sqrt(3)tan(θ/2)-1=0

{ "query": { "display": "$$\\sqrt{3}\\tan\\left(\\frac{θ}{2}\\right)-1=0$$", "symbolab_question": "EQUATION#\\sqrt{3}\\tan(\\frac{θ}{2})-1=0" }, "solution": { "level": "PERFORMED", "subject": "Trigonometry", "topic": "Trig Equations", "subTopic": "Trig Equations", "default": "θ=\\frac{π}{3}+2πn", "degrees": "θ=60^{\\circ }+360^{\\circ }n", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\sqrt{3}\\tan\\left(\\frac{θ}{2}\\right)-1=0{\\quad:\\quad}θ=\\frac{π}{3}+2πn$$", "input": "\\sqrt{3}\\tan\\left(\\frac{θ}{2}\\right)-1=0", "steps": [ { "type": "interim", "title": "Move $$1\\:$$to the right side", "input": "\\sqrt{3}\\tan\\left(\\frac{θ}{2}\\right)-1=0", "result": "\\sqrt{3}\\tan\\left(\\frac{θ}{2}\\right)=1", "steps": [ { "type": "step", "primary": "Add $$1$$ to both sides", "result": "\\sqrt{3}\\tan\\left(\\frac{θ}{2}\\right)-1+1=0+1" }, { "type": "step", "primary": "Simplify", "result": "\\sqrt{3}\\tan\\left(\\frac{θ}{2}\\right)=1" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": "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" } }, { "type": "interim", "title": "Divide both sides by $$\\sqrt{3}$$", "input": "\\sqrt{3}\\tan\\left(\\frac{θ}{2}\\right)=1", "result": "\\tan\\left(\\frac{θ}{2}\\right)=\\frac{\\sqrt{3}}{3}", "steps": [ { "type": "step", "primary": "Divide both sides by $$\\sqrt{3}$$", "result": "\\frac{\\sqrt{3}\\tan\\left(\\frac{θ}{2}\\right)}{\\sqrt{3}}=\\frac{1}{\\sqrt{3}}" }, { "type": "interim", "title": "Simplify", "input": "\\frac{\\sqrt{3}\\tan\\left(\\frac{θ}{2}\\right)}{\\sqrt{3}}=\\frac{1}{\\sqrt{3}}", "result": "\\tan\\left(\\frac{θ}{2}\\right)=\\frac{\\sqrt{3}}{3}", "steps": [ { "type": "interim", "title": "Simplify $$\\frac{\\sqrt{3}\\tan\\left(\\frac{θ}{2}\\right)}{\\sqrt{3}}:{\\quad}\\tan\\left(\\frac{θ}{2}\\right)$$", "input": "\\frac{\\sqrt{3}\\tan\\left(\\frac{θ}{2}\\right)}{\\sqrt{3}}", "steps": [ { "type": "step", "primary": "Cancel the common factor: $$\\sqrt{3}$$", "result": "=\\tan\\left(\\frac{θ}{2}\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s74DwjiBDVBIb8+yh6fkkMMZHMbk0d4X4JX0M/23jGJLrZBo3PxfB4k0oRw4WVCq5hRXYii9G6PtJS0VMzV5tSBP2i9gqKNBiEkMJvG7+cA4njFvV2K1IzNUnyWDNHQh50hSnSDepaV5uvVmXr38PwQe5AIz++qluupTlLFEcE9J2oFlfShkXzCtmXNyfHy+cdlB8b/gjlz0z92MC/a5fIyvHUnw3hcHAW7PdIoLTZlf8BLYoznbQT9RDNBW3uT3ST" } }, { "type": "interim", "title": "Simplify $$\\frac{1}{\\sqrt{3}}:{\\quad}\\frac{\\sqrt{3}}{3}$$", "input": "\\frac{1}{\\sqrt{3}}", "steps": [ { "type": "step", "primary": "Multiply by the conjugate $$\\frac{\\sqrt{3}}{\\sqrt{3}}$$", "result": "=\\frac{1\\cdot\\:\\sqrt{3}}{\\sqrt{3}\\sqrt{3}}", "meta": { "title": { "extension": "To rationalize the denominator, multiply numerator and denominator by the conjugate of the radical $$\\sqrt{3}$$" } } }, { "type": "step", "primary": "$$1\\cdot\\:\\sqrt{3}=\\sqrt{3}$$" }, { "type": "interim", "title": "$$\\sqrt{3}\\sqrt{3}=3$$", "input": "\\sqrt{3}\\sqrt{3}", "result": "=\\frac{\\sqrt{3}}{3}", "steps": [ { "type": "step", "primary": "Apply radical rule: $$\\sqrt{a}\\sqrt{a}=a$$", "secondary": [ "$$\\sqrt{3}\\sqrt{3}=3$$" ], "result": "=3", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7KTpmXttz9tzhFROsvdfdpl9t0rGzXfqIeCX9GJLyBFHMwViaLUXkeD+JukROhWdjqqRb86vK1jAPxwJmTBc7jxuwBfr4jqpPqXmFXXvxZ1uqe0B1E2GM2M0eTcXoyACf" } } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s78B+yE0cCl8hT7YErNdnZGYH5WPs65D84evxXUHATCeerju+5Z51e/ZZSD3gRHwjBnvjDY21b5XBQ44AG3rKeoOdIubRDLxVLz5gHGSWn9gxN5Aod6Hr1Lp2e/29KhSgUgQHNT2nz+ImvmnBFu276+bHHzIiMhw6o+pGfy6C0EZIkt3WiGR7ZaCaXvz77bMjS" } }, { "type": "step", "result": "\\tan\\left(\\frac{θ}{2}\\right)=\\frac{\\sqrt{3}}{3}" } ], "meta": { "interimType": "Generic Simplify 0Eq" } } ], "meta": { "interimType": "Divide Both Sides Specific 1Eq", "gptData": "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" } }, { "type": "interim", "title": "General solutions for $$\\tan\\left(\\frac{θ}{2}\\right)=\\frac{\\sqrt{3}}{3}$$", "result": "\\frac{θ}{2}=\\frac{π}{6}+πn", "steps": [ { "type": "step", "primary": "$$\\tan\\left(x\\right)$$ periodicity table with $$πn$$ cycle:<br/>$$\\begin{array}{|c|c|}\\hline x&\\tan(x)\\\\\\hline 0&0\\\\\\hline \\frac{π}{6}&\\frac{\\sqrt{3}}{3}\\\\\\hline \\frac{π}{4}&1\\\\\\hline \\frac{π}{3}&\\sqrt{3}\\\\\\hline \\frac{π}{2}&\\pm\\infty\\\\\\hline \\frac{2π}{3}&-\\sqrt{3}\\\\\\hline \\frac{3π}{4}&-1\\\\\\hline \\frac{5π}{6}&-\\frac{\\sqrt{3}}{3}\\\\\\hline &\\\\\\hline \\end{array}$$" }, { "type": "step", "result": "\\frac{θ}{2}=\\frac{π}{6}+πn" } ], "meta": { "interimType": "Trig General Solutions tan 1Eq" } }, { "type": "interim", "title": "Solve $$\\frac{θ}{2}=\\frac{π}{6}+πn:{\\quad}θ=\\frac{π}{3}+2πn$$", "input": "\\frac{θ}{2}=\\frac{π}{6}+πn", "steps": [ { "type": "interim", "title": "Multiply both sides by $$2$$", "input": "\\frac{θ}{2}=\\frac{π}{6}+πn", "result": "θ=\\frac{π}{3}+2πn", "steps": [ { "type": "step", "primary": "Multiply both sides by $$2$$", "result": "\\frac{2θ}{2}=2\\cdot\\:\\frac{π}{6}+2πn" }, { "type": "interim", "title": "Simplify", "input": "\\frac{2θ}{2}=2\\cdot\\:\\frac{π}{6}+2πn", "result": "θ=\\frac{π}{3}+2πn", "steps": [ { "type": "interim", "title": "Simplify $$\\frac{2θ}{2}:{\\quad}θ$$", "input": "\\frac{2θ}{2}", "steps": [ { "type": "step", "primary": "Divide the numbers: $$\\frac{2}{2}=1$$", "result": "=θ" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7diSwzpX9ldlv2DLvSBEhWwCWKUbvV6WK3fDUgFtg3Q+kEtLScKDV+HqoDFDPEKGuZEt3ZXAiqUE0HIXrrrezJHZKl2mV4AlztiApTAFigz0ksNded2fMa8XATFfdmqDhsIjaxJ4DvjTb2fbKjbvtlQ==" } }, { "type": "interim", "title": "Simplify $$2\\cdot\\:\\frac{π}{6}+2πn:{\\quad}\\frac{π}{3}+2πn$$", "input": "2\\cdot\\:\\frac{π}{6}+2πn", "steps": [ { "type": "interim", "title": "$$2\\cdot\\:\\frac{π}{6}=\\frac{π}{3}$$", "input": "2\\cdot\\:\\frac{π}{6}", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{π2}{6}" }, { "type": "step", "primary": "Cancel the common factor: $$2$$", "result": "=\\frac{π}{3}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7/OsC643lXZbU+VEjF1qvijqt2Xgtb3t95fujkffGgBwJQJZuTAY5js+oqjdT8ksl3Afg9kvCPt7oFJyoNy6n1P8//6/nV5O4fb8Xgwi7maq51BdaKQokHx8SZGXrDp2idDC0pmI5+Hy3BfDLTjeEfpXHEOoAtuxLo7UJ0uiKECg=" } }, { "type": "step", "result": "=\\frac{π}{3}+2πn" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7/OsC643lXZbU+VEjF1qvisVU6smAwBv1KCInXQhw8QsgJ/ZZA32ZInFBpDtxBfiKXYGCmiBF99lesmXZ9iIfJxnN4vsPuZ+fPOarwFlCxZaBBTEk/JQ2cZ9WKuRzClU7hdlJPcZzzyhBqDm8dNgG19MzzULovGGcpsEyVazFzb+k0BiTxph8dpeQmrsKbNMH" } }, { "type": "step", "result": "θ=\\frac{π}{3}+2πn" } ], "meta": { "interimType": "Generic Simplify 0Eq" } } ], "meta": { "interimType": "Multiply Both Sides Specific 1Eq", "gptData": "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" } } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve Title 1Eq" } }, { "type": "step", "result": "θ=\\frac{π}{3}+2πn" } ], "meta": { "solvingClass": "Trig Equations", "practiceLink": "/practice/trigonometry-practice#area=main&subtopic=Trig%20Equations", "practiceTopic": "Trig Equations" } }, "plot_output": { "meta": { "plotInfo": { "variable": "θ", "plotRequest": "\\sqrt{3}\\tan(\\frac{θ}{2})-1" }, "showViewLarger": true } }, "meta": { "showVerify": true } }