intercepts of sin(x)

{ "query": { "display": "intercepts $$\\sin\\left(x\\right)$$", "symbolab_question": "CONIC#intercepts \\sin(x)" }, "solution": { "level": "PERFORMED", "subject": "Functions & Graphing", "topic": "Functions", "subTopic": "intercepts", "default": "\\mathrm{X\\:Intercepts}: (2πn,0),(π+2πn,0),\\mathrm{Y\\:Intercepts}: (0,0)", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "Axis interception points of $$\\sin\\left(x\\right):\\quad\\:$$X Intercepts$$:\\:\\left(2πn,\\:0\\right),\\:\\left(π+2πn,\\:0\\right),\\:$$Y Intercepts$$:\\:\\left(0,\\:0\\right)$$", "steps": [ { "type": "interim", "title": "$$x-$$axis interception points of $$\\sin\\left(x\\right):{\\quad}\\left(2πn,\\:0\\right),\\:\\left(π+2πn,\\:0\\right)$$", "input": "\\sin\\left(x\\right)", "steps": [ { "type": "definition", "title": "x-axis interception points definition", "text": "x-intercept is a point on the graph where $$y=0$$" }, { "type": "interim", "title": "$$\\sin\\left(x\\right)=0:{\\quad}x=2πn,\\:x=π+2πn$$", "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": "step", "result": "\\left(2πn,\\:0\\right),\\:\\left(π+2πn,\\:0\\right)" } ], "meta": { "solvingClass": "Function Intersect", "interimType": "Interception X Points Top 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xyoU2cWyPLDgE1QLLHeauuc3PHQdChPJ2JhfqHT+ZU0OMXsqdoP/+t8mkG9iMyF4x8bt5g+KB4e7b6i2FR+k9P7M4JjEGoarAVOX48gw80TVIIX0FvvIrdePEGQu2OGIgY2DlwPZY/s+3QwhHk6uKuROlJ/rSje5O2ouNdRzeRIsAYw6enetreyrE67exYQOWfA==" } }, { "type": "interim", "title": "$$y-$$axis interception point of $$\\sin\\left(x\\right):{\\quad}\\left(0,\\:0\\right)$$", "input": "\\sin\\left(x\\right)", "steps": [ { "type": "definition", "title": "y-axis interception points definition", "text": "$$y$$-intercept is the point on the graph where $$x=0$$" }, { "type": "interim", "title": "Solve $$y=\\sin\\left(0\\right):{\\quad}0$$", "input": "\\sin\\left(0\\right)", "steps": [ { "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": "Generic Solve Title 1Eq" } }, { "type": "step", "result": "\\left(0,\\:0\\right)" } ], "meta": { "solvingClass": "Function Intersect", "interimType": "Interception Y Points Top 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xyoU2cWyPLDgE1QLLHeauuc3PHQdChPJ2JhfqHT+ZU0OMoY5LPa3x5862ED2Fb21Kz+w/rOyd5aPv89U4pRBmcfmXi2YjWAn0vmBaB/n2kg2RB0Vl0EbK8erWVizRrRpVBIeinSqlAqsBAw0XnsxmzcX+96iP7mHv7o1+fFq9dx9pX/yuz1zibiY5YG0jTR2ZWg==" } }, { "type": "step", "result": "\\mathrm{X\\:Intercepts}:\\:\\left(2πn,\\:0\\right),\\:\\left(π+2πn,\\:0\\right),\\:\\mathrm{Y\\:Intercepts}:\\:\\left(0,\\:0\\right)" } ], "meta": { "solvingClass": "Function Intersect" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "\\sin(x)" }, "showViewLarger": true } }, "meta": { "showVerify": true } }