18cos(x)+9sin(2x)=0

{ "query": { "display": "$$18\\cos\\left(x\\right)+9\\sin\\left(2x\\right)=0$$", "symbolab_question": "EQUATION#18\\cos(x)+9\\sin(2x)=0" }, "solution": { "level": "PERFORMED", "subject": "Trigonometry", "topic": "Trig Equations", "subTopic": "Trig Equations", "default": "x=\\frac{π}{2}+2πn,x=\\frac{3π}{2}+2πn", "degrees": "x=90^{\\circ }+360^{\\circ }n,x=270^{\\circ }+360^{\\circ }n", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$18\\cos\\left(x\\right)+9\\sin\\left(2x\\right)=0{\\quad:\\quad}x=\\frac{π}{2}+2πn,\\:x=\\frac{3π}{2}+2πn$$", "input": "18\\cos\\left(x\\right)+9\\sin\\left(2x\\right)=0", "steps": [ { "type": "interim", "title": "Rewrite using trig identities", "input": "18\\cos\\left(x\\right)+9\\sin\\left(2x\\right)", "result": "18\\cos\\left(x\\right)+18\\cos\\left(x\\right)\\sin\\left(x\\right)=0", "steps": [ { "type": "step", "primary": "Use the Double Angle identity: $$\\sin\\left(2x\\right)=2\\sin\\left(x\\right)\\cos\\left(x\\right)$$", "result": "=18\\cos\\left(x\\right)+9\\cdot\\:2\\sin\\left(x\\right)\\cos\\left(x\\right)" }, { "type": "step", "primary": "Simplify", "result": "=18\\cos\\left(x\\right)+18\\sin\\left(x\\right)\\cos\\left(x\\right)", "meta": { "solvingClass": "Solver" } } ], "meta": { "interimType": "Trig Rewrite Using Trig identities 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7pONWwiSACf/+/GT+4n7sP1CWpgPh/2SuaN7ooXJmtAI52FciCV6Q/ZuTzBHIPdDy6bFjjx5IuxvIy7/nJMTFRsLgcgrrcmkH6sZs7VXsfRCXglwnmROl9cGKdo0Ogfgy8sE+RHrZoyUlw3VI3I73nd3gzTogg2oR6Q1DQXdsHBGBBTEk/JQ2cZ9WKuRzClU7DcFgmo2GmeG2Kh9EaSPQP8xQp8yk2EdviTvfhz6ruO+9RQa+VFzMcO8uN2y3ZuWk" } }, { "type": "interim", "title": "Factor $$18\\cos\\left(x\\right)+18\\cos\\left(x\\right)\\sin\\left(x\\right):{\\quad}18\\cos\\left(x\\right)\\left(\\sin\\left(x\\right)+1\\right)$$", "input": "18\\cos\\left(x\\right)+18\\cos\\left(x\\right)\\sin\\left(x\\right)", "steps": [ { "type": "step", "primary": "Factor out common term $$18\\cos\\left(x\\right)$$", "result": "=18\\cos\\left(x\\right)\\left(1+\\sin\\left(x\\right)\\right)", "meta": { "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Factor Title 1Eq" } }, { "type": "step", "result": "18\\cos\\left(x\\right)\\left(\\sin\\left(x\\right)+1\\right)=0" }, { "type": "step", "primary": "Solving each part separately", "result": "\\cos\\left(x\\right)=0\\lor\\:\\sin\\left(x\\right)+1=0" }, { "type": "interim", "title": "$$\\cos\\left(x\\right)=0{\\quad:\\quad}x=\\frac{π}{2}+2πn,\\:x=\\frac{3π}{2}+2πn$$", "input": "\\cos\\left(x\\right)=0", "steps": [ { "type": "interim", "title": "General solutions for $$\\cos\\left(x\\right)=0$$", "result": "x=\\frac{π}{2}+2πn,\\:x=\\frac{3π}{2}+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}{2}+2πn,\\:x=\\frac{3π}{2}+2πn" } ], "meta": { "interimType": "Trig General Solutions cos 1Eq" } } ], "meta": { "solvingClass": "Trig Equations", "interimType": "Trig Equations" } }, { "type": "interim", "title": "$$\\sin\\left(x\\right)+1=0{\\quad:\\quad}x=\\frac{3π}{2}+2πn$$", "input": "\\sin\\left(x\\right)+1=0", "steps": [ { "type": "interim", "title": "Move $$1\\:$$to the right side", "input": "\\sin\\left(x\\right)+1=0", "result": "\\sin\\left(x\\right)=-1", "steps": [ { "type": "step", "primary": "Subtract $$1$$ from both sides", "result": "\\sin\\left(x\\right)+1-1=0-1" }, { "type": "step", "primary": "Simplify", "result": "\\sin\\left(x\\right)=-1" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": "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" } }, { "type": "interim", "title": "General solutions for $$\\sin\\left(x\\right)=-1$$", "result": "x=\\frac{3π}{2}+2πn", "steps": [ { "type": "step", "primary": "$$\\sin\\left(x\\right)$$ periodicity table with $$2πn$$ cycle:<br/>$$\\begin{array}{|c|c|c|c|}\\hline x&\\sin(x)&x&\\sin(x)\\\\\\hline 0&0&π&0\\\\\\hline \\frac{π}{6}&\\frac{1}{2}&\\frac{7π}{6}&-\\frac{1}{2}\\\\\\hline \\frac{π}{4}&\\frac{\\sqrt{2}}{2}&\\frac{5π}{4}&-\\frac{\\sqrt{2}}{2}\\\\\\hline \\frac{π}{3}&\\frac{\\sqrt{3}}{2}&\\frac{4π}{3}&-\\frac{\\sqrt{3}}{2}\\\\\\hline \\frac{π}{2}&1&\\frac{3π}{2}&-1\\\\\\hline \\frac{2π}{3}&\\frac{\\sqrt{3}}{2}&\\frac{5π}{3}&-\\frac{\\sqrt{3}}{2}\\\\\\hline \\frac{3π}{4}&\\frac{\\sqrt{2}}{2}&\\frac{7π}{4}&-\\frac{\\sqrt{2}}{2}\\\\\\hline \\frac{5π}{6}&\\frac{1}{2}&\\frac{11π}{6}&-\\frac{1}{2}\\\\\\hline \\end{array}$$" }, { "type": "step", "result": "x=\\frac{3π}{2}+2πn" } ], "meta": { "interimType": "Trig General Solutions sin 1Eq" } } ], "meta": { "solvingClass": "Trig Equations", "interimType": "Trig Equations" } }, { "type": "step", "primary": "Combine all the solutions", "result": "x=\\frac{π}{2}+2πn,\\:x=\\frac{3π}{2}+2πn" } ], "meta": { "solvingClass": "Trig Equations", "practiceLink": "/practice/trigonometry-practice#area=main&subtopic=Trig%20Equations", "practiceTopic": "Trig Equations" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "18\\cos(x)+9\\sin(2x)" }, "showViewLarger": true } }, "meta": { "showVerify": true } }