cos(x+2pi)=1-cos(x-2pi)

{ "query": { "display": "$$\\cos\\left(x+2π\\right)=1-\\cos\\left(x-2π\\right)$$", "symbolab_question": "EQUATION#\\cos(x+2π)=1-\\cos(x-2π)" }, "solution": { "level": "PERFORMED", "subject": "Trigonometry", "topic": "Trig Equations", "subTopic": "Trig Equations", "default": "x=\\frac{π}{3}+2πn,x=\\frac{5π}{3}+2πn", "degrees": "x=60^{\\circ }+360^{\\circ }n,x=300^{\\circ }+360^{\\circ }n", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\cos\\left(x+2π\\right)=1-\\cos\\left(x-2π\\right){\\quad:\\quad}x=\\frac{π}{3}+2πn,\\:x=\\frac{5π}{3}+2πn$$", "input": "\\cos\\left(x+2π\\right)=1-\\cos\\left(x-2π\\right)", "steps": [ { "type": "interim", "title": "Rewrite using trig identities", "input": "\\cos\\left(x+2π\\right)=1-\\cos\\left(x-2π\\right)", "result": "\\cos\\left(x\\right)=1-\\cos\\left(x\\right)", "steps": [ { "type": "interim", "title": "Rewrite using trig identities", "input": "\\cos\\left(x+2π\\right)", "steps": [ { "type": "step", "primary": "Use the Angle Sum identity: $$\\cos\\left(s+t\\right)=\\cos\\left(s\\right)\\cos\\left(t\\right)-\\sin\\left(s\\right)\\sin\\left(t\\right)$$", "result": "=\\cos\\left(x\\right)\\cos\\left(2π\\right)-\\sin\\left(x\\right)\\sin\\left(2π\\right)" }, { "type": "interim", "title": "Simplify $$\\cos\\left(x\\right)\\cos\\left(2π\\right)-\\sin\\left(x\\right)\\sin\\left(2π\\right):{\\quad}\\cos\\left(x\\right)$$", "input": "\\cos\\left(x\\right)\\cos\\left(2π\\right)-\\sin\\left(x\\right)\\sin\\left(2π\\right)", "result": "=\\cos\\left(x\\right)", "steps": [ { "type": "interim", "title": "$$\\cos\\left(x\\right)\\cos\\left(2π\\right)=\\cos\\left(x\\right)$$", "input": "\\cos\\left(x\\right)\\cos\\left(2π\\right)", "steps": [ { "type": "interim", "title": "$$\\cos\\left(2π\\right)=1$$", "input": "\\cos\\left(2π\\right)", "result": "=1\\cdot\\:\\cos\\left(x\\right)", "steps": [ { "type": "interim", "title": "$$\\cos\\left(2π\\right)=\\cos\\left(0\\right)$$", "input": "\\cos\\left(2π\\right)", "result": "=\\cos\\left(0\\right)", "steps": [ { "type": "step", "primary": "Rewrite $$2π$$ as $$2π+0$$", "result": "=\\cos\\left(2π+0\\right)" }, { "type": "step", "primary": "Apply the periodicity of $$\\cos$$: $$\\cos\\left(x+2π\\right)=\\cos\\left(x\\right)$$", "secondary": [ "$$\\cos\\left(2π+0\\right)=\\cos\\left(0\\right)$$" ], "result": "=\\cos\\left(0\\right)" } ], "meta": { "interimType": "N/A" } }, { "type": "interim", "title": "Use the following trivial identity:$${\\quad}\\cos\\left(0\\right)=1$$", "input": "\\cos\\left(0\\right)", "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": "=1" } ], "meta": { "interimType": "Trig Trivial Angle Value Title 0Eq" } }, { "type": "step", "result": "=1" } ], "meta": { "solvingClass": "Trig Evaluate", "interimType": "Trig Evaluate" } }, { "type": "step", "primary": "Multiply: $$\\cos\\left(x\\right)\\cdot\\:1=\\cos\\left(x\\right)$$", "result": "=\\cos\\left(x\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7OgJyajQjkqczngxvLtluw2J0QnpTkWUeUX1vhCYygEXMwViaLUXkeD+JukROhWdjOI1L/CTheMTHCWPaD5CWECT5fJOXu5XMmfm19A2dj5R/kAAJTXYxAIguexWJnAnjVGBHZHtaZ2awcLQM8+d9Vw==" } }, { "type": "interim", "title": "$$\\sin\\left(x\\right)\\sin\\left(2π\\right)=0$$", "input": "\\sin\\left(x\\right)\\sin\\left(2π\\right)", "steps": [ { "type": "interim", "title": "$$\\sin\\left(2π\\right)=0$$", "input": "\\sin\\left(2π\\right)", "result": "=0\\cdot\\:\\sin\\left(x\\right)", "steps": [ { "type": "interim", "title": "$$\\sin\\left(2π\\right)=\\sin\\left(0\\right)$$", "input": "\\sin\\left(2π\\right)", "result": "=\\sin\\left(0\\right)", "steps": [ { "type": "step", "primary": "Rewrite $$2π$$ as $$2π+0$$", "result": "=\\sin\\left(2π+0\\right)" }, { "type": "step", "primary": "Apply the periodicity of $$\\sin$$: $$\\sin\\left(x+2π\\right)=\\sin\\left(x\\right)$$", "secondary": [ "$$\\sin\\left(2π+0\\right)=\\sin\\left(0\\right)$$" ], "result": "=\\sin\\left(0\\right)" } ], "meta": { "interimType": "N/A" } }, { "type": "interim", "title": "Use the following trivial identity:$${\\quad}\\sin\\left(0\\right)=0$$", "input": "\\sin\\left(0\\right)", "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": "=0" } ], "meta": { "interimType": "Trig Trivial Angle Value Title 0Eq" } }, { "type": "step", "result": "=0" } ], "meta": { "solvingClass": "Trig Evaluate", "interimType": "Trig Evaluate" } }, { "type": "step", "primary": "Apply rule $$0\\cdot\\:a=0$$", "result": "=0" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7Db5kOPGdwaYYCr65H3kEuGJ0QnpTkWUeUX1vhCYygEXMwViaLUXkeD+JukROhWdjguVaeA0vkFtAR3gzqI9IZyVJNXQsKhPgoF38xXsuJbhZo4hvSPRe+lv8mAK3o5sZ" } }, { "type": "step", "result": "=\\cos\\left(x\\right)-0" }, { "type": "step", "primary": "$$\\cos\\left(x\\right)-0=\\cos\\left(x\\right)$$", "result": "=\\cos\\left(x\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "primary": "Use the Angle Difference identity: $$\\cos\\left(s-t\\right)=\\cos\\left(s\\right)\\cos\\left(t\\right)+\\sin\\left(s\\right)\\sin\\left(t\\right)$$", "result": "=\\cos\\left(x\\right)\\cos\\left(2π\\right)+\\sin\\left(x\\right)\\sin\\left(2π\\right)" }, { "type": "interim", "title": "Simplify $$\\cos\\left(x\\right)\\cos\\left(2π\\right)+\\sin\\left(x\\right)\\sin\\left(2π\\right):{\\quad}\\cos\\left(x\\right)$$", "input": "\\cos\\left(x\\right)\\cos\\left(2π\\right)+\\sin\\left(x\\right)\\sin\\left(2π\\right)", "result": "=\\cos\\left(x\\right)", "steps": [ { "type": "interim", "title": "$$\\cos\\left(x\\right)\\cos\\left(2π\\right)=\\cos\\left(x\\right)$$", "input": "\\cos\\left(x\\right)\\cos\\left(2π\\right)", "steps": [ { "type": "interim", "title": "$$\\cos\\left(2π\\right)=1$$", "input": "\\cos\\left(2π\\right)", "result": "=1\\cdot\\:\\cos\\left(x\\right)", "steps": [ { "type": "interim", "title": "$$\\cos\\left(2π\\right)=\\cos\\left(0\\right)$$", "input": "\\cos\\left(2π\\right)", "result": "=\\cos\\left(0\\right)", "steps": [ { "type": "step", "primary": "Rewrite $$2π$$ as $$2π+0$$", "result": "=\\cos\\left(2π+0\\right)" }, { "type": "step", "primary": "Apply the periodicity of $$\\cos$$: $$\\cos\\left(x+2π\\right)=\\cos\\left(x\\right)$$", "secondary": [ "$$\\cos\\left(2π+0\\right)=\\cos\\left(0\\right)$$" ], "result": "=\\cos\\left(0\\right)" } ], "meta": { "interimType": "N/A" } }, { "type": "interim", "title": "Use the following trivial identity:$${\\quad}\\cos\\left(0\\right)=1$$", "input": "\\cos\\left(0\\right)", "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": "=1" } ], "meta": { "interimType": "Trig Trivial Angle Value Title 0Eq" } }, { "type": "step", "result": "=1" } ], "meta": { "solvingClass": "Trig Evaluate", "interimType": "Trig Evaluate" } }, { "type": "step", "primary": "Multiply: $$\\cos\\left(x\\right)\\cdot\\:1=\\cos\\left(x\\right)$$", "result": "=\\cos\\left(x\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7OgJyajQjkqczngxvLtluw2J0QnpTkWUeUX1vhCYygEXMwViaLUXkeD+JukROhWdjOI1L/CTheMTHCWPaD5CWECT5fJOXu5XMmfm19A2dj5R/kAAJTXYxAIguexWJnAnjVGBHZHtaZ2awcLQM8+d9Vw==" } }, { "type": "interim", "title": "$$\\sin\\left(x\\right)\\sin\\left(2π\\right)=0$$", "input": "\\sin\\left(x\\right)\\sin\\left(2π\\right)", "steps": [ { "type": "interim", "title": "$$\\sin\\left(2π\\right)=0$$", "input": "\\sin\\left(2π\\right)", "result": "=0\\cdot\\:\\sin\\left(x\\right)", "steps": [ { "type": "interim", "title": "$$\\sin\\left(2π\\right)=\\sin\\left(0\\right)$$", "input": "\\sin\\left(2π\\right)", "result": "=\\sin\\left(0\\right)", "steps": [ { "type": "step", "primary": "Rewrite $$2π$$ as $$2π+0$$", "result": "=\\sin\\left(2π+0\\right)" }, { "type": "step", "primary": "Apply the periodicity of $$\\sin$$: $$\\sin\\left(x+2π\\right)=\\sin\\left(x\\right)$$", "secondary": [ "$$\\sin\\left(2π+0\\right)=\\sin\\left(0\\right)$$" ], "result": "=\\sin\\left(0\\right)" } ], "meta": { "interimType": "N/A" } }, { "type": "interim", "title": "Use the following trivial identity:$${\\quad}\\sin\\left(0\\right)=0$$", "input": "\\sin\\left(0\\right)", "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": "=0" } ], "meta": { "interimType": "Trig Trivial Angle Value Title 0Eq" } }, { "type": "step", "result": "=0" } ], "meta": { "solvingClass": "Trig Evaluate", "interimType": "Trig Evaluate" } }, { "type": "step", "primary": "Apply rule $$0\\cdot\\:a=0$$", "result": "=0" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7Db5kOPGdwaYYCr65H3kEuGJ0QnpTkWUeUX1vhCYygEXMwViaLUXkeD+JukROhWdjguVaeA0vkFtAR3gzqI9IZyVJNXQsKhPgoF38xXsuJbhZo4hvSPRe+lv8mAK3o5sZ" } }, { "type": "step", "result": "=\\cos\\left(x\\right)+0" }, { "type": "step", "primary": "$$\\cos\\left(x\\right)+0=\\cos\\left(x\\right)$$", "result": "=\\cos\\left(x\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } } ], "meta": { "interimType": "Trig Rewrite Using Trig identities 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7lRPp6QndqgNT+L6IZutAbEzHVzMTiTnWoC/siJRUoB+CVLSU5ij4wRfoGmSCTiRZbY6oju7y7D/ZMNk4pHwyUciNG73ekXpR3ysC7LzMiyaOCEHUlfcve4S33S3RljNjRSpN33oxZMojoqvYhvSJADvwe1LTQX8NfWw0aV9gJcwxGToczixRQljlXtEwLPT7P7VbYO5NtSsySBWEYH1uxw==" } }, { "type": "step", "result": "\\cos\\left(x\\right)=1-\\cos\\left(x\\right)" } ], "meta": { "interimType": "Trig Rewrite Using Trig identities 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7lRPp6QndqgNT+L6IZutAbDCSJ46PRYvZxkMQQtSdW7Cnj/gg/7th1F/Fkr+a2NdugGFMQkUTMNnEnFExdO5KvDLzJz4rkYnjv1/W3glakSm1s1RQOlcS2PC6wMP3eHirjTfYkmSBZXtOFE/XVkxIvuy8ZUEuuWz81gZsRV5LThe9dDZkXCs5+Cxg1YNvJNPUe9yYF2Q5qSv3gDlLQoBFjj5yGUAwgawpYTH/Vvtg3vE=" } }, { "type": "step", "primary": "Subtract $$1-\\cos\\left(x\\right)$$ from both sides", "result": "2\\cos\\left(x\\right)-1=0" }, { "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": "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" } }, { "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", "practiceLink": "/practice/trigonometry-practice#area=main&subtopic=Trig%20Equations", "practiceTopic": "Trig Equations" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "\\cos(x+2π)-1+\\cos(x-2π)" }, "showViewLarger": true } }, "meta": { "showVerify": true } }