[sin(4x)cos(x)-cos(4x)sin(x)]^2=1

{ "query": { "display": "$$[\\sin\\left(4x\\right)\\cos\\left(x\\right)-\\cos\\left(4x\\right)\\sin\\left(x\\right)]^{2}=1$$", "symbolab_question": "EQUATION#[\\sin(4x)\\cos(x)-\\cos(4x)\\sin(x)]^{2}=1" }, "solution": { "level": "PERFORMED", "subject": "Trigonometry", "topic": "Trig Equations", "subTopic": "Trig Equations", "default": "x=\\frac{π}{2}+\\frac{2πn}{3},x=\\frac{π}{6}+\\frac{2πn}{3}", "degrees": "x=90^{\\circ }+120^{\\circ }n,x=30^{\\circ }+120^{\\circ }n", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$[\\sin\\left(4x\\right)\\cos\\left(x\\right)-\\cos\\left(4x\\right)\\sin\\left(x\\right)]^{2}=1{\\quad:\\quad}x=\\frac{π}{2}+\\frac{2πn}{3},\\:x=\\frac{π}{6}+\\frac{2πn}{3}$$", "input": "[\\sin\\left(4x\\right)\\cos\\left(x\\right)-\\cos\\left(4x\\right)\\sin\\left(x\\right)]^{2}=1", "steps": [ { "type": "step", "primary": "Subtract $$1$$ from both sides", "result": "\\left(\\sin\\left(4x\\right)\\cos\\left(x\\right)-\\cos\\left(4x\\right)\\sin\\left(x\\right)\\right)^{2}-1=0" }, { "type": "interim", "title": "Factor $$\\left(\\sin\\left(4x\\right)\\cos\\left(x\\right)-\\cos\\left(4x\\right)\\sin\\left(x\\right)\\right)^{2}-1:{\\quad}\\left(\\sin\\left(4x\\right)\\cos\\left(x\\right)-\\cos\\left(4x\\right)\\sin\\left(x\\right)+1\\right)\\left(\\sin\\left(4x\\right)\\cos\\left(x\\right)-\\cos\\left(4x\\right)\\sin\\left(x\\right)-1\\right)$$", "input": "\\left(\\sin\\left(4x\\right)\\cos\\left(x\\right)-\\cos\\left(4x\\right)\\sin\\left(x\\right)\\right)^{2}-1", "steps": [ { "type": "step", "primary": "Rewrite $$1$$ as $$1^{2}$$", "result": "=\\left(\\sin\\left(4x\\right)\\cos\\left(x\\right)-\\cos\\left(4x\\right)\\sin\\left(x\\right)\\right)^{2}-1^{2}" }, { "type": "step", "primary": "Apply Difference of Two Squares Formula: $$x^{2}-y^{2}=\\left(x+y\\right)\\left(x-y\\right)$$", "secondary": [ "$$\\left(\\sin\\left(4x\\right)\\cos\\left(x\\right)-\\cos\\left(4x\\right)\\sin\\left(x\\right)\\right)^{2}-1^{2}=\\left(\\left(\\sin\\left(4x\\right)\\cos\\left(x\\right)-\\cos\\left(4x\\right)\\sin\\left(x\\right)\\right)+1\\right)\\left(\\left(\\sin\\left(4x\\right)\\cos\\left(x\\right)-\\cos\\left(4x\\right)\\sin\\left(x\\right)\\right)-1\\right)$$" ], "result": "=\\left(\\left(\\sin\\left(4x\\right)\\cos\\left(x\\right)-\\cos\\left(4x\\right)\\sin\\left(x\\right)\\right)+1\\right)\\left(\\left(\\sin\\left(4x\\right)\\cos\\left(x\\right)-\\cos\\left(4x\\right)\\sin\\left(x\\right)\\right)-1\\right)", "meta": { "practiceLink": "/practice/factoring-practice#area=main&subtopic=Difference%20of%20Two%20Squares", "practiceTopic": "Factor Difference of Squares" } }, { "type": "step", "primary": "Refine", "result": "=\\left(\\sin\\left(4x\\right)\\cos\\left(x\\right)-\\cos\\left(4x\\right)\\sin\\left(x\\right)+1\\right)\\left(\\sin\\left(4x\\right)\\cos\\left(x\\right)-\\cos\\left(4x\\right)\\sin\\left(x\\right)-1\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Factor Title 1Eq" } }, { "type": "step", "result": "\\left(\\sin\\left(4x\\right)\\cos\\left(x\\right)-\\cos\\left(4x\\right)\\sin\\left(x\\right)+1\\right)\\left(\\sin\\left(4x\\right)\\cos\\left(x\\right)-\\cos\\left(4x\\right)\\sin\\left(x\\right)-1\\right)=0" }, { "type": "step", "primary": "Solving each part separately", "result": "\\sin\\left(4x\\right)\\cos\\left(x\\right)-\\cos\\left(4x\\right)\\sin\\left(x\\right)+1=0\\lor\\:\\sin\\left(4x\\right)\\cos\\left(x\\right)-\\cos\\left(4x\\right)\\sin\\left(x\\right)-1=0" }, { "type": "interim", "title": "$$\\sin\\left(4x\\right)\\cos\\left(x\\right)-\\cos\\left(4x\\right)\\sin\\left(x\\right)+1=0{\\quad:\\quad}x=\\frac{π}{2}+\\frac{2πn}{3}$$", "input": "\\sin\\left(4x\\right)\\cos\\left(x\\right)-\\cos\\left(4x\\right)\\sin\\left(x\\right)+1=0", "steps": [ { "type": "interim", "title": "Rewrite using trig identities", "input": "\\sin\\left(4x\\right)\\cos\\left(x\\right)-\\cos\\left(4x\\right)\\sin\\left(x\\right)+1", "result": "1+\\sin\\left(4x-x\\right)=0", "steps": [ { "type": "step", "primary": "Use the Angle Difference identity: $$\\sin\\left(s\\right)\\cos\\left(t\\right)-\\cos\\left(s\\right)\\sin\\left(t\\right)=\\sin\\left(s-t\\right)$$", "result": "=1+\\sin\\left(4x-x\\right)" } ], "meta": { "interimType": "Trig Rewrite Using Trig identities 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7Awm8OHaUUE5X73KqyTFjFF0euHOiHrIFGWRnhcWSixBwtnzN5EZc5gmCV/9IhOSTOdhXIglekP2bk8wRyD3Q8umxY48eSLsbyMu/5yTExUbC4HIK63JpB+rGbO1V7H0Ql4JcJ5kTpfXBinaNDoH4MqjeYDvvoH78PuWqTSVt21eBBTEk/JQ2cZ9WKuRzClU7DcFgmo2GmeG2Kh9EaSPQP8xQp8yk2EdviTvfhz6ruO+9RQa+VFzMcO8uN2y3ZuWk" } }, { "type": "interim", "title": "Move $$1\\:$$to the right side", "input": "1+\\sin\\left(4x-x\\right)=0", "result": "\\sin\\left(4x-x\\right)=-1", "steps": [ { "type": "step", "primary": "Subtract $$1$$ from both sides", "result": "1+\\sin\\left(4x-x\\right)-1=0-1" }, { "type": "step", "primary": "Simplify", "result": "\\sin\\left(4x-x\\right)=-1" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": "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" } }, { "type": "interim", "title": "General solutions for $$\\sin\\left(4x-x\\right)=-1$$", "result": "4x-x=\\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": "4x-x=\\frac{3π}{2}+2πn" } ], "meta": { "interimType": "Trig General Solutions sin 1Eq" } }, { "type": "interim", "title": "Solve $$4x-x=\\frac{3π}{2}+2πn:{\\quad}x=\\frac{π}{2}+\\frac{2πn}{3}$$", "input": "4x-x=\\frac{3π}{2}+2πn", "steps": [ { "type": "step", "primary": "Add similar elements: $$4x-x=3x$$", "result": "3x=\\frac{3π}{2}+2πn" }, { "type": "interim", "title": "Divide both sides by $$3$$", "input": "3x=\\frac{3π}{2}+2πn", "result": "x=\\frac{π}{2}+\\frac{2πn}{3}", "steps": [ { "type": "step", "primary": "Divide both sides by $$3$$", "result": "\\frac{3x}{3}=\\frac{\\frac{3π}{2}}{3}+\\frac{2πn}{3}" }, { "type": "interim", "title": "Simplify", "input": "\\frac{3x}{3}=\\frac{\\frac{3π}{2}}{3}+\\frac{2πn}{3}", "result": "x=\\frac{π}{2}+\\frac{2πn}{3}", "steps": [ { "type": "interim", "title": "Simplify $$\\frac{3x}{3}:{\\quad}x$$", "input": "\\frac{3x}{3}", "steps": [ { "type": "step", "primary": "Divide the numbers: $$\\frac{3}{3}=1$$", "result": "=x" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7QJV6FEydUSv+hKOzRRB7mi061ljBSPJeENOw2efoSWt2cZ6LAXMcKee6PSyjLEFrRSpN33oxZMojoqvYhvSJACLkpiVA7S8UT8ieezZKu6ot8gw9eOFDGZzXCUlLc4C0" } }, { "type": "interim", "title": "Simplify $$\\frac{\\frac{3π}{2}}{3}+\\frac{2πn}{3}:{\\quad}\\frac{π}{2}+\\frac{2πn}{3}$$", "input": "\\frac{\\frac{3π}{2}}{3}+\\frac{2πn}{3}", "steps": [ { "type": "interim", "title": "$$\\frac{\\frac{3π}{2}}{3}=\\frac{π}{2}$$", "input": "\\frac{\\frac{3π}{2}}{3}", "steps": [ { "type": "step", "primary": "Apply the fraction rule: $$\\frac{\\frac{b}{c}}{a}=\\frac{b}{c\\:\\cdot\\:a}$$", "result": "=\\frac{3π}{2\\cdot\\:3}" }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:3=6$$", "result": "=\\frac{3π}{6}" }, { "type": "step", "primary": "Cancel the common factor: $$3$$", "result": "=\\frac{π}{2}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7tiTmKkmeEqHWILbBb88ajaaxrMgtxqxItjWpVJaLAobdd47a0hQ8flDbGsI5To1d+bRvmdH+XexJq0YRmlk3drKAEC1immJ8buNmGLd/2Z+IDZ1VqNwwAzGjxOWVq2SRJ2ijmmTW7DUH5cQghTxZhzAsLO/twYclA9e0Ao3wf0k=" } }, { "type": "step", "result": "=\\frac{π}{2}+\\frac{2πn}{3}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7tiTmKkmeEqHWILbBb88ajdrNDsI7OTwDDhLH4ub/7pk2BuryoOOEqPAk8viwO4hR3XeO2tIUPH5Q2xrCOU6NXfm0b5nR/l3sSatGEZpZN3bF85/p6iRd7AWrGKOnhJOgE6v794USxh0EoAWD1ttBCosaY2cs9YZ/08feWmaOgkhPpaTWPScnc3H2GfYFvoQGjzD7MOXPIGfDOjIDyfQd7LxnamvDLMW8R+aRNfa0VTmJqVxX90jlMfh9fKn6dzC4" } }, { "type": "step", "result": "x=\\frac{π}{2}+\\frac{2πn}{3}" } ], "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": "x=\\frac{π}{2}+\\frac{2πn}{3}" } ], "meta": { "solvingClass": "Trig Equations", "interimType": "Trig Equations" } }, { "type": "interim", "title": "$$\\sin\\left(4x\\right)\\cos\\left(x\\right)-\\cos\\left(4x\\right)\\sin\\left(x\\right)-1=0{\\quad:\\quad}x=\\frac{π}{6}+\\frac{2πn}{3}$$", "input": "\\sin\\left(4x\\right)\\cos\\left(x\\right)-\\cos\\left(4x\\right)\\sin\\left(x\\right)-1=0", "steps": [ { "type": "interim", "title": "Rewrite using trig identities", "input": "\\sin\\left(4x\\right)\\cos\\left(x\\right)-\\cos\\left(4x\\right)\\sin\\left(x\\right)-1", "result": "-1+\\sin\\left(4x-x\\right)=0", "steps": [ { "type": "step", "primary": "Use the Angle Difference identity: $$\\sin\\left(s\\right)\\cos\\left(t\\right)-\\cos\\left(s\\right)\\sin\\left(t\\right)=\\sin\\left(s-t\\right)$$", "result": "=-1+\\sin\\left(4x-x\\right)" } ], "meta": { "interimType": "Trig Rewrite Using Trig identities 0Eq", "gptData": 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"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" } }, { "type": "interim", "title": "General solutions for $$\\sin\\left(4x-x\\right)=1$$", "result": "4x-x=\\frac{π}{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": "4x-x=\\frac{π}{2}+2πn" } ], "meta": { "interimType": "Trig General Solutions sin 1Eq" } }, { "type": "interim", "title": "Solve $$4x-x=\\frac{π}{2}+2πn:{\\quad}x=\\frac{π}{6}+\\frac{2πn}{3}$$", "input": "4x-x=\\frac{π}{2}+2πn", "steps": [ { "type": "step", "primary": "Add similar elements: $$4x-x=3x$$", "result": "3x=\\frac{π}{2}+2πn" }, { "type": "interim", "title": "Divide both sides by $$3$$", "input": "3x=\\frac{π}{2}+2πn", "result": "x=\\frac{π}{6}+\\frac{2πn}{3}", "steps": [ { "type": "step", "primary": "Divide both sides by $$3$$", "result": "\\frac{3x}{3}=\\frac{\\frac{π}{2}}{3}+\\frac{2πn}{3}" }, { "type": "interim", "title": "Simplify", "input": "\\frac{3x}{3}=\\frac{\\frac{π}{2}}{3}+\\frac{2πn}{3}", "result": "x=\\frac{π}{6}+\\frac{2πn}{3}", "steps": [ { "type": "interim", "title": "Simplify $$\\frac{3x}{3}:{\\quad}x$$", "input": "\\frac{3x}{3}", "steps": [ { "type": "step", "primary": "Divide the numbers: $$\\frac{3}{3}=1$$", "result": "=x" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7QJV6FEydUSv+hKOzRRB7mi061ljBSPJeENOw2efoSWt2cZ6LAXMcKee6PSyjLEFrRSpN33oxZMojoqvYhvSJACLkpiVA7S8UT8ieezZKu6ot8gw9eOFDGZzXCUlLc4C0" } }, { "type": "interim", "title": "Simplify $$\\frac{\\frac{π}{2}}{3}+\\frac{2πn}{3}:{\\quad}\\frac{π}{6}+\\frac{2πn}{3}$$", "input": "\\frac{\\frac{π}{2}}{3}+\\frac{2πn}{3}", "steps": [ { "type": "interim", "title": "$$\\frac{\\frac{π}{2}}{3}=\\frac{π}{6}$$", "input": "\\frac{\\frac{π}{2}}{3}", "steps": [ { "type": "step", "primary": "Apply the fraction rule: $$\\frac{\\frac{b}{c}}{a}=\\frac{b}{c\\:\\cdot\\:a}$$", "result": "=\\frac{π}{2\\cdot\\:3}" }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:3=6$$", "result": "=\\frac{π}{6}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7tiTmKkmeEqHWILbBb88ajdU2utO1NDv4gIlh/66PqgrNGoPE9TME3q+OPmgkv2RQV7wanvTgjKqnID/UMq66G9DvBptQbBir7GwZ+GlV8S++/XUfFY44oQy7yrU6sldlPl7acUDUp+uHLnpG1rz26ek1nlqWS2nkMNwVvpLj2uw=" } }, { "type": "step", "result": "=\\frac{π}{6}+\\frac{2πn}{3}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7tiTmKkmeEqHWILbBb88ajf4x2k3NLD/hOs5Mdx7C6jSddNoa0JibEfgUXb69v4MPzRqDxPUzBN6vjj5oJL9kUFe8Gp704IyqpyA/1DKuuhsItVpJWhR3QDi6bni0CS279yPegwOu0B0iWFG/q1vMfJjcBIL5pmo83UMFZRSzJsUt8Vz4ri5I/oyZLRAY9v9SVOJ2fOgPewbgX5w12u6Ro/xmJzKLCps2zTe0VCJi6hO/Mg94S0N9we//Py6WzxN6" } }, { "type": "step", "result": "x=\\frac{π}{6}+\\frac{2πn}{3}" } ], "meta": { "interimType": "Generic Simplify 0Eq" } } ], "meta": { "interimType": "Divide Both Sides Specific 1Eq", "gptData": 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"x=\\frac{π}{2}+\\frac{2πn}{3},\\:x=\\frac{π}{6}+\\frac{2πn}{3}" } ], "meta": { "solvingClass": "Trig Equations", "practiceLink": "/practice/trigonometry-practice#area=main&subtopic=Trig%20Equations", "practiceTopic": "Trig Equations" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "[\\sin(4x)\\cos(x)-\\cos(4x)\\sin(x)]^{2}-1" }, "showViewLarger": true } }, "meta": { "showVerify": true } }