(3sin(t))*(3cos(t))=0

{ "query": { "display": "$$\\left(3\\sin\\left(t\\right)\\right)\\cdot\\:\\left(3\\cos\\left(t\\right)\\right)=0$$", "symbolab_question": "EQUATION#(3\\sin(t))\\cdot (3\\cos(t))=0" }, "solution": { "level": "PERFORMED", "subject": "Trigonometry", "topic": "Trig Equations", "subTopic": "Trig Equations", "default": "t=2πn,t=π+2πn,t=\\frac{π}{2}+2πn,t=\\frac{3π}{2}+2πn", "degrees": "t=0^{\\circ }+360^{\\circ }n,t=180^{\\circ }+360^{\\circ }n,t=90^{\\circ }+360^{\\circ }n,t=270^{\\circ }+360^{\\circ }n", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\left(3\\sin\\left(t\\right)\\right)\\left(3\\cos\\left(t\\right)\\right)=0{\\quad:\\quad}t=2πn,\\:t=π+2πn,\\:t=\\frac{π}{2}+2πn,\\:t=\\frac{3π}{2}+2πn$$", "input": "\\left(3\\sin\\left(t\\right)\\right)\\left(3\\cos\\left(t\\right)\\right)=0", "steps": [ { "type": "step", "primary": "Solving each part separately", "result": "\\sin\\left(t\\right)=0\\lor\\:\\cos\\left(t\\right)=0" }, { "type": "interim", "title": "$$\\sin\\left(t\\right)=0{\\quad:\\quad}t=2πn,\\:t=π+2πn$$", "input": "\\sin\\left(t\\right)=0", "steps": [ { "type": "interim", "title": "General solutions for $$\\sin\\left(t\\right)=0$$", "result": "t=0+2πn,\\:t=π+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": "t=0+2πn,\\:t=π+2πn" } ], "meta": { "interimType": "Trig General Solutions sin 1Eq" } }, { "type": "interim", "title": "Solve $$t=0+2πn:{\\quad}t=2πn$$", "input": "t=0+2πn", "steps": [ { "type": "step", "primary": "$$0+2πn=2πn$$", "result": "t=2πn" } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve Title 1Eq" } }, { "type": "step", "result": "t=2πn,\\:t=π+2πn" } ], "meta": { "solvingClass": "Trig Equations", "interimType": "Trig Equations" } }, { "type": "interim", "title": "$$\\cos\\left(t\\right)=0{\\quad:\\quad}t=\\frac{π}{2}+2πn,\\:t=\\frac{3π}{2}+2πn$$", "input": "\\cos\\left(t\\right)=0", "steps": [ { "type": "interim", "title": "General solutions for $$\\cos\\left(t\\right)=0$$", "result": "t=\\frac{π}{2}+2πn,\\:t=\\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": "t=\\frac{π}{2}+2πn,\\:t=\\frac{3π}{2}+2πn" } ], "meta": { "interimType": "Trig General Solutions cos 1Eq" } } ], "meta": { "solvingClass": "Trig Equations", "interimType": "Trig Equations" } }, { "type": "step", "primary": "Combine all the solutions", "result": "t=2πn,\\:t=π+2πn,\\:t=\\frac{π}{2}+2πn,\\:t=\\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": "t", "plotRequest": "(3\\sin(t))\\cdot (3\\cos(t))" }, "showViewLarger": true } }, "meta": { "showVerify": true } }