sin(2p+1)=-1

{ "query": { "display": "$$\\sin\\left(2p+1\\right)=-1$$", "symbolab_question": "EQUATION#\\sin(2p+1)=-1" }, "solution": { "level": "PERFORMED", "subject": "Trigonometry", "topic": "Trig Equations", "subTopic": "Trig Equations", "default": "p=πn-\\frac{1}{2}+\\frac{3π}{4}", "degrees": "p=106.35211…^{\\circ }+180^{\\circ }n", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\sin\\left(2p+1\\right)=-1{\\quad:\\quad}p=πn-\\frac{1}{2}+\\frac{3π}{4}$$", "input": "\\sin\\left(2p+1\\right)=-1", "steps": [ { "type": "interim", "title": "General solutions for $$\\sin\\left(2p+1\\right)=-1$$", "result": "2p+1=\\frac{3π}{2}+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": "2p+1=\\frac{3π}{2}+2πn" } ], "meta": { "interimType": "Trig General Solutions sin 1Eq" } }, { "type": "interim", "title": "Solve $$2p+1=\\frac{3π}{2}+2πn:{\\quad}p=πn-\\frac{1}{2}+\\frac{3π}{4}$$", "input": "2p+1=\\frac{3π}{2}+2πn", "steps": [ { "type": "interim", "title": "Move $$1\\:$$to the right side", "input": "2p+1=\\frac{3π}{2}+2πn", "result": "2p=\\frac{3π}{2}+2πn-1", "steps": [ { "type": "step", "primary": "Subtract $$1$$ from both sides", "result": "2p+1-1=\\frac{3π}{2}+2πn-1" }, { "type": "step", "primary": "Simplify", "result": "2p=\\frac{3π}{2}+2πn-1" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": "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" } }, { "type": "interim", "title": "Divide both sides by $$2$$", "input": "2p=\\frac{3π}{2}+2πn-1", "result": "p=πn-\\frac{1}{2}+\\frac{3π}{4}", "steps": [ { "type": "step", "primary": "Divide both sides by $$2$$", "result": "\\frac{2p}{2}=\\frac{\\frac{3π}{2}}{2}+\\frac{2πn}{2}-\\frac{1}{2}" }, { "type": "interim", "title": "Simplify", "input": "\\frac{2p}{2}=\\frac{\\frac{3π}{2}}{2}+\\frac{2πn}{2}-\\frac{1}{2}", "result": "p=πn-\\frac{1}{2}+\\frac{3π}{4}", "steps": [ { "type": "interim", "title": "Simplify $$\\frac{2p}{2}:{\\quad}p$$", "input": "\\frac{2p}{2}", "steps": [ { "type": "step", "primary": "Divide the numbers: $$\\frac{2}{2}=1$$", "result": "=p" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7GIZANqUgzR8yHruTN5GrUC061ljBSPJeENOw2efoSWvqvR7bPoVUSAwyzxDPDyaVRSpN33oxZMojoqvYhvSJACLkpiVA7S8UT8ieezZKu6qRige2M8QtLks6QRM9Mk9w" } }, { "type": "interim", "title": "Simplify $$\\frac{\\frac{3π}{2}}{2}+\\frac{2πn}{2}-\\frac{1}{2}:{\\quad}πn-\\frac{1}{2}+\\frac{3π}{4}$$", "input": "\\frac{\\frac{3π}{2}}{2}+\\frac{2πn}{2}-\\frac{1}{2}", "steps": [ { "type": "step", "primary": "Group like terms", "result": "=-\\frac{1}{2}+\\frac{2πn}{2}+\\frac{\\frac{3π}{2}}{2}" }, { "type": "interim", "title": "$$\\frac{2πn}{2}=πn$$", "input": "\\frac{2πn}{2}", "steps": [ { "type": "step", "primary": "Divide the numbers: $$\\frac{2}{2}=1$$", "result": "=πn" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7YVyjdFRW0jf8Bxzp61JzqHyRHuGw7+tM5METTDj6vVECnBmppnUV+797UqKeeyBQmGP7cF4dEaUuqswlNsH4qzvxhqyDNGf7EUmy3SORhvXPUvVtHPbCNTQhn1k5UbDm" } }, { "type": "interim", "title": "$$\\frac{\\frac{3π}{2}}{2}=\\frac{3π}{4}$$", "input": "\\frac{\\frac{3π}{2}}{2}", "steps": [ { "type": "step", "primary": "Apply the fraction rule: $$\\frac{\\frac{b}{c}}{a}=\\frac{b}{c\\:\\cdot\\:a}$$", "result": "=\\frac{3π}{2\\cdot\\:2}" }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:2=4$$", "result": "=\\frac{3π}{4}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7tiTmKkmeEqHWILbBb88ajZJTpMqJaq9bfSnb9kPhskfdd47a0hQ8flDbGsI5To1drfMRPynLSfP3VYe1OQldREkIgLCzuVxsE8bZJAi+zLMWtOub2dAxgaWacwKjDMkBc6krFpzteeOSStKZ2st+3LSXRhuEYkItk0ZmqvQzatokt3WiGR7ZaCaXvz77bMjS" } }, { "type": "step", "result": "=-\\frac{1}{2}+πn+\\frac{3π}{4}" }, { "type": "step", "primary": "Group like terms", "result": "=πn-\\frac{1}{2}+\\frac{3π}{4}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7tiTmKkmeEqHWILbBb88ajZ5yKgO88C1F1xZtWMQHGNUbMukakkexRTIhgb8cmbe53b0iJfUSOxSKdl4JAxKIkHWD310L1+P2yDQQfMEhENHGUjFexNEZ5qRksVKuypQg6rstwdw6Xlf9HA8qAvP64lO1T0snFOPqKXL+S6MxVmmY3ASC+aZqPN1DBWUUsybFLfFc+K4uSP6MmS0QGPb/UkKnOz8rKq05FNExPFI613UvytJr/CIONGNz+HAE9KWRU3GaCyMopm906ImCzX1YIA==" } }, { "type": "step", "result": "p=πn-\\frac{1}{2}+\\frac{3π}{4}" } ], "meta": { "interimType": "Generic Simplify 0Eq" } } ], "meta": { "interimType": "Divide Both Sides Specific 1Eq", "gptData": "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" } } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve Title 1Eq" } }, { "type": "step", "result": "p=πn-\\frac{1}{2}+\\frac{3π}{4}" } ], "meta": { "solvingClass": "Trig Equations", "practiceLink": "/practice/trigonometry-practice#area=main&subtopic=Trig%20Equations", "practiceTopic": "Trig Equations" } }, "plot_output": { "meta": { "plotInfo": { "variable": "p", "plotRequest": "\\sin(2p+1)+1" }, "showViewLarger": true } }, "meta": { "showVerify": true } }