left(x\\right)=0{\\quad:\\quad}x=2πn,\\:x=π+2πn$$", "input": "\\sin\\left(x\\right)=0", "steps": [ { "type": "interim", "title": "General solutions for $$\\sin\\left(x\\right)=0$$", "result": "x=0+2πn,\\:x=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": "x=0+2πn,\\:x=π+2πn" } ], "meta": { "interimType": "Trig General Solutions sin 1Eq" } }, { "type": "interim", "title": "Solve $$x=0+2πn:{\\quad}x=2πn$$", "input": "x=0+2πn", "steps": [ { "type": "step", "primary": "$$0+2πn=2πn$$", "result": "x=2πn" } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve Title 1Eq" } }, { "type": "step", "result": "x=2πn,\\:x=π+2πn" } ], "meta": { "solvingClass": "Trig Equations", "interimType": "Trig Equations" } }, { "type": "interim", "title": "$$2\\cos\\left(x\\right)-1=0{\\quad:\\quad}x=\\frac{π}{3}+2πn,\\:x=\\frac{5π}{3}+2πn$$", "input": "2\\cos\\left(x\\right)-1=0", "steps": [ { "type": "interim", "title": "Move $$1\\:$$to the right side", "input": "2\\cos\\left(x\\right)-1=0", "result": "2\\cos\\left(x\\right)=1", "steps": [ { "type": "step", "primary": "Add $$1$$ to both sides", "result": "2\\cos\\left(x\\right)-1+1=0+1" }, { "type": "step", "primary": "Simplify", "result": "2\\cos\\left(x\\right)=1" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": "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" } }, { "type": "interim", "title": "Divide both sides by $$2$$", "input": "2\\cos\\left(x\\right)=1", "result": "\\cos\\left(x\\right)=\\frac{1}{2}", "steps": [ { "type": "step", "primary": "Divide both sides by $$2$$", "result": "\\frac{2\\cos\\left(x\\right)}{2}=\\frac{1}{2}" }, { "type": "step", "primary": "Simplify", "result": "\\cos\\left(x\\right)=\\frac{1}{2}" } ], "meta": { "interimType": "Divide Both Sides Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s791AXYceO9Pc4CRAMY99YBc+Pqip+BMTSVx/OhjiEJApPnie/F7TDTqTxkucfXTe/L2LifiwEM4YYN0r1csQ0N26DDJHWnEbFgS5QJ7Yus3MjkYhWaYooxcLQLbyMdQYurTH6CH2AfoyKEr0+/Uxe+cZoxkl7XKCNfz0BLTb8L1fdFpY8+TJQURh2lEt0g234o2OhmBoV9GPJfBtlImn8h6aOQILKf053R9DIJaaWk7OAsiZT63gHQrxAXdE3jN0NdUkTAmTMqpaBfPwBiUB95XYpdcFaa/ISm/camJKLkN1+OlgaZLYE9gR5RL4J0wxEZodDCpUQ5bR2h/6GBALERYRaMe7VuinvQ0IJdEqmbZ9H30koNqZdcC0ISYHUsNsGlsMxPN+Du666aYYZH5j60ZIcOPtBjEwf5NB6rJgRkL46ykpGEGxsjZM9XQCCmA2ItQunUuCLvpwGYAhvio+mTW9a4n83XX1fDim1PknhUaB6pfF1z6umzUJTJvt+ojYZc7mh9SH8T2tyRGdJc3t6behdx4O9Aw/as2/krPxNmHE=" } }, { "type": "interim", "title": "General solutions for $$\\cos\\left(x\\right)=\\frac{1}{2}$$", "result": "x=\\frac{π}{3}+2πn,\\:x=\\frac{5π}{3}+2πn", "steps": [ { "type": "step", "primary": "$$\\cos\\left(x\\right)$$ periodicity table with $$2πn$$ cycle:<br/>$$\\begin{array}{|c|c|c|c|}\\hline x&\\cos(x)&x&\\cos(x)\\\\\\hline 0&1&π&-1\\\\\\hline \\frac{π}{6}&\\frac{\\sqrt{3}}{2}&\\frac{7π}{6}&-\\frac{\\sqrt{3}}{2}\\\\\\hline \\frac{π}{4}&\\frac{\\sqrt{2}}{2}&\\frac{5π}{4}&-\\frac{\\sqrt{2}}{2}\\\\\\hline \\frac{π}{3}&\\frac{1}{2}&\\frac{4π}{3}&-\\frac{1}{2}\\\\\\hline \\frac{π}{2}&0&\\frac{3π}{2}&0\\\\\\hline \\frac{2π}{3}&-\\frac{1}{2}&\\frac{5π}{3}&\\frac{1}{2}\\\\\\hline \\frac{3π}{4}&-\\frac{\\sqrt{2}}{2}&\\frac{7π}{4}&\\frac{\\sqrt{2}}{2}\\\\\\hline \\frac{5π}{6}&-\\frac{\\sqrt{3}}{2}&\\frac{11π}{6}&\\frac{\\sqrt{3}}{2}\\\\\\hline \\end{array}$$" }, { "type": "step", "result": "x=\\frac{π}{3}+2πn,\\:x=\\frac{5π}{3}+2πn" } ], "meta": { "interimType": "Trig General Solutions cos 1Eq" } } ], "meta": { "solvingClass": "Trig Equations", "interimType": "Trig Equations" } }, { "type": "step", "primary": "Combine all the solutions", "result": "x=2πn,\\:x=π+2πn,\\:x=\\frac{π}{3}+2πn,\\:x=\\frac{5π}{3}+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(2x)-\\sin(x)" }, "showViewLarger": true } }, "meta": { "showVerify": true } }