sqrt(3)tan(θ)+1=0

{ "query": { "display": "$$\\sqrt{3}\\tan\\left(θ\\right)+1=0$$", "symbolab_question": "EQUATION#\\sqrt{3}\\tan(θ)+1=0" }, "solution": { "level": "PERFORMED", "subject": "Trigonometry", "topic": "Trig Equations", "subTopic": "Trig Equations", "default": "θ=\\frac{5π}{6}+πn", "degrees": "θ=150^{\\circ }+180^{\\circ }n", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\sqrt{3}\\tan\\left(θ\\right)+1=0{\\quad:\\quad}θ=\\frac{5π}{6}+πn$$", "input": "\\sqrt{3}\\tan\\left(θ\\right)+1=0", "steps": [ { "type": "interim", "title": "Move $$1\\:$$to the right side", "input": "\\sqrt{3}\\tan\\left(θ\\right)+1=0", "result": "\\sqrt{3}\\tan\\left(θ\\right)=-1", "steps": [ { "type": "step", "primary": "Subtract $$1$$ from both sides", "result": "\\sqrt{3}\\tan\\left(θ\\right)+1-1=0-1" }, { "type": "step", "primary": "Simplify", "result": "\\sqrt{3}\\tan\\left(θ\\right)=-1" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7ECl81GL+sHp+UX1O7owT7YOkcEWulI8fXKLzTVBJnpMcjlLRK1jUV206qo4+vRN7dVvpw9V+GYcSm3wiKyN75USr8KQrDJv3SA1inZ0lCbT0e7R1I+ApxaZG8Mlfu7sSyi3z+IFRrI9aSJQroKx3dbxWYflv4gzAwKOIgiO7RVHcKamhYRnpfHqiIh8phHTzSK7rXfHkjIFKPh6UchcADOuzplkwci0ctYEiL1YBIK5pDVeK92dMkVb/ckOCbkIaxBVMhfEdue6qsYh7lS/4QsMhSOmsID5cim3F9v8Rs44HgIRL2cbEPZLm634cOSI7G1YrInBT7eibt4cYhvHwDzRQJEdnJEoPNB8oHGc09FO8leuFFBMpmtXRConWh4AZ+97UWQsZjLul80dbuB7w9a1I43qi0dE6UozwHb1lpIYEklGISy6Xet4gkSoeMnmllG1kkk5QNJr+25y/LNkvyL8yD3hLQ33B7/8/LpbPE3o=" } }, { "type": "interim", "title": "Divide both sides by $$\\sqrt{3}$$", "input": "\\sqrt{3}\\tan\\left(θ\\right)=-1", "result": "\\tan\\left(θ\\right)=-\\frac{\\sqrt{3}}{3}", "steps": [ { "type": "step", "primary": "Divide both sides by $$\\sqrt{3}$$", "result": "\\frac{\\sqrt{3}\\tan\\left(θ\\right)}{\\sqrt{3}}=\\frac{-1}{\\sqrt{3}}" }, { "type": "interim", "title": "Simplify", "input": "\\frac{\\sqrt{3}\\tan\\left(θ\\right)}{\\sqrt{3}}=\\frac{-1}{\\sqrt{3}}", "result": "\\tan\\left(θ\\right)=-\\frac{\\sqrt{3}}{3}", "steps": [ { "type": "interim", "title": "Simplify $$\\frac{\\sqrt{3}\\tan\\left(θ\\right)}{\\sqrt{3}}:{\\quad}\\tan\\left(θ\\right)$$", "input": "\\frac{\\sqrt{3}\\tan\\left(θ\\right)}{\\sqrt{3}}", "steps": [ { "type": "step", "primary": "Cancel the common factor: $$\\sqrt{3}$$", "result": "=\\tan\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s74DwjiBDVBIb8+yh6fkkMMYmJyq1aLOU5AvxErppUUjhbqvHcXS1YJzMCVe3rDX+wCUCWbkwGOY7PqKo3U/JLJaljYExoPuHUP+srvIBoYoFFKk3fejFkyiOiq9iG9IkAIuSmJUDtLxRPyJ57Nkq7qib97z4ciRBzgXtcnYl8cinMtAjEnFCFc8OXHUm5jgsFMJa9jzdDvGquramb/qO0hA==" } }, { "type": "interim", "title": "Simplify $$\\frac{-1}{\\sqrt{3}}:{\\quad}-\\frac{\\sqrt{3}}{3}$$", "input": "\\frac{-1}{\\sqrt{3}}", "steps": [ { "type": "step", "primary": "Apply the fraction rule: $$\\frac{-a}{b}=-\\frac{a}{b}$$", "result": "=-\\frac{1}{\\sqrt{3}}" }, { "type": "interim", "title": "Rationalize $$-\\frac{1}{\\sqrt{3}}:{\\quad}-\\frac{\\sqrt{3}}{3}$$", "input": "-\\frac{1}{\\sqrt{3}}", "result": "=-\\frac{\\sqrt{3}}{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": { "interimType": "Rationalize Title 1Eq" } } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7tkkS6M9CCT9LrMl7pDU/tluq8dxdLVgnMwJV7esNf7AJQJZuTAY5js+oqjdT8kslDa1hUvtzgBcwD7zey8pjX7ynZksDFGW8KDVRRFssbMXWwPs1+Gw97t4MeuaNjSYTwPBBw3byB5RPtpSy6YLt2l15CaX671zydgCHN23vogCJqVxX90jlMfh9fKn6dzC4" } }, { "type": "step", "result": "\\tan\\left(θ\\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 $$\\tan\\left(θ\\right)=-\\frac{\\sqrt{3}}{3}$$", "result": "θ=\\frac{5π}{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": "θ=\\frac{5π}{6}+πn" } ], "meta": { "interimType": "Trig General Solutions tan 1Eq" } } ], "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(θ)+1" }, "showViewLarger": true } }, "meta": { "showVerify": true } }