sin(x+30)=cos(2x)

{ "query": { "display": "$$\\sin\\left(x+30^{\\circ\\:}\\right)=\\cos\\left(2x\\right)$$", "symbolab_question": "EQUATION#\\sin(x+30^{\\circ })=\\cos(2x)" }, "solution": { "level": "PERFORMED", "subject": "Trigonometry", "topic": "Trig Equations", "subTopic": "Trig Equations", "default": "x=\\frac{1080^{\\circ }n+180^{\\circ }}{9},x=-\\frac{180^{\\circ }+1080^{\\circ }n}{3}", "radians": "x=\\frac{π}{9}+\\frac{6π}{9}n,x=-\\frac{π}{3}-\\frac{6π}{3}n", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\sin\\left(x+30^{\\circ\\:}\\right)=\\cos\\left(2x\\right){\\quad:\\quad}x=\\frac{1080^{\\circ\\:}n+180^{\\circ\\:}}{9},\\:x=-\\frac{180^{\\circ\\:}+1080^{\\circ\\:}n}{3}$$", "input": "\\sin\\left(x+30^{\\circ\\:}\\right)=\\cos\\left(2x\\right)", "steps": [ { "type": "interim", "title": "Rewrite using trig identities", "input": "\\sin\\left(x+30^{\\circ\\:}\\right)=\\cos\\left(2x\\right)", "result": "\\sin\\left(x+30^{\\circ\\:}\\right)=\\sin\\left(90^{\\circ\\:}-2x\\right)", "steps": [ { "type": "step", "primary": "Use the following identity: $$\\cos\\left(x\\right)=\\sin\\left(90^{\\circ\\:}-x\\right)$$", "result": "\\sin\\left(x+30^{\\circ\\:}\\right)=\\sin\\left(90^{\\circ\\:}-2x\\right)" } ], "meta": { "interimType": "Trig Rewrite Using Trig identities 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7Awm8OHaUUE5X73KqyTFjFOCHuG2gReUlWHvk//vmG3ASvn79xfScPqbTe394pe/APosKaRMq/pBPumoBrLj89/lZJuHdgx059G6Z6iwp6k/6DZLE6vHKrQJpnGEY59NT+RM9wO0tsHPMAwRhvS0f4D4WbuhI5igEeTT3Kz9NTNiYI73+I6tAoE/DsLqqq+MdfSaJu/fYRZontWwBsFxv2Rhy8Sn4CsnsiBBCKzJJgMfi9rijpllC3oQrllyLV3norqDtxpmeslF5YSvi9T16bw==" } }, { "type": "interim", "title": "Apply trig inverse properties", "input": "\\sin\\left(x+30^{\\circ\\:}\\right)=\\sin\\left(90^{\\circ\\:}-2x\\right)", "result": "x+30^{\\circ\\:}=90^{\\circ\\:}-2x+360^{\\circ\\:}n,\\:x+30^{\\circ\\:}=180^{\\circ\\:}-\\left(90^{\\circ\\:}-2x\\right)+360^{\\circ\\:}n", "steps": [ { "type": "step", "primary": "$$\\sin\\left(x\\right)=\\sin\\left(y\\right)\\quad\\Rightarrow\\quad\\:x=y+2{\\pi}n,\\:x=\\pi-y+2{\\pi}n$$", "result": "x+30^{\\circ\\:}=90^{\\circ\\:}-2x+360^{\\circ\\:}n,\\:x+30^{\\circ\\:}=180^{\\circ\\:}-\\left(90^{\\circ\\:}-2x\\right)+360^{\\circ\\:}n" } ], "meta": { "interimType": "Trig Apply Inverse Props 0Eq" } }, { "type": "interim", "title": "$$x+30^{\\circ\\:}=90^{\\circ\\:}-2x+360^{\\circ\\:}n{\\quad:\\quad}x=\\frac{1080^{\\circ\\:}n+180^{\\circ\\:}}{9}$$", "input": "x+30^{\\circ\\:}=90^{\\circ\\:}-2x+360^{\\circ\\:}n", "steps": [ { "type": "interim", "title": "Move $$30^{\\circ\\:}\\:$$to the right side", "input": "x+30^{\\circ\\:}=90^{\\circ\\:}-2x+360^{\\circ\\:}n", "result": "x=-2x+360^{\\circ\\:}n+60^{\\circ\\:}", "steps": [ { "type": "step", "primary": "Subtract $$30^{\\circ\\:}$$ from both sides", "result": "x+30^{\\circ\\:}-30^{\\circ\\:}=90^{\\circ\\:}-2x+360^{\\circ\\:}n-30^{\\circ\\:}" }, { "type": "interim", "title": "Simplify", "input": "x+30^{\\circ\\:}-30^{\\circ\\:}=90^{\\circ\\:}-2x+360^{\\circ\\:}n-30^{\\circ\\:}", "result": "x=-2x+360^{\\circ\\:}n+60^{\\circ\\:}", "steps": [ { "type": "interim", "title": "Simplify $$x+30^{\\circ\\:}-30^{\\circ\\:}:{\\quad}x$$", "input": "x+30^{\\circ\\:}-30^{\\circ\\:}", "steps": [ { "type": "step", "primary": "Add similar elements: $$30^{\\circ\\:}-30^{\\circ\\:}=0$$" }, { "type": "step", "result": "=x" } ], "meta": { "interimType": "Generic Simplify Specific 1Eq" } }, { "type": "interim", "title": "Simplify $$90^{\\circ\\:}-2x+360^{\\circ\\:}n-30^{\\circ\\:}:{\\quad}-2x+360^{\\circ\\:}n+60^{\\circ\\:}$$", "input": "90^{\\circ\\:}-2x+360^{\\circ\\:}n-30^{\\circ\\:}", "steps": [ { "type": "step", "primary": "Group like terms", "result": "=-2x+360^{\\circ\\:}n+90^{\\circ\\:}-30^{\\circ\\:}" }, { "type": "interim", "title": "Least Common Multiplier of $$2,\\:6:{\\quad}6$$", "input": "2,\\:6", "steps": [ { "type": "definition", "title": "Least Common Multiplier (LCM)", "text": "The LCM of $$a,\\:b$$ is the smallest positive number that is divisible by both $$a$$ and $$b$$" }, { "type": "interim", "title": "Prime factorization of $$2:{\\quad}2$$", "input": "2", "steps": [ { "type": "step", "primary": "$$2$$ is a prime number, therefore no factorization is possible", "result": "=2" } ], "meta": { "solvingClass": "Composite Integer", "interimType": "Prime Fac 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xyoU2cWyPLDgE1QLLHeauuc3PHQdChPJ2JhfqHT+ZU0OMrfn8NOj0LUzuzje6xTyxRl8ZboA8wPLg0yhI4RzfjFw/y9DKGIPglJ+qMi9xDu2KE1OovxZAaXg7BtrFPk4UcCzRnGgMN6CYRfod7Mq0dp1+G9v2aKasChgV65VW8cTW" } }, { "type": "interim", "title": "Prime factorization of $$6:{\\quad}2\\cdot\\:3$$", "input": "6", "steps": [ { "type": "step", "primary": "$$6\\:$$divides by $$2\\quad\\:6=3\\cdot\\:2$$", "result": "=2\\cdot\\:3" }, { "type": "step", "primary": "$$2,\\:3$$ are all prime numbers, therefore no further factorization is possible", "result": "=2\\cdot\\:3" } ], "meta": { "solvingClass": "Composite Integer", "interimType": "Prime Fac 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xyoU2cWyPLDgE1QLLHeauuc3PHQdChPJ2JhfqHT+ZU0OMrfn8NOj0LUzuzje6xTyxRuUHkFwKrCGUG/pR2kioRow/y9DKGIPglJ+qMi9xDu2KE1OovxZAaXg7BtrFPk4UcCzRnGgMN6CYRfod7Mq0dp1AjXz67i9oO9i25G22wINi" } }, { "type": "step", "primary": "Multiply each factor the greatest number of times it occurs in either $$2$$ or $$6$$", "result": "=2\\cdot\\:3" }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:3=6$$", "result": "=6" } ], "meta": { "solvingClass": "LCM", "interimType": "LCM Top 1Eq" } }, { "type": "interim", "title": "Adjust Fractions based on the LCM", "steps": [ { "type": "step", "primary": "Multiply each numerator by the same amount needed to multiply its<br/>corresponding denominator to turn it into the LCM $$6$$" }, { "type": "step", "primary": "For $$90^{\\circ\\:}:\\:$$multiply the denominator and numerator by $$3$$", "result": "90^{\\circ\\:}=\\frac{180^{\\circ\\:}3}{2\\cdot\\:3}=90^{\\circ\\:}" } ], "meta": { "interimType": "LCD Adjust Fractions 1Eq" } }, { "type": "step", "result": "=90^{\\circ\\:}-30^{\\circ\\:}" }, { "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{180^{\\circ\\:}3-180^{\\circ\\:}}{6}" }, { "type": "step", "primary": "Add similar elements: $$540^{\\circ\\:}-180^{\\circ\\:}=360^{\\circ\\:}$$", "result": "=60^{\\circ\\:}" }, { "type": "step", "primary": "Cancel the common factor: $$2$$", "result": "=-2x+360^{\\circ\\:}n+60^{\\circ\\:}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7H0c8m9fFtCO4ZPS5ct0nxS0M/KeH0DK7MmzbV5Q9HQS7JDHPBrz0hChW/oZRQaWX3oZCq59Hq2va8/E5S/sf73LAfWVoqNf4kpgJnJXMOWNuFoVr6rJv9/tqdcXfVOSJjGuvQmPeg32BAwLOwQDVq9bA+zX4bD3u3gx65o2NJhNzoFNVvppNBx7RPIydnBGK6ACX6Dsg+xhW8LiJsKAKI4cD0K8NkM+CJlGI5I5ow+/7s+p4kEfMJ9G7SA2I7SWG" } }, { "type": "step", "result": "x=-2x+360^{\\circ\\:}n+60^{\\circ\\:}" } ], "meta": { "interimType": "Generic Simplify 0Eq" } } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": "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" } }, { "type": "interim", "title": "Move $$2x\\:$$to the left side", "input": "x=-2x+360^{\\circ\\:}n+60^{\\circ\\:}", "result": "3x=360^{\\circ\\:}n+60^{\\circ\\:}", "steps": [ { "type": "step", "primary": "Add $$2x$$ to both sides", "result": "x+2x=-2x+360^{\\circ\\:}n+60^{\\circ\\:}+2x" }, { "type": "step", "primary": "Simplify", "result": "3x=360^{\\circ\\:}n+60^{\\circ\\:}" } ], "meta": { "interimType": "Move to the Left Title 1Eq", "gptData": "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" } }, { "type": "interim", "title": "Divide both sides by $$3$$", "input": "3x=360^{\\circ\\:}n+60^{\\circ\\:}", "result": "x=\\frac{1080^{\\circ\\:}n+180^{\\circ\\:}}{9}", "steps": [ { "type": "step", "primary": "Divide both sides by $$3$$", "result": "\\frac{3x}{3}=\\frac{360^{\\circ\\:}n}{3}+\\frac{60^{\\circ\\:}}{3}" }, { "type": "interim", "title": "Simplify", "input": "\\frac{3x}{3}=\\frac{360^{\\circ\\:}n}{3}+\\frac{60^{\\circ\\:}}{3}", "result": "x=\\frac{1080^{\\circ\\:}n+180^{\\circ\\:}}{9}", "steps": [ { "type": "interim", "title": "Simplify $$\\frac{3x}{3}:{\\quad}x$$", "input": "\\frac{3x}{3}", "steps": [ { "type": "step", "primary": "Divide the numbers: $$\\frac{3}{3}=1$$", "result": "=x" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7QJV6FEydUSv+hKOzRRB7mi061ljBSPJeENOw2efoSWt2cZ6LAXMcKee6PSyjLEFrRSpN33oxZMojoqvYhvSJACLkpiVA7S8UT8ieezZKu6ot8gw9eOFDGZzXCUlLc4C0" } }, { "type": "interim", "title": "Simplify $$\\frac{360^{\\circ\\:}n}{3}+\\frac{60^{\\circ\\:}}{3}:{\\quad}\\frac{1080^{\\circ\\:}n+180^{\\circ\\:}}{9}$$", "input": "\\frac{360^{\\circ\\:}n}{3}+\\frac{60^{\\circ\\:}}{3}", "steps": [ { "type": "step", "primary": "Apply rule $$\\frac{a}{c}\\pm\\frac{b}{c}=\\frac{a\\pm\\:b}{c}$$", "result": "=\\frac{360^{\\circ\\:}n+60^{\\circ\\:}}{3}" }, { "type": "interim", "title": "Join $$360^{\\circ\\:}n+60^{\\circ\\:}:{\\quad}\\frac{1080^{\\circ\\:}n+180^{\\circ\\:}}{3}$$", "input": "360^{\\circ\\:}n+60^{\\circ\\:}", "result": "=\\frac{\\frac{1080^{\\circ\\:}n+180^{\\circ\\:}}{3}}{3}", "steps": [ { "type": "step", "primary": "Convert element to fraction: $$360^{\\circ\\:}n=\\frac{360^{\\circ\\:}n3}{3}$$", "result": "=\\frac{360^{\\circ\\:}n\\cdot\\:3}{3}+60^{\\circ\\:}" }, { "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{360^{\\circ\\:}n\\cdot\\:3+180^{\\circ\\:}}{3}" }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:3=6$$", "result": "=\\frac{1080^{\\circ\\:}n+180^{\\circ\\:}}{3}" } ], "meta": { "interimType": "Algebraic Manipulation Join Concise Title 1Eq" } }, { "type": "step", "primary": "Apply the fraction rule: $$\\frac{\\frac{b}{c}}{a}=\\frac{b}{c\\:\\cdot\\:a}$$", "result": "=\\frac{1080^{\\circ\\:}n+180^{\\circ\\:}}{3\\cdot\\:3}" }, { "type": "step", "primary": "Multiply the numbers: $$3\\cdot\\:3=9$$", "result": "=\\frac{1080^{\\circ\\:}n+180^{\\circ\\:}}{9}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s71VWBOBW9C7vXuDdGOZiiaJTk+/xbp5WFgskQtvTfDfXNrB09CeCE4OvwgVyTxRNpFmyfDD4/teN2iRTghyYaD6ORWLXkjysF58uLgjK3bCuCAG8B59WnlNa4wTKeX2Om0svr60zJKgi5t9ZnFrdOjB73846MSmbYxJ9vklkr2st6pfF1z6umzUJTJvt+ojYZY4LVEbDWfJhdqtpUA6/Oh9RjQMqNQ0EK9eenpdoDgjpYSOGIuqHbktEDCLYY0ZmXnpT4Zlk7HR+ndFsTk317kw==" } }, { "type": "step", "result": "x=\\frac{1080^{\\circ\\:}n+180^{\\circ\\:}}{9}" } ], "meta": { "interimType": "Generic Simplify 0Eq" } } ], "meta": { "interimType": "Divide Both Sides Specific 1Eq", "gptData": "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" } } ], "meta": { "solvingClass": "Equations", "interimType": "Equations" } }, { "type": "interim", "title": "$$x+30^{\\circ\\:}=180^{\\circ\\:}-\\left(90^{\\circ\\:}-2x\\right)+360^{\\circ\\:}n{\\quad:\\quad}x=-\\frac{180^{\\circ\\:}+1080^{\\circ\\:}n}{3}$$", "input": "x+30^{\\circ\\:}=180^{\\circ\\:}-\\left(90^{\\circ\\:}-2x\\right)+360^{\\circ\\:}n", "steps": [ { "type": "interim", "title": "Expand $$180^{\\circ\\:}-\\left(90^{\\circ\\:}-2x\\right)+360^{\\circ\\:}n:{\\quad}180^{\\circ\\:}-90^{\\circ\\:}+2x+360^{\\circ\\:}n$$", "input": "180^{\\circ\\:}-\\left(90^{\\circ\\:}-2x\\right)+360^{\\circ\\:}n", "steps": [ { "type": "interim", "title": "$$-\\left(90^{\\circ\\:}-2x\\right):{\\quad}-90^{\\circ\\:}+2x$$", "input": "-\\left(90^{\\circ\\:}-2x\\right)", "result": "=180^{\\circ\\:}-90^{\\circ\\:}+2x+360^{\\circ\\:}n", "steps": [ { "type": "step", "primary": "Distribute parentheses", "result": "=-\\left(90^{\\circ\\:}\\right)-\\left(-2x\\right)" }, { "type": "step", "primary": "Apply minus-plus rules", "secondary": [ "$$-\\left(-a\\right)=a,\\:\\:\\:-\\left(a\\right)=-a$$" ], "result": "=-90^{\\circ\\:}+2x" } ], "meta": { "interimType": "N/A" } } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Expand Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7WafLUeHxfUY6g4hxuTM9uQd9hOHA/4psQrCFGg82owdirzMrlnspeydjkNMrfI9JMW3OQqtXUttDATdwih0A9OhMIALeFw1bUVSyoIjfWd+3aDrACz7snN7Cp96xiSn472wZm7kDUxdE6YSmfEbr2q59bjvikOqoMyATXnqkYQz3cOJPpe8NZ9T357qr1vPeqEB+X1sNtrzxbIvmwnSqpeePehRlGrQp7JqnaIzpOaY=" } }, { "type": "step", "result": "x+30^{\\circ\\:}=180^{\\circ\\:}-90^{\\circ\\:}+2x+360^{\\circ\\:}n" }, { "type": "interim", "title": "Move $$30^{\\circ\\:}\\:$$to the right side", "input": "x+30^{\\circ\\:}=180^{\\circ\\:}-90^{\\circ\\:}+2x+360^{\\circ\\:}n", "result": "x=2x+180^{\\circ\\:}+360^{\\circ\\:}n-120^{\\circ\\:}", "steps": [ { "type": "step", "primary": "Subtract $$30^{\\circ\\:}$$ from both sides", "result": "x+30^{\\circ\\:}-30^{\\circ\\:}=180^{\\circ\\:}-90^{\\circ\\:}+2x+360^{\\circ\\:}n-30^{\\circ\\:}" }, { "type": "interim", "title": "Simplify", "input": "x+30^{\\circ\\:}-30^{\\circ\\:}=180^{\\circ\\:}-90^{\\circ\\:}+2x+360^{\\circ\\:}n-30^{\\circ\\:}", "result": "x=2x+180^{\\circ\\:}+360^{\\circ\\:}n-120^{\\circ\\:}", "steps": [ { "type": "interim", "title": "Simplify $$x+30^{\\circ\\:}-30^{\\circ\\:}:{\\quad}x$$", "input": "x+30^{\\circ\\:}-30^{\\circ\\:}", "steps": [ { "type": "step", "primary": "Add similar elements: $$30^{\\circ\\:}-30^{\\circ\\:}=0$$" }, { "type": "step", "result": "=x" } ], "meta": { "interimType": "Generic Simplify Specific 1Eq" } }, { "type": "interim", "title": "Simplify $$180^{\\circ\\:}-90^{\\circ\\:}+2x+360^{\\circ\\:}n-30^{\\circ\\:}:{\\quad}2x+180^{\\circ\\:}+360^{\\circ\\:}n-120^{\\circ\\:}$$", "input": "180^{\\circ\\:}-90^{\\circ\\:}+2x+360^{\\circ\\:}n-30^{\\circ\\:}", "steps": [ { "type": "step", "primary": "Group like terms", "result": "=2x+180^{\\circ\\:}+360^{\\circ\\:}n-90^{\\circ\\:}-30^{\\circ\\:}" }, { "type": "interim", "title": "Least Common Multiplier of $$2,\\:6:{\\quad}6$$", "input": "2,\\:6", "steps": [ { "type": "definition", "title": "Least Common Multiplier (LCM)", "text": "The LCM of $$a,\\:b$$ is the smallest positive number that is divisible by both $$a$$ and $$b$$" }, { "type": "interim", "title": "Prime factorization of $$2:{\\quad}2$$", "input": "2", "steps": [ { "type": "step", "primary": "$$2$$ is a prime number, therefore no factorization is possible", "result": "=2" } ], "meta": { "solvingClass": "Composite Integer", "interimType": "Prime Fac 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xyoU2cWyPLDgE1QLLHeauuc3PHQdChPJ2JhfqHT+ZU0OMrfn8NOj0LUzuzje6xTyxRl8ZboA8wPLg0yhI4RzfjFw/y9DKGIPglJ+qMi9xDu2KE1OovxZAaXg7BtrFPk4UcCzRnGgMN6CYRfod7Mq0dp1+G9v2aKasChgV65VW8cTW" } }, { "type": "interim", "title": "Prime factorization of $$6:{\\quad}2\\cdot\\:3$$", "input": "6", "steps": [ { "type": "step", "primary": "$$6\\:$$divides by $$2\\quad\\:6=3\\cdot\\:2$$", "result": "=2\\cdot\\:3" }, { "type": "step", "primary": "$$2,\\:3$$ are all prime numbers, therefore no further factorization is possible", "result": "=2\\cdot\\:3" } ], "meta": { "solvingClass": "Composite Integer", "interimType": "Prime Fac 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xyoU2cWyPLDgE1QLLHeauuc3PHQdChPJ2JhfqHT+ZU0OMrfn8NOj0LUzuzje6xTyxRuUHkFwKrCGUG/pR2kioRow/y9DKGIPglJ+qMi9xDu2KE1OovxZAaXg7BtrFPk4UcCzRnGgMN6CYRfod7Mq0dp1AjXz67i9oO9i25G22wINi" } }, { "type": "step", "primary": "Multiply each factor the greatest number of times it occurs in either $$2$$ or $$6$$", "result": "=2\\cdot\\:3" }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:3=6$$", "result": "=6" } ], "meta": { "solvingClass": "LCM", "interimType": "LCM Top 1Eq" } }, { "type": "interim", "title": "Adjust Fractions based on the LCM", "steps": [ { "type": "step", "primary": "Multiply each numerator by the same amount needed to multiply its<br/>corresponding denominator to turn it into the LCM $$6$$" }, { "type": "step", "primary": "For $$90^{\\circ\\:}:\\:$$multiply the denominator and numerator by $$3$$", "result": "90^{\\circ\\:}=\\frac{180^{\\circ\\:}3}{2\\cdot\\:3}=90^{\\circ\\:}" } ], "meta": { "interimType": "LCD Adjust Fractions 1Eq" } }, { "type": "step", "result": "=-90^{\\circ\\:}-30^{\\circ\\:}" }, { "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{-180^{\\circ\\:}3-180^{\\circ\\:}}{6}" }, { "type": "step", "primary": "Add similar elements: $$-540^{\\circ\\:}-180^{\\circ\\:}=-720^{\\circ\\:}$$", "result": "=\\frac{-720^{\\circ\\:}}{6}" }, { "type": "step", "primary": "Apply the fraction rule: $$\\frac{-a}{b}=-\\frac{a}{b}$$", "result": "=-120^{\\circ\\:}" }, { "type": "step", "primary": "Cancel the common factor: $$2$$", "result": "=2x+180^{\\circ\\:}+360^{\\circ\\:}n-120^{\\circ\\:}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7e0XeQ5VqTwjR7KafmeQRC5u92gX8OxVpyvxiCzBuvWz+jkBJ+lEo87R/FqM8YMy+FOdistf9wE1i22KZRn0LvN13jtrSFDx+UNsawjlOjV0SuuxFDH/ljoNSyD5EdGwsK2JAVGatQR4XegF/BnRG51l+r4im7zAqY0VrTdGbpDSyzJrvm+pEIuupo7hmyZbhixpjZyz1hn/Tx95aZo6CSLvv6m4VhfTI3XTCpIa7a8ZDfII329yPQWg3C65zh/aWk23msMQamYsdczROVkdDG/xlN+FYeles8Xpl3voo4dGJqVxX90jlMfh9fKn6dzC4" } }, { "type": "step", "result": "x=2x+180^{\\circ\\:}+360^{\\circ\\:}n-120^{\\circ\\:}" } ], "meta": { "interimType": "Generic Simplify 0Eq" } } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": "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" } }, { "type": "interim", "title": "Move $$2x\\:$$to the left side", "input": "x=2x+180^{\\circ\\:}+360^{\\circ\\:}n-120^{\\circ\\:}", "result": "-x=180^{\\circ\\:}+360^{\\circ\\:}n-120^{\\circ\\:}", "steps": [ { "type": "step", "primary": "Subtract $$2x$$ from both sides", "result": "x-2x=2x+180^{\\circ\\:}+360^{\\circ\\:}n-120^{\\circ\\:}-2x" }, { "type": "step", "primary": "Simplify", "result": "-x=180^{\\circ\\:}+360^{\\circ\\:}n-120^{\\circ\\:}" } ], "meta": { "interimType": "Move to the Left Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7Ck3igSZBBMM7D9WaufA8dIBeEQG+qWePiaOjKVzfExb1yam5omAPNYp9Bm9jjePpsro9rsY00/dduM69foWdHX2sRChv64PCt6roGlGu5O+vRKrsbOYdvqYOmcefLn0CgKqJpxedGbNrfIwPkA+O5dY1Ye8eMlC7LBzA26yT9AuycR+cGWbZ0prnfmTLL4ORP+mA9LpY01/GMc9HL8xtagowYhSs3kL58m9lluLOmEVmy0iDBb9r9fy/qEBwLjHSAvTBjPxPbprHWbsfCdz3Lnsiq9XPg4fzd5GPh+wK83sXOCl7PMYCTYi5CcK9G/+TVzNZq8WyuIhsl1hTRr/Do2oS+4H1ZPiUpT7yW5mDm3cJVUlFsneEG+WJ8PQ/2jWrcgR5baU2V7XG2rGdYR9S7Ou24RE/ezAczu5417y8oS1azJHJVWZgKTWNfA2Ny6NLCTZFOTGDsoKURl4j5tKc6ryx0vkT2vjhml4Eml10FT4K7FCIVH5mNb6+dq+yhv8Bpj7cCVHWQK2LZhpeGMU06eRTlOQwxYRLnXIWxOQlPqV+UHjj5wCFuXTkPs3WjhedhQPEG7O1Z1Z0SY5cF7csulY82rMSGQ/moPqKg1Tiz1+L1z+ccaEiq89Gm211KLaWkmP/nEop0HkUfA7gb98efybEXmnEB6OAwF1d/w0eX6aTd0r2E3qk+TNU1/PySNy3XwifH5oYV9SfK23b3r1CZpO7sjfGI6LadYf8BT1wxbiwiNrEngO+NNvZ9sqNu+2V" } }, { "type": "interim", "title": "Divide both sides by $$-1$$", "input": "-x=180^{\\circ\\:}+360^{\\circ\\:}n-120^{\\circ\\:}", "result": "x=-\\frac{180^{\\circ\\:}+1080^{\\circ\\:}n}{3}", "steps": [ { "type": "step", "primary": "Divide both sides by $$-1$$", "result": "\\frac{-x}{-1}=\\frac{180^{\\circ\\:}}{-1}+\\frac{360^{\\circ\\:}n}{-1}-\\frac{120^{\\circ\\:}}{-1}" }, { "type": "interim", "title": "Simplify", "input": "\\frac{-x}{-1}=\\frac{180^{\\circ\\:}}{-1}+\\frac{360^{\\circ\\:}n}{-1}-\\frac{120^{\\circ\\:}}{-1}", "result": "x=-\\frac{180^{\\circ\\:}+1080^{\\circ\\:}n}{3}", "steps": [ { "type": "interim", "title": "Simplify $$\\frac{-x}{-1}:{\\quad}x$$", "input": "\\frac{-x}{-1}", "steps": [ { "type": "step", "primary": "Apply the fraction rule: $$\\frac{-a}{-b}=\\frac{a}{b}$$", "result": "=\\frac{x}{1}" }, { "type": "step", "primary": "Apply rule $$\\frac{a}{1}=a$$", "result": "=x" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7CuVG0pUqQLV87zgbXotwawCWKUbvV6WK3fDUgFtg3Q84452XiK4JmjK/ZyKR8NAKo3oe/oyhMy2+1TQhDBd2f2zM6E3fuZxF1XkKAYaRXCDDJ+sIB+fsqfndDw1x39hdvzIPeEtDfcHv/z8uls8Teg==" } }, { "type": "interim", "title": "Simplify $$\\frac{180^{\\circ\\:}}{-1}+\\frac{360^{\\circ\\:}n}{-1}-\\frac{120^{\\circ\\:}}{-1}:{\\quad}-\\frac{180^{\\circ\\:}+1080^{\\circ\\:}n}{3}$$", "input": "\\frac{180^{\\circ\\:}}{-1}+\\frac{360^{\\circ\\:}n}{-1}-\\frac{120^{\\circ\\:}}{-1}", "steps": [ { "type": "step", "primary": "Apply rule $$\\frac{a}{c}\\pm\\frac{b}{c}=\\frac{a\\pm\\:b}{c}$$", "result": "=\\frac{180^{\\circ\\:}+360^{\\circ\\:}n-120^{\\circ\\:}}{-1}" }, { "type": "step", "primary": "Apply the fraction rule: $$\\frac{a}{-b}=-\\frac{a}{b}$$", "result": "=-\\frac{180^{\\circ\\:}+360^{\\circ\\:}n-120^{\\circ\\:}}{1}" }, { "type": "interim", "title": "Join $$180^{\\circ\\:}+360^{\\circ\\:}n-120^{\\circ\\:}:{\\quad}\\frac{180^{\\circ\\:}+1080^{\\circ\\:}n}{3}$$", "input": "180^{\\circ\\:}+360^{\\circ\\:}n-120^{\\circ\\:}", "result": "=-\\frac{\\frac{180^{\\circ\\:}+1080^{\\circ\\:}n}{3}}{1}", "steps": [ { "type": "step", "primary": "Convert element to fraction: $$180^{\\circ\\:}=180^{\\circ\\:},\\:360^{\\circ\\:}n=\\frac{360^{\\circ\\:}n3}{3}$$", "result": "=180^{\\circ\\:}+\\frac{360^{\\circ\\:}n\\cdot\\:3}{3}-120^{\\circ\\:}" }, { "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{180^{\\circ\\:}3+360^{\\circ\\:}n\\cdot\\:3-360^{\\circ\\:}}{3}" }, { "type": "interim", "title": "$$180^{\\circ\\:}3+360^{\\circ\\:}n\\cdot\\:3-360^{\\circ\\:}=180^{\\circ\\:}+1080^{\\circ\\:}n$$", "input": "180^{\\circ\\:}3+360^{\\circ\\:}n\\cdot\\:3-360^{\\circ\\:}", "steps": [ { "type": "step", "primary": "Add similar elements: $$540^{\\circ\\:}-360^{\\circ\\:}=180^{\\circ\\:}$$", "result": "=180^{\\circ\\:}+2\\cdot\\:540^{\\circ\\:}n" }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:3=6$$", "result": "=180^{\\circ\\:}+1080^{\\circ\\:}n" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7vgi2vuzg4u3kLtjS+vzlDDUWzD36ECLtu3NIC6r0weZC5jjWYiwUCoBAR87ltTFpI+ggLrWINZyunz1qb7ayBnCQoYlYQ8U+Tfyx0kyzI8j+yrcoNnHLbPjwrBzMeSwvBAgEQUqX+oedQGeChW35/4xRfhq4bXv9lKhZM7D7OKesO//Bl1uwasUOljNzkVN00q+DboYjg0z/EzI+wZPtALnxAxkXepa3YEUkTROiLwTpeuQ25mYzXusEa85JqKYc2Sj4wKUzC6TLMs+EU3H9a7CI2sSeA74029n2yo277ZU=" } }, { "type": "step", "result": "=\\frac{180^{\\circ\\:}+1080^{\\circ\\:}n}{3}" } ], "meta": { "interimType": "Algebraic Manipulation Join Concise Title 1Eq" } }, { "type": "step", "primary": "Apply the fraction rule: $$\\frac{a}{1}=a$$", "result": "=-\\frac{180^{\\circ\\:}+1080^{\\circ\\:}n}{3}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7sctMNWyhnAEl4COXxkn3W72ueFR23N4er1Yqa8zgPSGl1EThFWruHeqbPpgoXgL08WYZnGqL4kM5rMBaoPUofs63xs5d7juPxU4mU4KR8HMAlilG71elit3w1IBbYN0PlbtMv58phRb2qRbzdA+fvFKsBASOdaf0dYinR76sHS/I5jV2VZshjTLyhezZtRlRCz0vIneOBcyt7N8S8M2gvx429vuTSxWa7B/X3D1oP03AWQmX+FAZQ57eQ8HwbCJCjckZMoDDg1MGiT3XboNjItacy7VbivW89pOsOsBSiAtvM6X8qDk80BBFBB7dSbtnCOG2prXIcTh3e2afaPUhgHonG8kFxpoONEvVG09kiyQ=" } }, { "type": "step", "result": "x=-\\frac{180^{\\circ\\:}+1080^{\\circ\\:}n}{3}" } ], "meta": { "interimType": "Generic Simplify 0Eq" } } ], "meta": { "interimType": "Divide Both Sides Specific 1Eq", "gptData": "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" } }, { "type": "step", "result": "x=\\frac{1080^{\\circ\\:}n+180^{\\circ\\:}}{9},\\:x=-\\frac{180^{\\circ\\:}+1080^{\\circ\\:}n}{3}" } ], "meta": { "solvingClass": "Equations", "interimType": "Equations" } }, { "type": "step", "result": "x=\\frac{1080^{\\circ\\:}n+180^{\\circ\\:}}{9},\\:x=-\\frac{180^{\\circ\\:}+1080^{\\circ\\:}n}{3}" } ], "meta": { "solvingClass": "Trig Equations", "practiceLink": "/practice/trigonometry-practice#area=main&subtopic=Trig%20Equations", "practiceTopic": "Trig Equations" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "\\sin(x+30^{\\circ })-\\cos(2x)" }, "showViewLarger": true } }, "meta": { "showVerify": true } }