(sin(60))/(174.36)=(sin(x))/(200)

{ "query": { "display": "$$\\frac{\\sin\\left(60^{\\circ\\:}\\right)}{174.36}=\\frac{\\sin\\left(x\\right)}{200}$$", "symbolab_question": "EQUATION#\\frac{\\sin(60^{\\circ })}{174.36}=\\frac{\\sin(x)}{200}" }, "solution": { "level": "PERFORMED", "subject": "Trigonometry", "topic": "Trig Equations", "subTopic": "Trig Equations", "default": "x=1.45563…+360^{\\circ }n,x=180^{\\circ }-1.45563…+360^{\\circ }n", "radians": "x=1.45563…+2πn,x=π-1.45563…+2πn", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\frac{\\sin\\left(60^{\\circ\\:}\\right)}{174.36}=\\frac{\\sin\\left(x\\right)}{200}{\\quad:\\quad}x=1.45563…+360^{\\circ\\:}n,\\:x=180^{\\circ\\:}-1.45563…+360^{\\circ\\:}n$$", "input": "\\frac{\\sin\\left(60^{\\circ\\:}\\right)}{174.36}=\\frac{\\sin\\left(x\\right)}{200}", "steps": [ { "type": "interim", "title": "$$\\sin\\left(60^{\\circ\\:}\\right)=\\frac{\\sqrt{3}}{2}$$", "input": "\\sin\\left(60^{\\circ\\:}\\right)", "steps": [ { "type": "interim", "title": "Use the following trivial identity:$${\\quad}\\sin\\left(60^{\\circ\\:}\\right)=\\frac{\\sqrt{3}}{2}$$", "input": "\\sin\\left(60^{\\circ\\:}\\right)", "steps": [ { "type": "step", "primary": "$$\\sin\\left(x\\right)$$ periodicity table with $$360^{\\circ\\:}n$$ cycle:<br/>$$\\begin{array}{|c|c|c|c|}\\hline x&\\sin(x)&x&\\sin(x)\\\\\\hline 0&0&180^{\\circ }&0\\\\\\hline 30^{\\circ }&\\frac{1}{2}&210^{\\circ }&-\\frac{1}{2}\\\\\\hline 45^{\\circ }&\\frac{\\sqrt{2}}{2}&225^{\\circ }&-\\frac{\\sqrt{2}}{2}\\\\\\hline 60^{\\circ }&\\frac{\\sqrt{3}}{2}&240^{\\circ }&-\\frac{\\sqrt{3}}{2}\\\\\\hline 90^{\\circ }&1&270^{\\circ }&-1\\\\\\hline 120^{\\circ }&\\frac{\\sqrt{3}}{2}&300^{\\circ }&-\\frac{\\sqrt{3}}{2}\\\\\\hline 135^{\\circ }&\\frac{\\sqrt{2}}{2}&315^{\\circ }&-\\frac{\\sqrt{2}}{2}\\\\\\hline 150^{\\circ }&\\frac{1}{2}&330^{\\circ }&-\\frac{1}{2}\\\\\\hline \\end{array}$$" }, { "type": "step", "result": "=\\frac{\\sqrt{3}}{2}" } ], "meta": { "interimType": "Trig Trivial Angle Value Title 0Eq" } }, { "type": "step", "result": "=\\frac{\\sqrt{3}}{2}" } ], "meta": { "solvingClass": "Trig Evaluate", "interimType": "Trig Evaluate" } }, { "type": "step", "result": "\\frac{\\frac{\\sqrt{3}}{2}}{174.36}=\\frac{\\sin\\left(x\\right)}{200}" }, { "type": "step", "primary": "Switch sides", "result": "\\frac{\\sin\\left(x\\right)}{200}=\\frac{\\frac{\\sqrt{3}}{2}}{174.36}" }, { "type": "interim", "title": "Multiply both sides by $$100$$", "input": "\\frac{\\sin\\left(x\\right)}{200}=\\frac{\\frac{\\sqrt{3}}{2}}{174.36}", "result": "\\frac{\\sin\\left(x\\right)}{2}=\\frac{100\\sqrt{3}}{348.72}", "steps": [ { "type": "step", "primary": "Multiply both sides by $$100$$", "result": "\\frac{\\sin\\left(x\\right)}{200}\\cdot\\:100=\\frac{\\frac{\\sqrt{3}}{2}}{174.36}\\cdot\\:100" }, { "type": "interim", "title": "Simplify", "input": "\\frac{\\sin\\left(x\\right)}{200}\\cdot\\:100=\\frac{\\frac{\\sqrt{3}}{2}}{174.36}\\cdot\\:100", "result": "\\frac{\\sin\\left(x\\right)}{2}=\\frac{100\\sqrt{3}}{348.72}", "steps": [ { "type": "interim", "title": "Simplify $$\\frac{\\sin\\left(x\\right)}{200}\\cdot\\:100:{\\quad}\\frac{\\sin\\left(x\\right)}{2}$$", "input": "\\frac{\\sin\\left(x\\right)}{200}\\cdot\\:100", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{\\sin\\left(x\\right)\\cdot\\:100}{200}" }, { "type": "step", "primary": "Cancel the common factor: $$100$$", "result": "=\\frac{\\sin\\left(x\\right)}{2}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7rKk9+1UlbSbMyKyv9KIHF0b6oHApNQXciDT7dXHY/gvfXIaN3O3UVIl2G4Gtt+7AQslTDKxOR/6J+ZOGvUcaurNJ+JKdJ3esdBiDP1Qb8I7/P/+v51eTuH2/F4MIu5mqcoXe7Y4hvhsUDcQ7xpGVJR8GRsQZ74Pf700BVFDXkFj1HFHFZyz3Fv2gLKVuGdBGyyWtvaX0Z/BXOIGETcXlPA==" } }, { "type": "interim", "title": "Simplify $$\\frac{\\frac{\\sqrt{3}}{2}}{174.36}\\cdot\\:100:{\\quad}\\frac{100\\sqrt{3}}{348.72}$$", "input": "\\frac{\\frac{\\sqrt{3}}{2}}{174.36}\\cdot\\:100", "steps": [ { "type": "interim", "title": "$$\\frac{\\frac{\\sqrt{3}}{2}}{174.36}=\\frac{\\sqrt{3}}{348.72}$$", "input": "\\frac{\\frac{\\sqrt{3}}{2}}{174.36}", "steps": [ { "type": "step", "primary": "Apply the fraction rule: $$\\frac{\\frac{b}{c}}{a}=\\frac{b}{c\\:\\cdot\\:a}$$", "result": "=\\frac{\\sqrt{3}}{2\\cdot\\:174.36}" }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:174.36=348.72$$", "result": "=\\frac{\\sqrt{3}}{348.72}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7tiTmKkmeEqHWILbBb88aje3SDTsC9UBpv1iNeF8j4vm4f3CxR0SIZUZq7s6gwc9Fq47vuWedXv2WUg94ER8IwZ74w2NtW+VwUOOABt6ynqC4/VRHPKp4uDV2IWN2eS4oB5NXi7bTNbXVXk8s7Avpa+yhAcZFhmtydWsQoArslaJGDhxbjFbgladzWb9fKPkWsqut9e4WT+TSUSQ923x9vzC2wuyELBaDFPhIStW/D+I=" } }, { "type": "step", "result": "=100\\cdot\\:\\frac{\\sqrt{3}}{348.72}" }, { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{\\sqrt{3}\\cdot\\:100}{348.72}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7tiTmKkmeEqHWILbBb88aje3SDTsC9UBpv1iNeF8j4vnYA9+J54fZQ87TUZYowF6aAJYpRu9XpYrd8NSAW2DdD/KxLrO04AooUAReaJjhZCYP+kvM2XKVN0dIVqTnX29Gy8Y/S8Mx4Hd4GqYyyDJUlj/L0MoYg+CUn6oyL3EO7YppEjsYKnQdDP7MPDbdrF10aANtvnqfoyqubx/FzK7ajzVQn0Dd+FxZDRYQPY4AlXT74/Vc5NjeV2m/v2Itex4czYrpw7xGsZ/J+5wYn0nspA==" } }, { "type": "step", "result": "\\frac{\\sin\\left(x\\right)}{2}=\\frac{100\\sqrt{3}}{348.72}" } ], "meta": { "interimType": "Generic Simplify 0Eq" } } ], "meta": { "interimType": "Multiply Both Sides Specific 1Eq", "gptData": "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" } }, { "type": "interim", "title": "Multiply both sides by $$2$$", "input": "\\frac{\\sin\\left(x\\right)}{2}=\\frac{100\\sqrt{3}}{348.72}", "result": "\\sin\\left(x\\right)=0.99337…", "steps": [ { "type": "step", "primary": "Multiply both sides by $$2$$", "result": "\\frac{2\\sin\\left(x\\right)}{2}=\\frac{2\\cdot\\:100\\sqrt{3}}{348.72}" }, { "type": "interim", "title": "Simplify", "input": "\\frac{2\\sin\\left(x\\right)}{2}=\\frac{2\\cdot\\:100\\sqrt{3}}{348.72}", "result": "\\sin\\left(x\\right)=0.99337…", "steps": [ { "type": "interim", "title": "Simplify $$\\frac{2\\sin\\left(x\\right)}{2}:{\\quad}\\sin\\left(x\\right)$$", "input": "\\frac{2\\sin\\left(x\\right)}{2}", "steps": [ { "type": "step", "primary": "Divide the numbers: $$\\frac{2}{2}=1$$", "result": "=\\sin\\left(x\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s71+6656DMR94zeG/NTorKddPsJsufb28cY226iO1FOs2rju+5Z51e/ZZSD3gRHwjBXFggB3asoHCJiWnl7pJbzIEFMST8lDZxn1Yq5HMKVTtAb51BceE+a5ppiVMbLOTZpjU0UHMtqClGVl7qQf9jQY8BPOx0wlsgFN8qUa6AzA0=" } }, { "type": "interim", "title": "Simplify $$\\frac{2\\cdot\\:100\\sqrt{3}}{348.72}:{\\quad}0.99337…$$", "input": "\\frac{2\\cdot\\:100\\sqrt{3}}{348.72}", "steps": [ { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:100=200$$", "result": "=\\frac{200\\sqrt{3}}{348.72}" }, { "type": "step", "primary": "Convert element to a decimal form", "secondary": [ "$$\\sqrt{3}=1.73205…$$" ], "result": "=\\frac{200\\cdot\\:1.73205…}{348.72}" }, { "type": "step", "primary": "Multiply the numbers: $$200\\cdot\\:1.73205…=346.41016…$$", "result": "=\\frac{346.41016…}{348.72}" }, { "type": "step", "primary": "Divide the numbers: $$\\frac{346.41016…}{348.72}=0.99337…$$", "result": "=0.99337…" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s76wwMXmpSw8uHSAcJv7PDwkBg9VSEu1lDmDnmE2Cib7ooFE/9DbBjIX3H1AUYiFKvq47vuWedXv2WUg94ER8Iwdy+l8SYWKMhnNmLM0ppKUVFKk3fejFkyiOiq9iG9IkAIuSmJUDtLxRPyJ57Nkq7qn5JgplxPxcPDrGN5ckNcGEGFhNtcWICAr3Q8FYRKqcnIrVCwyomV8gmoUs55zT79A==" } }, { "type": "step", "result": "\\sin\\left(x\\right)=0.99337…" } ], "meta": { "interimType": "Generic Simplify 0Eq" } } ], "meta": { "interimType": "Multiply Both Sides Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYkjaL1+oeMTpmayd0D+HAtfzrbIwNq+/0ZZdHW5VwkXAORi5s0Y11ti2hoex67S78eX+O4MWGdqTzDAO/M8/S09enbsh7NlTyc0Ww8E77CcEEnlQHZNykwVgDw6TiWSXSafw73G5Z/YM1lJ31Kl95qI7lGm6t34nhX6Er5si7J+4U9ik5WINGrsWotLzS7/BjMilG2DUqMrP9Xo55x7KtrBiWYHCew+iATXBdG4jYMBGNKrzlt585vT4drujVcPj7NoqOaxYYh7w3ib/OUq23+Rbe+FlaZIgZ0JdESVuh8G3eRzM3vynqxJKIg3jtF9ZCY+dRwkanZaJUwyYScC1TCzvlUs9ssN4GudQG+VOP3Gnyt8LFQm5UpsbsZ5ifTkhFGRpIJqMaDZWJwDpG5SWwcsd7ZG6RIQIR6lm1Ekl71S7RSpN33oxZMojoqvYhvSJAOmO3MAGBYB0j1IfnmTb390sN3+9TPIoWhnXg9uO7NbHvzIPeEtDfcHv/z8uls8Teg==" } }, { "type": "interim", "title": "Apply trig inverse properties", "input": "\\sin\\left(x\\right)=0.99337…", "result": "x=\\arcsin\\left(0.99337…\\right)+360^{\\circ\\:}n,\\:x=180^{\\circ\\:}-\\arcsin\\left(0.99337…\\right)+360^{\\circ\\:}n", "steps": [ { "type": "step", "primary": "General solutions for $$\\sin\\left(x\\right)=0.99337…$$", "secondary": [ "$$\\sin\\left(x\\right)=a\\quad\\Rightarrow\\quad\\:x=\\arcsin\\left(a\\right)+360^{\\circ\\:}n,\\:\\quad\\:x=180^{\\circ\\:}-\\arcsin\\left(a\\right)+360^{\\circ\\:}n$$" ], "result": "x=\\arcsin\\left(0.99337…\\right)+360^{\\circ\\:}n,\\:x=180^{\\circ\\:}-\\arcsin\\left(0.99337…\\right)+360^{\\circ\\:}n" } ], "meta": { "interimType": "Trig Apply Inverse Props 0Eq" } }, { "type": "step", "primary": "Show solutions in decimal form", "result": "x=1.45563…+360^{\\circ\\:}n,\\:x=180^{\\circ\\:}-1.45563…+360^{\\circ\\:}n" } ], "meta": { "solvingClass": "Trig Equations", "practiceLink": "/practice/trigonometry-practice#area=main&subtopic=Trig%20Equations", "practiceTopic": "Trig Equations" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "\\frac{\\sin(60^{\\circ })}{174.36}-\\frac{\\sin(x)}{200}" }, "showViewLarger": true } }, "meta": { "showVerify": true } }