sin(x+16)=cos(11x+2)

{ "query": { "display": "$$\\sin\\left(x+16^{\\circ\\:}\\right)=\\cos\\left(11x+2\\right)$$", "symbolab_question": "EQUATION#\\sin(x+16^{\\circ })=\\cos(11x+2)" }, "solution": { "level": "PERFORMED", "subject": "Trigonometry", "topic": "Trig Equations", "subTopic": "Trig Equations", "default": "x=\\frac{32400^{\\circ }n+6660^{\\circ }-180}{1080},x=-\\frac{6660^{\\circ }+32400^{\\circ }n+180}{900}", "radians": "x=\\frac{\\frac{37π}{5}}{216}-\\frac{1}{6}+\\frac{180π}{1080}n,x=-\\frac{1}{5}-\\frac{\\frac{37π}{5}}{180}-\\frac{180π}{900}n", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\sin\\left(x+16^{\\circ\\:}\\right)=\\cos\\left(11x+2\\right){\\quad:\\quad}x=\\frac{32400^{\\circ\\:}n+6660^{\\circ\\:}-180}{1080},\\:x=-\\frac{6660^{\\circ\\:}+32400^{\\circ\\:}n+180}{900}$$", "input": "\\sin\\left(x+16^{\\circ\\:}\\right)=\\cos\\left(11x+2\\right)", "steps": [ { "type": "interim", "title": "Rewrite using trig identities", "input": "\\sin\\left(x+16^{\\circ\\:}\\right)=\\cos\\left(11x+2\\right)", "result": "\\sin\\left(x+16^{\\circ\\:}\\right)=\\sin\\left(90^{\\circ\\:}-\\left(11x+2\\right)\\right)", "steps": [ { "type": "step", "primary": "Use the following identity: $$\\cos\\left(x\\right)=\\sin\\left(90^{\\circ\\:}-x\\right)$$", "result": "\\sin\\left(x+16^{\\circ\\:}\\right)=\\sin\\left(90^{\\circ\\:}-\\left(11x+2\\right)\\right)" } ], "meta": { "interimType": "Trig Rewrite Using Trig identities 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7Awm8OHaUUE5X73KqyTFjFKz74GCfU6cV8p9/aSGweuO9T4qUeMwh0IeICQnl77RNsrkXJxYJ4fAIo4WDY4askW8mWMl+hBjAHADIIzgA6jy2oyMT9jq45AQhAkHfGusOk09dCZeqOKuHyyOKZxyWCriyG1iS1738v5NTxHlex7PBB2nx6Xf0OQ7lHdC+dUaR+yNSiytksEPlFcowSvFIuYEFMST8lDZxn1Yq5HMKVTsNwWCajYaZ4bYqH0RpI9A/zFCnzKTYR2+JO9+HPqu4771FBr5UXMxw7y43bLdm5aQ=" } }, { "type": "interim", "title": "Apply trig inverse properties", "input": "\\sin\\left(x+16^{\\circ\\:}\\right)=\\sin\\left(90^{\\circ\\:}-\\left(11x+2\\right)\\right)", "result": "x+16^{\\circ\\:}=90^{\\circ\\:}-\\left(11x+2\\right)+360^{\\circ\\:}n,\\:x+16^{\\circ\\:}=180^{\\circ\\:}-\\left(90^{\\circ\\:}-\\left(11x+2\\right)\\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+16^{\\circ\\:}=90^{\\circ\\:}-\\left(11x+2\\right)+360^{\\circ\\:}n,\\:x+16^{\\circ\\:}=180^{\\circ\\:}-\\left(90^{\\circ\\:}-\\left(11x+2\\right)\\right)+360^{\\circ\\:}n" } ], "meta": { "interimType": "Trig Apply Inverse Props 0Eq" } }, { "type": "interim", "title": "$$x+16^{\\circ\\:}=90^{\\circ\\:}-\\left(11x+2\\right)+360^{\\circ\\:}n{\\quad:\\quad}x=\\frac{32400^{\\circ\\:}n+6660^{\\circ\\:}-180}{1080}$$", "input": "x+16^{\\circ\\:}=90^{\\circ\\:}-\\left(11x+2\\right)+360^{\\circ\\:}n", "steps": [ { "type": "interim", "title": "Expand $$90^{\\circ\\:}-\\left(11x+2\\right)+360^{\\circ\\:}n:{\\quad}90^{\\circ\\:}-11x-2+360^{\\circ\\:}n$$", "input": "90^{\\circ\\:}-\\left(11x+2\\right)+360^{\\circ\\:}n", "steps": [ { "type": "interim", "title": "$$-\\left(11x+2\\right):{\\quad}-11x-2$$", "input": "-\\left(11x+2\\right)", "result": "=90^{\\circ\\:}-11x-2+360^{\\circ\\:}n", "steps": [ { "type": "step", "primary": "Distribute parentheses", "result": "=-\\left(11x\\right)-\\left(2\\right)" }, { "type": "step", "primary": "Apply minus-plus rules", "secondary": [ "$$+\\left(-a\\right)=-a$$" ], "result": "=-11x-2" } ], "meta": { "interimType": "N/A" } } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Expand Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7eSNf6wMIawzTaYPPsynoxDgTq8Xg5nLfvPHIZir45UEzQLPJKK8X7B/Jt7M+7tggvrAvJLHXtxC93VuzexEnHOrqSfLUa3Knc6asOLlsttk7v9Ju9Z9hXz0Xh9kSNnZ4/G2ANG+ldkUkzmezrPTTg95e1OW8hOYegwjPj3xws30bqXZRbCFLmJxWKGR3giZEi3Y0yS++5PyfLVcpDF8wUA==" } }, { "type": "step", "result": "x+16^{\\circ\\:}=90^{\\circ\\:}-11x-2+360^{\\circ\\:}n" }, { "type": "interim", "title": "Move $$16^{\\circ\\:}\\:$$to the right side", "input": "x+16^{\\circ\\:}=90^{\\circ\\:}-11x-2+360^{\\circ\\:}n", "result": "x=-11x+360^{\\circ\\:}n+74^{\\circ\\:}-2", "steps": [ { "type": "step", "primary": "Subtract $$16^{\\circ\\:}$$ from both sides", "result": "x+16^{\\circ\\:}-16^{\\circ\\:}=90^{\\circ\\:}-11x-2+360^{\\circ\\:}n-16^{\\circ\\:}" }, { "type": "interim", "title": "Simplify", "input": "x+16^{\\circ\\:}-16^{\\circ\\:}=90^{\\circ\\:}-11x-2+360^{\\circ\\:}n-16^{\\circ\\:}", "result": "x=-11x+360^{\\circ\\:}n+74^{\\circ\\:}-2", "steps": [ { "type": "interim", "title": "Simplify $$x+16^{\\circ\\:}-16^{\\circ\\:}:{\\quad}x$$", "input": "x+16^{\\circ\\:}-16^{\\circ\\:}", "steps": [ { "type": "step", "primary": "Add similar elements: $$16^{\\circ\\:}-16^{\\circ\\:}=0$$" }, { "type": "step", "result": "=x" } ], "meta": { "interimType": "Generic Simplify Specific 1Eq" } }, { "type": "interim", "title": "Simplify $$90^{\\circ\\:}-11x-2+360^{\\circ\\:}n-16^{\\circ\\:}:{\\quad}-11x+360^{\\circ\\:}n+74^{\\circ\\:}-2$$", "input": "90^{\\circ\\:}-11x-2+360^{\\circ\\:}n-16^{\\circ\\:}", "steps": [ { "type": "step", "primary": "Group like terms", "result": "=-11x+360^{\\circ\\:}n+90^{\\circ\\:}-16^{\\circ\\:}-2" }, { "type": "interim", "title": "Combine the fractions $$90^{\\circ\\:}-16^{\\circ\\:}:{\\quad}74^{\\circ\\:}$$", "input": "90^{\\circ\\:}-16^{\\circ\\:}", "steps": [ { "type": "interim", "title": "Least Common Multiplier of $$2,\\:45:{\\quad}90$$", "input": "2,\\:45", "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 $$45:{\\quad}3\\cdot\\:3\\cdot\\:5$$", "input": "45", "steps": [ { "type": "step", "primary": "$$45\\:$$divides by $$3\\quad\\:45=15\\cdot\\:3$$", "result": "=3\\cdot\\:15" }, { "type": "step", "primary": "$$15\\:$$divides by $$3\\quad\\:15=5\\cdot\\:3$$", "result": "=3\\cdot\\:3\\cdot\\:5" }, { "type": "step", "primary": "$$3,\\:5$$ are all prime numbers, therefore no further factorization is possible", "result": "=3\\cdot\\:3\\cdot\\:5" } ], "meta": { "solvingClass": "Composite Integer", "interimType": "Prime Fac 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xyoU2cWyPLDgE1QLLHeauuc3PHQdChPJ2JhfqHT+ZU0OMrfn8NOj0LUzuzje6xTyxRsG/uC0ndYtZpJL4uAxK7FIDLsMYp+HRYGJJwTT47bdaB4gitN/2ICkrV6ivfiR3BLFRzd4QlsM8ugKm4vxBIEDXL2BGGFCjut3dXcmbc+9v" } }, { "type": "step", "primary": "Multiply each factor the greatest number of times it occurs in either $$2$$ or $$45$$", "result": "=2\\cdot\\:3\\cdot\\:3\\cdot\\:5" }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:3\\cdot\\:3\\cdot\\:5=90$$", "result": "=90" } ], "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 $$90$$" }, { "type": "step", "primary": "For $$90^{\\circ\\:}:\\:$$multiply the denominator and numerator by $$45$$", "result": "90^{\\circ\\:}=\\frac{180^{\\circ\\:}45}{2\\cdot\\:45}=90^{\\circ\\:}" }, { "type": "step", "primary": "For $$16^{\\circ\\:}:\\:$$multiply the denominator and numerator by $$2$$", "result": "16^{\\circ\\:}=\\frac{720^{\\circ\\:}2}{45\\cdot\\:2}=16^{\\circ\\:}" } ], "meta": { "interimType": "LCD Adjust Fractions 1Eq" } }, { "type": "step", "result": "=90^{\\circ\\:}-16^{\\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\\:}45-1440^{\\circ\\:}}{90}" }, { "type": "step", "primary": "Add similar elements: $$8100^{\\circ\\:}-1440^{\\circ\\:}=6660^{\\circ\\:}$$", "result": "=74^{\\circ\\:}" } ], "meta": { "interimType": "LCD Top 1Eq" } }, { "type": "step", "result": "=-11x+360^{\\circ\\:}n+74^{\\circ\\:}-2" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79WdsPCQCuhcNttbO+j+ynUHJ6scC3ClTy9xtWNP8gRj6upGU7Z4m+FtP876nmntvAJYpRu9XpYrd8NSAW2DdD0sFsmp3up4XGIgAJuEzDZYtDPynh9AyuzJs21eUPR0EIrsy4Z0aeNd3eKHDxBNaUGRLd2VwIqlBNByF6663syR2SpdpleAJc7YgKUwBYoM979Yt4fSBOwKnf/QGvOgdUEHJ6scC3ClTy9xtWNP8gRj6upGU7Z4m+FtP876nmntvsIjaxJ4DvjTb2fbKjbvtlQ==" } }, { "type": "step", "result": "x=-11x+360^{\\circ\\:}n+74^{\\circ\\:}-2" } ], "meta": { "interimType": "Generic Simplify 0Eq" } } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7gyluZm1QVe+2PlxOzU13UpZ6VZENqy6Q18eaCsPX7EQgPUKsp8sgB2YjtC0yhSw6dbWjuGoliOL0Xcdx8f83/PLG/zedZxJCWxEdiNiKyCZamLayY/knzwBrvnCVPbClRCbesps1tFJ5fLjekMVOrPFmiWKOq7QFU+wqaX0tybYeM6OK8Zs1/sgc9hyPhMcO5oXwki6OmSGuZ7Ilst/Z9h8Cnw2971iGgyksG5vVKZvY/0bU2CZMB0TKZ7zPAOv1ClxEt6mM4wzMJqKI4wTswaPnShcBEo+1Tm61nmq+mhoxGToczixRQljlXtEwLPT7lLLW349w4sAIVmDehqoASy+t9AtHPI5vSfuE6WhLWQGj2uGvTHxQ0C1/mvtvhTAKQHM3/Sh2fX2EhOJRB7h6IgGp0sDY54aoJihini85+3qvW7zyUp0x/7ynFKlE/7SK/6F/U1Mq20zR8FmY77UVGfBgIPZaNfDR36b4MwFMpdSijls42SZ7m+UdmKSFbltgIMR4ndGwMNeCQxf6z7xDs9E/q6QKNiOyJ4YvYkVFrz8MCgLu2d7Tf0YCz45c8kcPIuTqDTaa6kFdlIEFBhEqhZb44akLOVySZB79hkFBOoJXKwDrETn4lYD/sLVsodGn7Yb8ChKEd5LBzR2wpK2+pHMAyH8W6/z6tr1o7FwXaZj/hnO7jQ5+axTIg8r+vZ4ikvZyWCUp99t5AToCyySUh7KT4FflvT6UoCpKHk3Zh6RR8gORhBVlty5T+JwSvNOlXrWFWHS3uOVcS6EA2kKBIQ==" } }, { "type": "interim", "title": "Move $$11x\\:$$to the left side", "input": "x=-11x+360^{\\circ\\:}n+74^{\\circ\\:}-2", "result": "12x=360^{\\circ\\:}n+74^{\\circ\\:}-2", "steps": [ { "type": "step", "primary": "Add $$11x$$ to both sides", "result": "x+11x=-11x+360^{\\circ\\:}n+74^{\\circ\\:}-2+11x" }, { "type": "step", "primary": "Simplify", "result": "12x=360^{\\circ\\:}n+74^{\\circ\\:}-2" } ], "meta": { "interimType": "Move to the Left Title 1Eq", "gptData": 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"steps": [ { "type": "step", "primary": "Divide both sides by $$12$$", "result": "\\frac{12x}{12}=\\frac{360^{\\circ\\:}n}{12}+\\frac{74^{\\circ\\:}}{12}-\\frac{2}{12}" }, { "type": "interim", "title": "Simplify", "input": "\\frac{12x}{12}=\\frac{360^{\\circ\\:}n}{12}+\\frac{74^{\\circ\\:}}{12}-\\frac{2}{12}", "result": "x=\\frac{32400^{\\circ\\:}n+6660^{\\circ\\:}-180}{1080}", "steps": [ { "type": "interim", "title": "Simplify $$\\frac{12x}{12}:{\\quad}x$$", "input": "\\frac{12x}{12}", "steps": [ { "type": "step", "primary": "Divide the numbers: $$\\frac{12}{12}=1$$", "result": "=x" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7jx0bPoTfDSyxds17kYoNyXyRHuGw7+tM5METTDj6vVEed1oZgJr3Rrt+25B4RF18ZEt3ZXAiqUE0HIXrrrezJHZKl2mV4AlztiApTAFigz28oEBSPDPQa5rxqD6vLXZlialcV/dI5TH4fXyp+ncwuA==" } }, { "type": "interim", "title": "Simplify $$\\frac{360^{\\circ\\:}n}{12}+\\frac{74^{\\circ\\:}}{12}-\\frac{2}{12}:{\\quad}\\frac{32400^{\\circ\\:}n+6660^{\\circ\\:}-180}{1080}$$", "input": "\\frac{360^{\\circ\\:}n}{12}+\\frac{74^{\\circ\\:}}{12}-\\frac{2}{12}", "steps": [ { "type": "step", "primary": "Apply rule $$\\frac{a}{c}\\pm\\frac{b}{c}=\\frac{a\\pm\\:b}{c}$$", "result": "=\\frac{360^{\\circ\\:}n+74^{\\circ\\:}-2}{12}" }, { "type": "interim", "title": "Join $$360^{\\circ\\:}n+74^{\\circ\\:}-2:{\\quad}\\frac{32400^{\\circ\\:}n+6660^{\\circ\\:}-180}{90}$$", "input": "360^{\\circ\\:}n+74^{\\circ\\:}-2", "result": "=\\frac{\\frac{32400^{\\circ\\:}n+6660^{\\circ\\:}-180}{90}}{12}", "steps": [ { "type": "step", "primary": "Convert element to fraction: $$360^{\\circ\\:}n=\\frac{360^{\\circ\\:}n90}{90},\\:2=\\frac{2\\cdot\\:90}{90}$$", "result": "=\\frac{360^{\\circ\\:}n\\cdot\\:90}{90}+74^{\\circ\\:}-\\frac{2\\cdot\\:90}{90}" }, { "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\\:90+6660^{\\circ\\:}-2\\cdot\\:90}{90}" }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:90=180$$", "result": "=\\frac{32400^{\\circ\\:}n+6660^{\\circ\\:}-180}{90}" } ], "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{32400^{\\circ\\:}n+6660^{\\circ\\:}-180}{90\\cdot\\:12}" }, { "type": "step", "primary": "Multiply the numbers: $$90\\cdot\\:12=1080$$", "result": "=\\frac{32400^{\\circ\\:}n+6660^{\\circ\\:}-180}{1080}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s71VWBOBW9C7vXuDdGOZiiaMpTU9cKxDyidq1FzmFikS/4NlIi7CjctVSccwwbhU/c1iq2Fp94WutV6qJMrDhJi8AYOMuP4JW3uWQsjElsZW2jkVi15I8rBefLi4Iyt2wrjMj7b6OLkBlQ++Tq9B/+rxyIS5ryS+pEFikMK9eSj8nWJ8yDa0nNjdL8C9hueI5k/z//r+dXk7h9vxeDCLuZqnKF3u2OIb4bFA3EO8aRlSXB1zhrdjmZRkPts7p6sjSHqjLY20pLGWpRJUjHB+gfZszEN7Lr5BoDS1xL5QTr6gerGspXHnmDGd3IwIdnzW1y9Wy1uVC9VfQHI9ugf/+4/A==" } }, { "type": "step", "result": "x=\\frac{32400^{\\circ\\:}n+6660^{\\circ\\:}-180}{1080}" } ], "meta": { "interimType": "Generic Simplify 0Eq" } } ], "meta": { "interimType": "Divide Both Sides Specific 1Eq", "gptData": 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"$$x+16^{\\circ\\:}=180^{\\circ\\:}-\\left(90^{\\circ\\:}-\\left(11x+2\\right)\\right)+360^{\\circ\\:}n{\\quad:\\quad}x=-\\frac{6660^{\\circ\\:}+32400^{\\circ\\:}n+180}{900}$$", "input": "x+16^{\\circ\\:}=180^{\\circ\\:}-\\left(90^{\\circ\\:}-\\left(11x+2\\right)\\right)+360^{\\circ\\:}n", "steps": [ { "type": "interim", "title": "Expand $$180^{\\circ\\:}-\\left(90^{\\circ\\:}-\\left(11x+2\\right)\\right)+360^{\\circ\\:}n:{\\quad}180^{\\circ\\:}-90^{\\circ\\:}+11x+2+360^{\\circ\\:}n$$", "input": "180^{\\circ\\:}-\\left(90^{\\circ\\:}-\\left(11x+2\\right)\\right)+360^{\\circ\\:}n", "steps": [ { "type": "interim", "title": "$$-\\left(11x+2\\right):{\\quad}-11x-2$$", "input": "-\\left(11x+2\\right)", "steps": [ { "type": "step", "primary": "Distribute parentheses", "result": "=-\\left(11x\\right)-\\left(2\\right)" }, { "type": "step", "primary": "Apply minus-plus rules", "secondary": [ "$$+\\left(-a\\right)=-a$$" ], "result": "=-11x-2" } ], "meta": { "interimType": "N/A" } }, { "type": "step", "result": "=180^{\\circ\\:}-\\left(-11x+90^{\\circ\\:}-2\\right)+360^{\\circ\\:}n" }, { "type": "interim", "title": "$$-\\left(90^{\\circ\\:}-11x-2\\right):{\\quad}-90^{\\circ\\:}+11x+2$$", "input": "-\\left(90^{\\circ\\:}-11x-2\\right)", "result": "=180^{\\circ\\:}-90^{\\circ\\:}+11x+2+360^{\\circ\\:}n", "steps": [ { "type": "step", "primary": "Distribute parentheses", "result": "=-\\left(90^{\\circ\\:}\\right)-\\left(-11x\\right)-\\left(-2\\right)" }, { "type": "step", "primary": "Apply minus-plus rules", "secondary": [ "$$-\\left(-a\\right)=a,\\:\\:\\:-\\left(a\\right)=-a$$" ], "result": "=-90^{\\circ\\:}+11x+2" } ], "meta": { "interimType": "N/A" } } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Expand Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7WafLUeHxfUY6g4hxuTM9uQd9hOHA/4psQrCFGg82owfKdFw+ZLHPxGwZJMv8o1vlM0CzySivF+wfybezPu7YIL6wLySx17cQvd1bs3sRJxzq6kny1Gtyp3OmrDi5bLbZO7/SbvWfYV89F4fZEjZ2ePxtgDRvpXZFJM5ns6z004N+q0KITVBFu9N57jkDQ9thzDMS7VCjxuCPI7NLwHCGxpizVG2GlDm07bpDTQwMoJuLdjTJL77k/J8tVykMXzBQ" } }, { "type": "step", "result": "x+16^{\\circ\\:}=180^{\\circ\\:}-90^{\\circ\\:}+11x+2+360^{\\circ\\:}n" }, { "type": "interim", "title": "Move $$16^{\\circ\\:}\\:$$to the right side", "input": "x+16^{\\circ\\:}=180^{\\circ\\:}-90^{\\circ\\:}+11x+2+360^{\\circ\\:}n", "result": "x=11x+180^{\\circ\\:}+360^{\\circ\\:}n-106^{\\circ\\:}+2", "steps": [ { "type": "step", "primary": "Subtract $$16^{\\circ\\:}$$ from both sides", "result": "x+16^{\\circ\\:}-16^{\\circ\\:}=180^{\\circ\\:}-90^{\\circ\\:}+11x+2+360^{\\circ\\:}n-16^{\\circ\\:}" }, { "type": "interim", "title": "Simplify", "input": "x+16^{\\circ\\:}-16^{\\circ\\:}=180^{\\circ\\:}-90^{\\circ\\:}+11x+2+360^{\\circ\\:}n-16^{\\circ\\:}", "result": "x=11x+180^{\\circ\\:}+360^{\\circ\\:}n-106^{\\circ\\:}+2", "steps": [ { "type": "interim", "title": "Simplify $$x+16^{\\circ\\:}-16^{\\circ\\:}:{\\quad}x$$", "input": "x+16^{\\circ\\:}-16^{\\circ\\:}", "steps": [ { "type": "step", "primary": "Add similar elements: $$16^{\\circ\\:}-16^{\\circ\\:}=0$$" }, { "type": "step", "result": "=x" } ], "meta": { "interimType": "Generic Simplify Specific 1Eq" } }, { "type": "interim", "title": "Simplify $$180^{\\circ\\:}-90^{\\circ\\:}+11x+2+360^{\\circ\\:}n-16^{\\circ\\:}:{\\quad}11x+180^{\\circ\\:}+360^{\\circ\\:}n-106^{\\circ\\:}+2$$", "input": "180^{\\circ\\:}-90^{\\circ\\:}+11x+2+360^{\\circ\\:}n-16^{\\circ\\:}", "steps": [ { "type": "step", "primary": "Group like terms", "result": "=11x+180^{\\circ\\:}+360^{\\circ\\:}n-90^{\\circ\\:}-16^{\\circ\\:}+2" }, { "type": "interim", "title": "Combine the fractions $$-90^{\\circ\\:}-16^{\\circ\\:}:{\\quad}-106^{\\circ\\:}$$", "input": "-90^{\\circ\\:}-16^{\\circ\\:}", "steps": [ { "type": "interim", "title": "Least Common Multiplier of $$2,\\:45:{\\quad}90$$", "input": "2,\\:45", "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 $$45:{\\quad}3\\cdot\\:3\\cdot\\:5$$", "input": "45", "steps": [ { "type": "step", "primary": "$$45\\:$$divides by $$3\\quad\\:45=15\\cdot\\:3$$", "result": "=3\\cdot\\:15" }, { "type": "step", "primary": "$$15\\:$$divides by $$3\\quad\\:15=5\\cdot\\:3$$", "result": "=3\\cdot\\:3\\cdot\\:5" }, { "type": "step", "primary": "$$3,\\:5$$ are all prime numbers, therefore no further factorization is possible", "result": "=3\\cdot\\:3\\cdot\\:5" } ], "meta": { "solvingClass": "Composite Integer", "interimType": "Prime Fac 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xyoU2cWyPLDgE1QLLHeauuc3PHQdChPJ2JhfqHT+ZU0OMrfn8NOj0LUzuzje6xTyxRsG/uC0ndYtZpJL4uAxK7FIDLsMYp+HRYGJJwTT47bdaB4gitN/2ICkrV6ivfiR3BLFRzd4QlsM8ugKm4vxBIEDXL2BGGFCjut3dXcmbc+9v" } }, { "type": "step", "primary": "Multiply each factor the greatest number of times it occurs in either $$2$$ or $$45$$", "result": "=2\\cdot\\:3\\cdot\\:3\\cdot\\:5" }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:3\\cdot\\:3\\cdot\\:5=90$$", "result": "=90" } ], "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 $$90$$" }, { "type": "step", "primary": "For $$90^{\\circ\\:}:\\:$$multiply the denominator and numerator by $$45$$", "result": "90^{\\circ\\:}=\\frac{180^{\\circ\\:}45}{2\\cdot\\:45}=90^{\\circ\\:}" }, { "type": "step", "primary": "For $$16^{\\circ\\:}:\\:$$multiply the denominator and numerator by $$2$$", "result": "16^{\\circ\\:}=\\frac{720^{\\circ\\:}2}{45\\cdot\\:2}=16^{\\circ\\:}" } ], "meta": { "interimType": "LCD Adjust Fractions 1Eq" } }, { "type": "step", "result": "=-90^{\\circ\\:}-16^{\\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\\:}45-1440^{\\circ\\:}}{90}" }, { "type": "step", "primary": "Add similar elements: $$-8100^{\\circ\\:}-1440^{\\circ\\:}=-9540^{\\circ\\:}$$", "result": "=\\frac{-9540^{\\circ\\:}}{90}" }, { "type": "step", "primary": "Apply the fraction rule: $$\\frac{-a}{b}=-\\frac{a}{b}$$", "result": "=-106^{\\circ\\:}" } ], "meta": { "interimType": "LCD Top 1Eq" } }, { "type": "step", "result": "=11x+180^{\\circ\\:}+360^{\\circ\\:}n-106^{\\circ\\:}+2" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7e0XeQ5VqTwjR7KafmeQRCyzmd7/Fg0vR7wEgoxGYMYQDnjUqdGV7NkauDpMGsi4El3tYwDNKMPzNlDxasx9hKVXTSum/z5kLpMzXS1UJIexSFINPDe2QsacA5EdOdsUQgF4RAb6pZ4+Jo6MpXN8TFovGQ5emD4iDQLJjOb/pmw6p0f1ocMFiORufRrusW3gwgQUxJPyUNnGfVirkcwpVO6ETsmdBkf1q8Ln0WAB/L1DJmDaHjUkkwMw1qym9JfHRT9iDHnAFALVDvofr9ph+Ozju2D9a7f+hyfKyUh0827/RhyiZYjlvoUiqTdcYlWX3" } }, { "type": "step", "result": "x=11x+180^{\\circ\\:}+360^{\\circ\\:}n-106^{\\circ\\:}+2" } ], "meta": { "interimType": "Generic Simplify 0Eq" } } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": "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" } }, { "type": "interim", "title": "Move $$11x\\:$$to the left side", "input": "x=11x+180^{\\circ\\:}+360^{\\circ\\:}n-106^{\\circ\\:}+2", "result": "-10x=180^{\\circ\\:}+360^{\\circ\\:}n-106^{\\circ\\:}+2", "steps": [ { "type": "step", "primary": "Subtract $$11x$$ from both sides", "result": "x-11x=11x+180^{\\circ\\:}+360^{\\circ\\:}n-106^{\\circ\\:}+2-11x" }, { "type": "step", "primary": "Simplify", "result": "-10x=180^{\\circ\\:}+360^{\\circ\\:}n-106^{\\circ\\:}+2" } ], "meta": { "interimType": "Move to the Left Title 1Eq", "gptData": "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" } }, { "type": "interim", "title": "Divide both sides by $$-10$$", "input": "-10x=180^{\\circ\\:}+360^{\\circ\\:}n-106^{\\circ\\:}+2", "result": "x=-\\frac{6660^{\\circ\\:}+32400^{\\circ\\:}n+180}{900}", "steps": [ { "type": "step", "primary": "Divide both sides by $$-10$$", "result": "\\frac{-10x}{-10}=\\frac{180^{\\circ\\:}}{-10}+\\frac{360^{\\circ\\:}n}{-10}-\\frac{106^{\\circ\\:}}{-10}+\\frac{2}{-10}" }, { "type": "interim", "title": "Simplify", "input": "\\frac{-10x}{-10}=\\frac{180^{\\circ\\:}}{-10}+\\frac{360^{\\circ\\:}n}{-10}-\\frac{106^{\\circ\\:}}{-10}+\\frac{2}{-10}", "result": "x=-\\frac{6660^{\\circ\\:}+32400^{\\circ\\:}n+180}{900}", "steps": [ { "type": "interim", "title": "Simplify $$\\frac{-10x}{-10}:{\\quad}x$$", "input": "\\frac{-10x}{-10}", "steps": [ { "type": "step", "primary": "Apply the fraction rule: $$\\frac{-a}{-b}=\\frac{a}{b}$$", "result": "=\\frac{10x}{10}" }, { "type": "step", "primary": "Divide the numbers: $$\\frac{10}{10}=1$$", "result": "=x" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7poU1hNU7RESIJxKCWgdS1eP78sN6s+TxU1X701cSbd+jkVi15I8rBefLi4Iyt2wrNnlcfYSxPIgsp1FH5TTQYHKF3u2OIb4bFA3EO8aRlSWU8+6NvmvO4Ea/LVC6eTtzNfSityYr0BoM9brdfoLlvQ==" } }, { "type": "interim", "title": "Simplify $$\\frac{180^{\\circ\\:}}{-10}+\\frac{360^{\\circ\\:}n}{-10}-\\frac{106^{\\circ\\:}}{-10}+\\frac{2}{-10}:{\\quad}-\\frac{6660^{\\circ\\:}+32400^{\\circ\\:}n+180}{900}$$", "input": "\\frac{180^{\\circ\\:}}{-10}+\\frac{360^{\\circ\\:}n}{-10}-\\frac{106^{\\circ\\:}}{-10}+\\frac{2}{-10}", "steps": [ { "type": "step", "primary": "Group like terms", "result": "=\\frac{180^{\\circ\\:}}{-10}+\\frac{2}{-10}+\\frac{360^{\\circ\\:}n}{-10}-\\frac{106^{\\circ\\:}}{-10}" }, { "type": "step", "primary": "Apply rule $$\\frac{a}{c}\\pm\\frac{b}{c}=\\frac{a\\pm\\:b}{c}$$", "result": "=\\frac{180^{\\circ\\:}+2+360^{\\circ\\:}n-106^{\\circ\\:}}{-10}" }, { "type": "step", "primary": "Apply the fraction rule: $$\\frac{a}{-b}=-\\frac{a}{b}$$", "result": "=-\\frac{180^{\\circ\\:}+2+360^{\\circ\\:}n-106^{\\circ\\:}}{10}" }, { "type": "interim", "title": "Join $$180^{\\circ\\:}+2+360^{\\circ\\:}n-106^{\\circ\\:}:{\\quad}\\frac{6660^{\\circ\\:}+32400^{\\circ\\:}n+180}{90}$$", "input": "180^{\\circ\\:}+2+360^{\\circ\\:}n-106^{\\circ\\:}", "result": "=-\\frac{\\frac{6660^{\\circ\\:}+32400^{\\circ\\:}n+180}{90}}{10}", "steps": [ { "type": "step", "primary": "Convert element to fraction: $$180^{\\circ\\:}=180^{\\circ\\:},\\:2=\\frac{2\\cdot\\:90}{90},\\:360^{\\circ\\:}n=\\frac{360^{\\circ\\:}n90}{90}$$", "result": "=180^{\\circ\\:}+\\frac{2\\cdot\\:90}{90}+\\frac{360^{\\circ\\:}n\\cdot\\:90}{90}-106^{\\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\\:}90+2\\cdot\\:90+360^{\\circ\\:}n\\cdot\\:90-9540^{\\circ\\:}}{90}" }, { "type": "interim", "title": "$$180^{\\circ\\:}90+2\\cdot\\:90+360^{\\circ\\:}n\\cdot\\:90-9540^{\\circ\\:}=6660^{\\circ\\:}+32400^{\\circ\\:}n+180$$", "input": "180^{\\circ\\:}90+2\\cdot\\:90+360^{\\circ\\:}n\\cdot\\:90-9540^{\\circ\\:}", "steps": [ { "type": "step", "primary": "Group like terms", "result": "=16200^{\\circ\\:}-9540^{\\circ\\:}+2\\cdot\\:16200^{\\circ\\:}n+2\\cdot\\:90" }, { "type": "step", "primary": "Add similar elements: $$16200^{\\circ\\:}-9540^{\\circ\\:}=6660^{\\circ\\:}$$", "result": "=6660^{\\circ\\:}+2\\cdot\\:16200^{\\circ\\:}n+2\\cdot\\:90" }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:90=180$$", "result": "=6660^{\\circ\\:}+32400^{\\circ\\:}n+180" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7CUkrwY51qAb21omF5laqWbUutNCEcVBxfA1j2Piqaxh5tnqiDlgr3ez1unXp/QGv1pwvSSo5Nttq7wE/okCt6XMkXPY0hwc9e7xFBv7eZWOjkVi15I8rBefLi4Iyt2wrHuMru/zdCt+GTFhrYcIQRvyeGKW8A+3K681c2/IP+/gWjWQ/4S2pylUmHYw0qzOpgqXZcjeiDFdnaBFvMdpsgPZXL21HOKZAziRRRFHvPVjSr4NuhiODTP8TMj7Bk+0Aoqo40SM9+Jw3KJOZqq1B7AjuOC5RmPf3SPvkFZv5X/04SOGuRDgxd8VRAu9osJgFIVTCSEp6aZS84750wh/+ew==" } }, { "type": "step", "result": "=\\frac{6660^{\\circ\\:}+32400^{\\circ\\:}n+180}{90}" } ], "meta": { "interimType": "Algebraic Manipulation Join Concise Title 1Eq" } }, { "type": "interim", "title": "Simplify $$\\frac{\\frac{6660^{\\circ\\:}+32400^{\\circ\\:}n+180}{90}}{10}:{\\quad}\\frac{6660^{\\circ\\:}+32400^{\\circ\\:}n+180}{900}$$", "input": "\\frac{\\frac{6660^{\\circ\\:}+32400^{\\circ\\:}n+180}{90}}{10}", "result": "=-\\frac{6660^{\\circ\\:}+32400^{\\circ\\:}n+180}{900}", "steps": [ { "type": "step", "primary": "Apply the fraction rule: $$\\frac{\\frac{b}{c}}{a}=\\frac{b}{c\\:\\cdot\\:a}$$", "result": "=\\frac{6660^{\\circ\\:}+32400^{\\circ\\:}n+180}{90\\cdot\\:10}" }, { "type": "step", "primary": "Multiply the numbers: $$90\\cdot\\:10=900$$", "result": "=\\frac{6660^{\\circ\\:}+32400^{\\circ\\:}n+180}{900}" } ], "meta": { "interimType": "Algebraic Manipulation Simplify Title 1Eq" } } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7sctMNWyhnAEl4COXxkn3W8PC0nYFZWrOTHC5S1qvVWxBNysRKahy0o8CzbYRpi987xv9grelwOcLKpuhaGsiqAG4LV51gLoWkJfp0lP2BGXGdjrl2wE+b9pvnLWF1RwsoOR8onEllUKZ8FQC+PKsSXm0yklMEFxxRaRLK8x7VKlGowkOT4CzmbWKAoqaKmpIodvFysI7ZA59frYmfijDsC+n1C/YhC6mOhid1tXWnZA/y9DKGIPglJ+qMi9xDu2KaRI7GCp0HQz+zDw23axddB7kkOqkHAamvSjC/cLewSuNdEeSrYQPQgq3uljbyFw8fpJGz/leaNa2R3urOrGMvkwUG9afPiR4UKjWa51rsD4KYeoOu7Uoth9BcgRE49oKFhOXdZ28SMd+gzlNhdmg41hFbxvmJLt4B+PxP0aK2cg=" } }, { "type": "step", "result": "x=-\\frac{6660^{\\circ\\:}+32400^{\\circ\\:}n+180}{900}" } ], "meta": { "interimType": "Generic Simplify 0Eq" } } ], "meta": { "interimType": "Divide Both Sides Specific 1Eq", "gptData": 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"solvingClass": "Equations", "interimType": "Equations" } }, { "type": "step", "result": "x=\\frac{32400^{\\circ\\:}n+6660^{\\circ\\:}-180}{1080},\\:x=-\\frac{6660^{\\circ\\:}+32400^{\\circ\\:}n+180}{900}" } ], "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+16^{\\circ })-\\cos(11x+2)" }, "showViewLarger": true } }, "meta": { "showVerify": true } }