arctan(-1/(sqrt(3)))

{ "query": { "display": "$$\\arctan\\left(-\\frac{1}{\\sqrt{3}}\\right)$$", "symbolab_question": "TRIG_EVALUATE#\\arctan(-\\frac{1}{\\sqrt{3}})" }, "solution": { "level": "PERFORMED", "subject": "Trigonometry", "topic": "Evaluate Functions", "subTopic": "Simplified", "default": "-\\frac{π}{6}", "decimal": "-30", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\arctan\\left(-\\frac{1}{\\sqrt{3}}\\right)=-\\frac{π}{6}$$", "input": "\\arctan\\left(-\\frac{1}{\\sqrt{3}}\\right)", "steps": [ { "type": "interim", "title": "$$\\arctan\\left(-\\frac{1}{\\sqrt{3}}\\right)=\\arctan\\left(-\\frac{\\sqrt{3}}{3}\\right)$$", "input": "\\arctan\\left(-\\frac{1}{\\sqrt{3}}\\right)", "steps": [ { "type": "interim", "title": "$$\\frac{1}{\\sqrt{3}}=\\frac{\\sqrt{3}}{3}$$", "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": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s78B+yE0cCl8hT7YErNdnZGYH5WPs65D84evxXUHATCeerju+5Z51e/ZZSD3gRHwjBnvjDY21b5XBQ44AG3rKeoOdIubRDLxVLz5gHGSWn9gz//NvXaLneG2moeSe54R9ofl+2Q1REnsr4pFHHZPAlAOqrxevU0S7BviPFPoEcZw6wiNrEngO+NNvZ9sqNu+2V" } }, { "type": "step", "result": "=\\arctan\\left(-\\frac{\\sqrt{3}}{3}\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7EO1WRSe7TxupdIW8DpQ6akR0O+mEG6vThMZ5+DrGpknTLx8mOdHYVzxX643JqKFIQslTDKxOR/6J+ZOGvUcauphMhI1dYkQcBy/Dfx/H0IxCNWox8otddtwDvOqnWCVPuK/CPOCJfkN3rVLMAJKBQwniDfxUdAs4y15bjJftwC5syiQ1IhsnfU83dsVfC+ZGfbhFjKUqusZ1g6Yd3VRGiE3WhGuA6u9cCyUkTGrB//4=" } }, { "type": "step", "result": "=\\arctan\\left(-\\frac{\\sqrt{3}}{3}\\right)" }, { "type": "step", "primary": "Use the following property: $$\\arctan\\left(-x\\right)=-\\arctan\\left(x\\right)$$", "secondary": [ "$$\\arctan\\left(-\\frac{\\sqrt{3}}{3}\\right)=-\\arctan\\left(\\frac{\\sqrt{3}}{3}\\right)$$" ], "result": "=-\\arctan\\left(\\frac{\\sqrt{3}}{3}\\right)" }, { "type": "interim", "title": "Use the following trivial identity:$${\\quad}\\arctan\\left(\\frac{\\sqrt{3}}{3}\\right)=\\frac{π}{6}$$", "input": "\\arctan\\left(\\frac{\\sqrt{3}}{3}\\right)", "steps": [ { "type": "step", "primary": "$$\\begin{array}{|c|c|c|}\\hline x&\\arctan(x)&\\arctan(x)\\\\\\hline 0&0&0^{\\circ}\\\\\\hline \\frac{\\sqrt{3}}{3}&\\frac{\\pi}{6}&30^{\\circ}\\\\\\hline 1&\\frac{\\pi}{4}&45^{\\circ}\\\\\\hline \\sqrt{3}&\\frac{\\pi}{3}&60^{\\circ}\\\\\\hline \\end{array}$$" }, { "type": "step", "result": "=\\frac{π}{6}" } ], "meta": { "interimType": "Trig Trivial Angle Value Title 0Eq" } }, { "type": "step", "result": "=-\\frac{π}{6}" } ], "meta": { "solvingClass": "Trig Evaluate", "practiceLink": "/practice/trigonometry-practice#area=main&subtopic=Evaluate%20Functions", "practiceTopic": "Evaluate Functions" } }, "meta": { "showVerify": true } }