cos(x)-sqrt(3-3cos^2(x))=0

{ "query": { "display": "$$\\cos\\left(x\\right)-\\sqrt{3-3\\cos^{2}\\left(x\\right)}=0$$", "symbolab_question": "EQUATION#\\cos(x)-\\sqrt{3-3\\cos^{2}(x)}=0" }, "solution": { "level": "PERFORMED", "subject": "Trigonometry", "topic": "Trig Equations", "subTopic": "Trig Equations", "default": "x=\\frac{π}{6}+2πn,x=\\frac{11π}{6}+2πn", "degrees": "x=30^{\\circ }+360^{\\circ }n,x=330^{\\circ }+360^{\\circ }n", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\cos\\left(x\\right)-\\sqrt{3-3\\cos^{2}\\left(x\\right)}=0{\\quad:\\quad}x=\\frac{π}{6}+2πn,\\:x=\\frac{11π}{6}+2πn$$", "input": "\\cos\\left(x\\right)-\\sqrt{3-3\\cos^{2}\\left(x\\right)}=0", "steps": [ { "type": "interim", "title": "Solve by substitution", "input": "\\cos\\left(x\\right)-\\sqrt{3-3\\cos^{2}\\left(x\\right)}=0", "result": "\\cos\\left(x\\right)=\\frac{\\sqrt{3}}{2}", "steps": [ { "type": "step", "primary": "Let: $$\\cos\\left(x\\right)=u$$", "result": "u-\\sqrt{3-3u^{2}}=0" }, { "type": "interim", "title": "$$u-\\sqrt{3-3u^{2}}=0{\\quad:\\quad}u=\\frac{\\sqrt{3}}{2}$$", "input": "u-\\sqrt{3-3u^{2}}=0", "steps": [ { "type": "interim", "title": "Remove square roots", "input": "u-\\sqrt{3-3u^{2}}=0", "result": "3-3u^{2}=u^{2}", "steps": [ { "type": "step", "primary": "Subtract $$u$$ from both sides", "result": "u-\\sqrt{3-3u^{2}}-u=0-u" }, { "type": "step", "primary": "Simplify", "result": "-\\sqrt{3-3u^{2}}=-u" }, { "type": "interim", "title": "Square both sides:$${\\quad}3-3u^{2}=u^{2}$$", "input": "u-\\sqrt{3-3u^{2}}=0", "result": "3-3u^{2}=u^{2}", "steps": [ { "type": "step", "result": "\\left(-\\sqrt{3-3u^{2}}\\right)^{2}=\\left(-u\\right)^{2}" }, { "type": "interim", "title": "Expand $$\\left(-\\sqrt{3-3u^{2}}\\right)^{2}:{\\quad}3-3u^{2}$$", "input": "\\left(-\\sqrt{3-3u^{2}}\\right)^{2}", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$\\left(-a\\right)^{n}=a^{n},\\:$$if $$n$$ is even", "secondary": [ "$$\\left(-\\sqrt{3-3u^{2}}\\right)^{2}=\\left(\\sqrt{3-3u^{2}}\\right)^{2}$$" ], "result": "=\\left(\\sqrt{3-3u^{2}}\\right)^{2}" }, { "type": "step", "primary": "Apply radical rule: $$\\sqrt{a}=a^{\\frac{1}{2}}$$", "result": "=\\left(\\left(3-3u^{2}\\right)^{\\frac{1}{2}}\\right)^{2}", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } }, { "type": "step", "primary": "Apply exponent rule: $$\\left(a^{b}\\right)^{c}=a^{bc}$$", "result": "=\\left(3-3u^{2}\\right)^{\\frac{1}{2}\\cdot\\:2}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "interim", "title": "$$\\frac{1}{2}\\cdot\\:2=1$$", "input": "\\frac{1}{2}\\cdot\\:2", "result": "=3-3u^{2}", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{1\\cdot\\:2}{2}" }, { "type": "step", "primary": "Cancel the common factor: $$2$$", "result": "=1" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l3vTdf410Ywhq1vZ0kzF8e30Fwl9QKPJxyO/TFRCb5Grju+5Z51e/ZZSD3gRHwjBE9/03SOiEv+BIHutWLr6nUfz18ijmoplMAomfJM9x8W1GdKgiNs+PolKvTuWzYk/" } } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Expand Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7qwccusS280xkvjRXw002Zlu5bkF56rz1y8YqXzYUSJ19WY+x1CE1XPI0qZBLPyqKs7PdN8ocStDSJ08Eeyb0kcjP9vZe0h5cDZXk0KEZ9KgScn/M2sLYBGgxClqzCxM2F5W0ZMok0qbUB2jwtEHdEYmC2eU+TxQpfuONS82fzGI=" } }, { "type": "interim", "title": "Expand $$\\left(-u\\right)^{2}:{\\quad}u^{2}$$", "input": "\\left(-u\\right)^{2}", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$\\left(-a\\right)^{n}=a^{n},\\:$$if $$n$$ is even", "secondary": [ "$$\\left(-u\\right)^{2}=u^{2}$$" ], "result": "=u^{2}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Expand Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7eYiQkslyurKFnXNb7JWRPbHnWavzWID9ABcLON52g+IF0f9e9MOLJfBD6LX95TzkeFr9fJWrn4Z0QR9jTkKwM0DLjX7DcYtB54q3geRNejFj/CGea1RaYy+tYejYDsWbvzIPeEtDfcHv/z8uls8Teg==" } }, { "type": "step", "result": "3-3u^{2}=u^{2}" } ], "meta": { "interimType": "Radicals Square Both Sides Title 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDIgNRdOAFHGNCe+Kt/Cl+fPa3PTeDYVM9NCmNlgVS/EX6B3nxM3r9utUBLxT/6HTwhkQY9Af7SiXjCPVqjDrZVVPTIH4EzjLo2mzzL6wGKcPC30sSftAIFS6Qkpy19IkppD+IQ+trcYYxyfOPi4jvW3hL3hU5yO5/xrtTXJ807Bw==" } } ], "meta": { "interimType": "Remove Square Roots 0Eq" } }, { "type": "interim", "title": "Solve $$3-3u^{2}=u^{2}:{\\quad}u=\\frac{\\sqrt{3}}{2},\\:u=-\\frac{\\sqrt{3}}{2}$$", "input": "3-3u^{2}=u^{2}", "steps": [ { "type": "interim", "title": "Move $$3\\:$$to the right side", "input": "3-3u^{2}=u^{2}", "result": "-3u^{2}=u^{2}-3", "steps": [ { "type": "step", "primary": "Subtract $$3$$ from both sides", "result": "3-3u^{2}-3=u^{2}-3" }, { "type": "step", "primary": "Simplify", "result": "-3u^{2}=u^{2}-3" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": "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" } }, { "type": "interim", "title": "Move $$u^{2}\\:$$to the left side", "input": "-3u^{2}=u^{2}-3", "result": "-4u^{2}=-3", "steps": [ { "type": "step", "primary": "Subtract $$u^{2}$$ from both sides", "result": "-3u^{2}-u^{2}=u^{2}-3-u^{2}" }, { "type": "step", "primary": "Simplify", "result": "-4u^{2}=-3" } ], "meta": { "interimType": "Move to the Left Title 1Eq", "gptData": "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" } }, { "type": "interim", "title": "Divide both sides by $$-4$$", "input": "-4u^{2}=-3", "result": "u^{2}=\\frac{3}{4}", "steps": [ { "type": "step", "primary": "Divide both sides by $$-4$$", "result": "\\frac{-4u^{2}}{-4}=\\frac{-3}{-4}" }, { "type": "step", "primary": "Simplify", "result": "u^{2}=\\frac{3}{4}" } ], "meta": { "interimType": "Divide Both Sides Specific 1Eq", "gptData": "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" } }, { "type": "step", "primary": "For $$x^{2}=f\\left(a\\right)$$ the solutions are $$x=\\sqrt{f\\left(a\\right)},\\:\\:-\\sqrt{f\\left(a\\right)}$$" }, { "type": "step", "result": "u=\\sqrt{\\frac{3}{4}},\\:u=-\\sqrt{\\frac{3}{4}}" }, { "type": "interim", "title": "$$\\sqrt{\\frac{3}{4}}=\\frac{\\sqrt{3}}{2}$$", "input": "\\sqrt{\\frac{3}{4}}", "steps": [ { "type": "step", "primary": "Apply radical rule: $$\\sqrt[n]{\\frac{a}{b}}=\\frac{\\sqrt[n]{a}}{\\sqrt[n]{b}},\\:\\quad$$ assuming $$a\\ge0,\\:b\\ge0$$", "result": "=\\frac{\\sqrt{3}}{\\sqrt{4}}", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } }, { "type": "interim", "title": "$$\\sqrt{4}=2$$", "input": "\\sqrt{4}", "result": "=\\frac{\\sqrt{3}}{2}", "steps": [ { "type": "step", "primary": "Factor the number: $$4=2^{2}$$", "result": "=\\sqrt{2^{2}}" }, { "type": "step", "primary": "Apply radical rule: $$\\sqrt[n]{a^n}=a$$", "secondary": [ "$$\\sqrt{2^{2}}=2$$" ], "result": "=2", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } } ], "meta": { "interimType": "N/A" } } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7FJIkkmi1CWuhEmDQGlA0N481GBhaFmRuYU8ivCdOkRGrju+5Z51e/ZZSD3gRHwjBnvjDY21b5XBQ44AG3rKeoERJNpqBgWYJo7POzHRJNgWKy+4oc4WkS69iOkw+tuw7OdycaE+yHbER8Bud6bqjpd6vq8ch6QORd3MMFeFuyk+wiNrEngO+NNvZ9sqNu+2V" } }, { "type": "interim", "title": "$$-\\sqrt{\\frac{3}{4}}=-\\frac{\\sqrt{3}}{2}$$", "input": "-\\sqrt{\\frac{3}{4}}", "steps": [ { "type": "interim", "title": "Simplify $$\\sqrt{\\frac{3}{4}}:{\\quad}\\frac{\\sqrt{3}}{2}$$", "input": "\\sqrt{\\frac{3}{4}}", "result": "=-\\frac{\\sqrt{3}}{2}", "steps": [ { "type": "step", "primary": "Apply radical rule: $$\\sqrt[n]{\\frac{a}{b}}=\\frac{\\sqrt[n]{a}}{\\sqrt[n]{b}},\\:\\quad$$ assuming $$a\\ge0,\\:b\\ge0$$", "result": "=\\frac{\\sqrt{3}}{\\sqrt{4}}", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } }, { "type": "interim", "title": "$$\\sqrt{4}=2$$", "input": "\\sqrt{4}", "result": "=\\frac{\\sqrt{3}}{2}", "steps": [ { "type": "step", "primary": "Factor the number: $$4=2^{2}$$", "result": "=\\sqrt{2^{2}}" }, { "type": "step", "primary": "Apply radical rule: $$\\sqrt[n]{a^n}=a$$", "secondary": [ "$$\\sqrt{2^{2}}=2$$" ], "result": "=2", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } } ], "meta": { "interimType": "N/A" } } ], "meta": { "interimType": "Algebraic Manipulation Simplify Title 1Eq" } } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7xVDfakA4g1ZczTHh+Arxm/DDmuD5U4GDJIF9k+W0I1gJQJZuTAY5js+oqjdT8kslDa1hUvtzgBcwD7zey8pjX+7J0HqxMc0mt58lG7zw6WRrXh3OSOXhogkv14GjeoU6kzLzxByUErH9eEiuIkZhcyb97z4ciRBzgXtcnYl8cilurCc0v6+SezXpJu+Sddf4" } }, { "type": "step", "result": "u=\\frac{\\sqrt{3}}{2},\\:u=-\\frac{\\sqrt{3}}{2}" } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve Title 1Eq" } }, { "type": "step", "result": "u=\\frac{\\sqrt{3}}{2},\\:u=-\\frac{\\sqrt{3}}{2}" }, { "type": "interim", "title": "Verify Solutions:$${\\quad}u=\\frac{\\sqrt{3}}{2}\\:$$True$$,\\:\\:u=-\\frac{\\sqrt{3}}{2}\\:$$False", "steps": [ { "type": "step", "primary": "Check the solutions by plugging them into $$u-\\sqrt{3-3u^{2}}=0$$<br/>Remove the ones that don't agree with the equation." }, { "type": "interim", "title": "Plug in $$u=\\frac{\\sqrt{3}}{2}:{\\quad}$$True", "input": "\\left(\\frac{\\sqrt{3}}{2}\\right)-\\sqrt{3-3\\left(\\frac{\\sqrt{3}}{2}\\right)^{2}}=0", "steps": [ { "type": "interim", "title": "$$\\left(\\frac{\\sqrt{3}}{2}\\right)-\\sqrt{3-3\\left(\\frac{\\sqrt{3}}{2}\\right)^{2}}=0$$", "input": "\\left(\\frac{\\sqrt{3}}{2}\\right)-\\sqrt{3-3\\left(\\frac{\\sqrt{3}}{2}\\right)^{2}}", "steps": [ { "type": "step", "primary": "Remove parentheses: $$\\left(a\\right)=a$$", "result": "=\\frac{\\sqrt{3}}{2}-\\sqrt{3-3\\left(\\frac{\\sqrt{3}}{2}\\right)^{2}}" }, { "type": "interim", "title": "$$\\sqrt{3-3\\left(\\frac{\\sqrt{3}}{2}\\right)^{2}}=\\frac{\\sqrt{3}}{2}$$", "input": "\\sqrt{3-3\\left(\\frac{\\sqrt{3}}{2}\\right)^{2}}", "steps": [ { "type": "interim", "title": "$$3\\left(\\frac{\\sqrt{3}}{2}\\right)^{2}=\\frac{9}{4}$$", "input": "3\\left(\\frac{\\sqrt{3}}{2}\\right)^{2}", "steps": [ { "type": "interim", "title": "$$\\left(\\frac{\\sqrt{3}}{2}\\right)^{2}=\\frac{3}{2^{2}}$$", "input": "\\left(\\frac{\\sqrt{3}}{2}\\right)^{2}", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$\\left(\\frac{a}{b}\\right)^{c}=\\frac{a^{c}}{b^{c}}$$", "result": "=\\frac{\\left(\\sqrt{3}\\right)^{2}}{2^{2}}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "interim", "title": "$$\\left(\\sqrt{3}\\right)^{2}:{\\quad}3$$", "steps": [ { "type": "step", "primary": "Apply radical rule: $$\\sqrt{a}=a^{\\frac{1}{2}}$$", "result": "=\\left(3^{\\frac{1}{2}}\\right)^{2}", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } }, { "type": "step", "primary": "Apply exponent rule: $$\\left(a^{b}\\right)^{c}=a^{bc}$$", "result": "=3^{\\frac{1}{2}\\cdot\\:2}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "interim", "title": "$$\\frac{1}{2}\\cdot\\:2=1$$", "input": "\\frac{1}{2}\\cdot\\:2", "result": "=3", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{1\\cdot\\:2}{2}" }, { "type": "step", "primary": "Cancel the common factor: $$2$$", "result": "=1" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l3vTdf410Ywhq1vZ0kzF8e30Fwl9QKPJxyO/TFRCb5Grju+5Z51e/ZZSD3gRHwjBE9/03SOiEv+BIHutWLr6nUfz18ijmoplMAomfJM9x8W1GdKgiNs+PolKvTuWzYk/" } } ], "meta": { "interimType": "N/A" } }, { "type": "step", "result": "=\\frac{3}{2^{2}}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7W+TF9DwfoU38VHY68TaD49ujMIZR3PB6kZB8bK0KmmkgJ/ZZA32ZInFBpDtxBfiKXYGCmiBF99lesmXZ9iIfJwVBF5cNIbPYgwV7iZRJjp3fRaH3KOLhpljC85zkkN7NluLRRQa3XzMDjwlm3w6d70ttWPhaZnAFOOZej7vzgjpYlSN7rzdW8vgDfTX46ChV" } }, { "type": "step", "result": "=3\\cdot\\:\\frac{3}{2^{2}}" }, { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{3\\cdot\\:3}{2^{2}}" }, { "type": "step", "primary": "Multiply the numbers: $$3\\cdot\\:3=9$$", "result": "=\\frac{9}{2^{2}}" }, { "type": "step", "primary": "$$2^{2}=4$$", "result": "=\\frac{9}{4}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7zNk+SA+mC9lleqKUpucpeLVhGw8/uyUpZUzHdz2Z03rehkKrn0era9rz8TlL+x/vBVZ9vx5jzfo/n1rSDQAgpjtf1sw6Oo0A9hZgrG2WO2nAzSq7csTjGCRetQiXje/WO4cVozyKeYsc8uMhOmoC2cDOePaceqcyaqMmhKbN+RAkt3WiGR7ZaCaXvz77bMjS" } }, { "type": "step", "result": "=\\sqrt{3-\\frac{9}{4}}" }, { "type": "interim", "title": "Join $$3-\\frac{9}{4}:{\\quad}\\frac{3}{4}$$", "input": "3-\\frac{9}{4}", "result": "=\\sqrt{\\frac{3}{4}}", "steps": [ { "type": "step", "primary": "Convert element to fraction: $$3=\\frac{3\\cdot\\:4}{4}$$", "result": "=\\frac{3\\cdot\\:4}{4}-\\frac{9}{4}" }, { "type": "step", "primary": "Since the denominators are equal, combine the fractions: $$\\frac{a}{c}\\pm\\frac{b}{c}=\\frac{a\\pm\\:b}{c}$$", "result": "=\\frac{3\\cdot\\:4-9}{4}" }, { "type": "interim", "title": "$$3\\cdot\\:4-9=3$$", "input": "3\\cdot\\:4-9", "steps": [ { "type": "step", "primary": "Multiply the numbers: $$3\\cdot\\:4=12$$", "result": "=12-9" }, { "type": "step", "primary": "Subtract the numbers: $$12-9=3$$", "result": "=3" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7gAfD+kFQxD1zlcEbtWKCiN6GQqufR6tr2vPxOUv7H++tj2bbSfUlSjnYqC784D/mVnEva1E6F4KI9o/Cnut2gxEkjbsFmiwCRLK0CzCktyY=" } }, { "type": "step", "result": "=\\frac{3}{4}" } ], "meta": { "interimType": "Algebraic Manipulation Join Concise Title 1Eq" } }, { "type": "step", "primary": "Apply radical rule: $$\\sqrt[n]{\\frac{a}{b}}=\\frac{\\sqrt[n]{a}}{\\sqrt[n]{b}},\\:\\quad$$ assuming $$a\\ge0,\\:b\\ge0$$", "result": "=\\frac{\\sqrt{3}}{\\sqrt{4}}", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } }, { "type": "interim", "title": "$$\\sqrt{4}=2$$", "input": "\\sqrt{4}", "result": "=\\frac{\\sqrt{3}}{2}", "steps": [ { "type": "step", "primary": "Factor the number: $$4=2^{2}$$", "result": "=\\sqrt{2^{2}}" }, { "type": "step", "primary": "Apply radical rule: $$\\sqrt[n]{a^n}=a$$", "secondary": [ "$$\\sqrt{2^{2}}=2$$" ], "result": "=2", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } } ], "meta": { "interimType": "N/A" } } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7FgYYtrZxftZ6x0LVOcyWAycg0t9KufQ5/Ecxx5MB742wUmhpn/55zsVQaiUSh8cwCUCWbkwGOY7PqKo3U/JLJYhkdlW7magKQ94V4KYzz7CtCZA9IvYdgMxbjpdehNrsOnNprf2LV4k2mPx0u5sFXPV0qdHtl37QbY1K0H1S2A2pDZ12mUdtc/QS3Uskt4ckg8+yKmP9yd6q8DLylj2Dfo8BPOx0wlsgFN8qUa6AzA0=" } }, { "type": "step", "result": "=\\frac{\\sqrt{3}}{2}-\\frac{\\sqrt{3}}{2}" }, { "type": "step", "primary": "Add similar elements: $$\\frac{\\sqrt{3}}{2}-\\frac{\\sqrt{3}}{2}=0$$", "result": "=0" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7W+TF9DwfoU38VHY68TaD40bpjMlHer8IAlEEolaflLU/Z293JjhTs4Oy/ZKlIxZ3nLrn8oweBNVImD807HpPtlXTSum/z5kLpMzXS1UJIeyqjfZqhkrOoAz8sRb7wXZWhmdb7B8weX1yesprU+vuFScg0t9KufQ5/Ecxx5MB740qW7EqYw0p2aBALBXY83pmg8+yKmP9yd6q8DLylj2Dfr6pTdnTIGQKIluKtl6JcHY=" } }, { "type": "step", "primary": "$$0=0$$" }, { "type": "step", "result": "\\mathrm{True}" } ], "meta": { "interimType": "Generic Plug 1Eq" } }, { "type": "interim", "title": "Plug in $$u=-\\frac{\\sqrt{3}}{2}:{\\quad}$$False", "input": "\\left(-\\frac{\\sqrt{3}}{2}\\right)-\\sqrt{3-3\\left(-\\frac{\\sqrt{3}}{2}\\right)^{2}}=0", "steps": [ { "type": "interim", "title": "$$\\left(-\\frac{\\sqrt{3}}{2}\\right)-\\sqrt{3-3\\left(-\\frac{\\sqrt{3}}{2}\\right)^{2}}=-\\sqrt{3}$$", "input": "\\left(-\\frac{\\sqrt{3}}{2}\\right)-\\sqrt{3-3\\left(-\\frac{\\sqrt{3}}{2}\\right)^{2}}", "steps": [ { "type": "step", "primary": "Remove parentheses: $$\\left(-a\\right)=-a$$", "result": "=-\\frac{\\sqrt{3}}{2}-\\sqrt{3-3\\left(-\\frac{\\sqrt{3}}{2}\\right)^{2}}" }, { "type": "interim", "title": "$$\\sqrt{3-3\\left(-\\frac{\\sqrt{3}}{2}\\right)^{2}}=\\frac{\\sqrt{3}}{2}$$", "input": "\\sqrt{3-3\\left(-\\frac{\\sqrt{3}}{2}\\right)^{2}}", "steps": [ { "type": "interim", "title": "$$3\\left(-\\frac{\\sqrt{3}}{2}\\right)^{2}=\\frac{9}{4}$$", "input": "3\\left(-\\frac{\\sqrt{3}}{2}\\right)^{2}", "steps": [ { "type": "interim", "title": "$$\\left(-\\frac{\\sqrt{3}}{2}\\right)^{2}=\\frac{3}{2^{2}}$$", "input": "\\left(-\\frac{\\sqrt{3}}{2}\\right)^{2}", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$\\left(-a\\right)^{n}=a^{n},\\:$$if $$n$$ is even", "secondary": [ "$$\\left(-\\frac{\\sqrt{3}}{2}\\right)^{2}=\\left(\\frac{\\sqrt{3}}{2}\\right)^{2}$$" ], "result": "=\\left(\\frac{\\sqrt{3}}{2}\\right)^{2}" }, { "type": "step", "primary": "Apply exponent rule: $$\\left(\\frac{a}{b}\\right)^{c}=\\frac{a^{c}}{b^{c}}$$", "result": "=\\frac{\\left(\\sqrt{3}\\right)^{2}}{2^{2}}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "interim", "title": "$$\\left(\\sqrt{3}\\right)^{2}:{\\quad}3$$", "steps": [ { "type": "step", "primary": "Apply radical rule: $$\\sqrt{a}=a^{\\frac{1}{2}}$$", "result": "=\\left(3^{\\frac{1}{2}}\\right)^{2}", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } }, { "type": "step", "primary": "Apply exponent rule: $$\\left(a^{b}\\right)^{c}=a^{bc}$$", "result": "=3^{\\frac{1}{2}\\cdot\\:2}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "interim", "title": "$$\\frac{1}{2}\\cdot\\:2=1$$", "input": "\\frac{1}{2}\\cdot\\:2", "result": "=3", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{1\\cdot\\:2}{2}" }, { "type": "step", "primary": "Cancel the common factor: $$2$$", "result": "=1" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l3vTdf410Ywhq1vZ0kzF8e30Fwl9QKPJxyO/TFRCb5Grju+5Z51e/ZZSD3gRHwjBE9/03SOiEv+BIHutWLr6nUfz18ijmoplMAomfJM9x8W1GdKgiNs+PolKvTuWzYk/" } } ], "meta": { "interimType": "N/A" } }, { "type": "step", "result": "=\\frac{3}{2^{2}}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s75s3xpOjgqYGSPN+T51/sNbVhGw8/uyUpZUzHdz2Z03rehkKrn0era9rz8TlL+x/vBVZ9vx5jzfo/n1rSDQAgpmQ9hOyW4HaqmPcC+KQqgPXzjtzz8vhq/9L/Uf6ItOLe3q+rxyHpA5F3cwwV4W7KT8QpWYC6V1ngvsxlSk8JWMPSXbpvCObEfBqS00fKTC1L" } }, { "type": "step", "result": "=3\\cdot\\:\\frac{3}{2^{2}}" }, { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{3\\cdot\\:3}{2^{2}}" }, { "type": "step", "primary": "Multiply the numbers: $$3\\cdot\\:3=9$$", "result": "=\\frac{9}{2^{2}}" }, { "type": "step", "primary": "$$2^{2}=4$$", "result": "=\\frac{9}{4}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7dsA6wQxo1vUYs1ku81RiOFlCzFZJV2W1So+1bM1nFpNV00rpv8+ZC6TM10tVCSHs0xDS+Y5aj0hl+F6LvDaAls4IjHGi+fpCujnpY11M729S6OvPjjJmWoVoa5Cn4OoNluLRRQa3XzMDjwlm3w6d75zybXaEL0ZiHYPQV2hAsJywiNrEngO+NNvZ9sqNu+2V" } }, { "type": "step", "result": "=\\sqrt{3-\\frac{9}{4}}" }, { "type": "interim", "title": "Join $$3-\\frac{9}{4}:{\\quad}\\frac{3}{4}$$", "input": "3-\\frac{9}{4}", "result": "=\\sqrt{\\frac{3}{4}}", "steps": [ { "type": "step", "primary": "Convert element to fraction: $$3=\\frac{3\\cdot\\:4}{4}$$", "result": "=\\frac{3\\cdot\\:4}{4}-\\frac{9}{4}" }, { "type": "step", "primary": "Since the denominators are equal, combine the fractions: $$\\frac{a}{c}\\pm\\frac{b}{c}=\\frac{a\\pm\\:b}{c}$$", "result": "=\\frac{3\\cdot\\:4-9}{4}" }, { "type": "interim", "title": "$$3\\cdot\\:4-9=3$$", "input": "3\\cdot\\:4-9", "steps": [ { "type": "step", "primary": "Multiply the numbers: $$3\\cdot\\:4=12$$", "result": "=12-9" }, { "type": "step", "primary": "Subtract the numbers: $$12-9=3$$", "result": "=3" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7gAfD+kFQxD1zlcEbtWKCiN6GQqufR6tr2vPxOUv7H++tj2bbSfUlSjnYqC784D/mVnEva1E6F4KI9o/Cnut2gxEkjbsFmiwCRLK0CzCktyY=" } }, { "type": "step", "result": "=\\frac{3}{4}" } ], "meta": { "interimType": "Algebraic Manipulation Join Concise Title 1Eq" } }, { "type": "step", "primary": "Apply radical rule: $$\\sqrt[n]{\\frac{a}{b}}=\\frac{\\sqrt[n]{a}}{\\sqrt[n]{b}},\\:\\quad$$ assuming $$a\\ge0,\\:b\\ge0$$", "result": "=\\frac{\\sqrt{3}}{\\sqrt{4}}", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } }, { "type": "interim", "title": "$$\\sqrt{4}=2$$", "input": "\\sqrt{4}", "result": "=\\frac{\\sqrt{3}}{2}", "steps": [ { "type": "step", "primary": "Factor the number: $$4=2^{2}$$", "result": "=\\sqrt{2^{2}}" }, { "type": "step", "primary": "Apply radical rule: $$\\sqrt[n]{a^n}=a$$", "secondary": [ "$$\\sqrt{2^{2}}=2$$" ], "result": "=2", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } } ], "meta": { "interimType": "N/A" } } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s74P6Z8BU3eXfkZzQFYQBJtIPPsipj/cneqvAy8pY9g36J9p1oKVW9FLQg9VZ/1GTOdYPfXQvX4/bINBB8wSEQ0ezNQwy21Qht4dZg+h1P+FDuydB6sTHNJrefJRu88OlkcubCnYZOJ5L8/2gsdymw1A5oy/cahRosOOVU2GeVr/iYl+XsdCSldAtkLX5O9HBzJv3vPhyJEHOBe1ydiXxyKW6sJzS/r5J7Nekm75J11/g=" } }, { "type": "step", "result": "=-\\frac{\\sqrt{3}}{2}-\\frac{\\sqrt{3}}{2}" }, { "type": "step", "primary": "Apply rule $$\\frac{a}{c}\\pm\\frac{b}{c}=\\frac{a\\pm\\:b}{c}$$", "result": "=\\frac{-\\sqrt{3}-\\sqrt{3}}{2}" }, { "type": "step", "primary": "Add similar elements: $$-\\sqrt{3}-\\sqrt{3}=-2\\sqrt{3}$$", "result": "=\\frac{-2\\sqrt{3}}{2}" }, { "type": "step", "primary": "Apply the fraction rule: $$\\frac{-a}{b}=-\\frac{a}{b}$$", "result": "=-\\frac{2\\sqrt{3}}{2}" }, { "type": "step", "primary": "Divide the numbers: $$\\frac{2}{2}=1$$", "result": "=-\\sqrt{3}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s75s3xpOjgqYGSPN+T51/sNbQlkt4FfcKFvud2iSg0pEzlElNxSxi36w/BVjVJ1EiPKvbIgbmuNkdl3VTO4fTlOQCWKUbvV6WK3fDUgFtg3Q+Vu0y/nymFFvapFvN0D5+8IQu4L24be2rtnf/ZXbdPZYUCF2W3GoK9dRMLS3rdcL/99m1nIQ4nHtGCmMPv4GFO+oFsE5ZAVkvSPp5lCMoAcEj+eLm1Sdr+9klmhmQtfjXPeiCTXOq2CWEySA4FXdN9sIjaxJ4DvjTb2fbKjbvtlQ==" } }, { "type": "step", "primary": "$$-\\sqrt{3}=0$$" }, { "type": "step", "result": "\\mathrm{False}" } ], "meta": { "interimType": "Generic Plug 1Eq" } } ], "meta": { "interimType": "Check Solutions Plug Preface 1Eq" } }, { "type": "step", "primary": "The solution is", "result": "u=\\frac{\\sqrt{3}}{2}" } ], "meta": { "solvingClass": "Equations", "interimType": "Equations" } }, { "type": "step", "primary": "Substitute back $$u=\\cos\\left(x\\right)$$", "result": "\\cos\\left(x\\right)=\\frac{\\sqrt{3}}{2}" } ], "meta": { "interimType": "Substitution Method 0Eq" } }, { "type": "interim", "title": "$$\\cos\\left(x\\right)=\\frac{\\sqrt{3}}{2}{\\quad:\\quad}x=\\frac{π}{6}+2πn,\\:x=\\frac{11π}{6}+2πn$$", "input": "\\cos\\left(x\\right)=\\frac{\\sqrt{3}}{2}", "steps": [ { "type": "interim", "title": "General solutions for $$\\cos\\left(x\\right)=\\frac{\\sqrt{3}}{2}$$", "result": "x=\\frac{π}{6}+2πn,\\:x=\\frac{11π}{6}+2πn", "steps": [ { "type": "step", "primary": "$$\\cos\\left(x\\right)$$ periodicity table with $$2πn$$ cycle:<br/>$$\\begin{array}{|c|c|c|c|}\\hline x&\\cos(x)&x&\\cos(x)\\\\\\hline 0&1&π&-1\\\\\\hline \\frac{π}{6}&\\frac{\\sqrt{3}}{2}&\\frac{7π}{6}&-\\frac{\\sqrt{3}}{2}\\\\\\hline \\frac{π}{4}&\\frac{\\sqrt{2}}{2}&\\frac{5π}{4}&-\\frac{\\sqrt{2}}{2}\\\\\\hline \\frac{π}{3}&\\frac{1}{2}&\\frac{4π}{3}&-\\frac{1}{2}\\\\\\hline \\frac{π}{2}&0&\\frac{3π}{2}&0\\\\\\hline \\frac{2π}{3}&-\\frac{1}{2}&\\frac{5π}{3}&\\frac{1}{2}\\\\\\hline \\frac{3π}{4}&-\\frac{\\sqrt{2}}{2}&\\frac{7π}{4}&\\frac{\\sqrt{2}}{2}\\\\\\hline \\frac{5π}{6}&-\\frac{\\sqrt{3}}{2}&\\frac{11π}{6}&\\frac{\\sqrt{3}}{2}\\\\\\hline \\end{array}$$" }, { "type": "step", "result": "x=\\frac{π}{6}+2πn,\\:x=\\frac{11π}{6}+2πn" } ], "meta": { "interimType": "Trig General Solutions cos 1Eq" } } ], "meta": { "interimType": "N/A" } }, { "type": "step", "primary": "Combine all the solutions", "result": "x=\\frac{π}{6}+2πn,\\:x=\\frac{11π}{6}+2πn" } ], "meta": { "solvingClass": "Trig Equations", "practiceLink": "/practice/trigonometry-practice#area=main&subtopic=Trig%20Equations", "practiceTopic": "Trig Equations" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "\\cos(x)-\\sqrt{3-3\\cos^{2}(x)}" }, "showViewLarger": true } }, "meta": { "showVerify": true } }