(15)/(sin(32))=(12)/(sin(b))

{ "query": { "display": "$$\\frac{15}{\\sin\\left(32^{\\circ\\:}\\right)}=\\frac{12}{\\sin\\left(b\\right)}$$", "symbolab_question": "EQUATION#\\frac{15}{\\sin(32^{\\circ })}=\\frac{12}{\\sin(b)}" }, "solution": { "level": "PERFORMED", "subject": "Trigonometry", "topic": "Trig Equations", "subTopic": "Trig Equations", "default": "b=0.43778…+360^{\\circ }n,b=180^{\\circ }-0.43778…+360^{\\circ }n", "radians": "b=0.43778…+2πn,b=π-0.43778…+2πn", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\frac{15}{\\sin\\left(32^{\\circ\\:}\\right)}=\\frac{12}{\\sin\\left(b\\right)}{\\quad:\\quad}b=0.43778…+360^{\\circ\\:}n,\\:b=180^{\\circ\\:}-0.43778…+360^{\\circ\\:}n$$", "input": "\\frac{15}{\\sin\\left(32^{\\circ\\:}\\right)}=\\frac{12}{\\sin\\left(b\\right)}", "steps": [ { "type": "interim", "title": "Cross multiply", "input": "\\frac{15}{\\sin\\left(32^{\\circ\\:}\\right)}=\\frac{12}{\\sin\\left(b\\right)}", "result": "15\\sin\\left(b\\right)=\\sin\\left(32^{\\circ\\:}\\right)\\cdot\\:12", "steps": [ { "type": "step", "primary": "Apply fraction cross multiply: if $$\\frac{a}{b}=\\frac{c}{d}$$ then $$a\\cdot\\:d=b\\cdot\\:c$$", "result": "15\\sin\\left(b\\right)=\\sin\\left(32^{\\circ\\:}\\right)\\cdot\\:12" } ], "meta": { "interimType": "Equation Cross Multiply Title 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjAjYcBSbJJKcO5nwp2W+NzBwjMiVYZJAv/rYaEWvJeI6yVdmhOjlkjCiyiA4A3MqtvMmMyCHSdC7aPMPQ3KzxJFnPdZdx0r8OK7cZR0Id8EaPjr1TkSBcI3XU/hnblBJuIOvG3xQumzZZPBJ5THLCLFTeQKHeh69S6dnv9vSoUoFIMFR8NMvrUkXo40CKv9zpq/Mg94S0N9we//Py6WzxN6" } }, { "type": "interim", "title": "Divide both sides by $$15$$", "input": "15\\sin\\left(b\\right)=\\sin\\left(32^{\\circ\\:}\\right)\\cdot\\:12", "result": "\\sin\\left(b\\right)=\\frac{4\\sin\\left(32^{\\circ\\:}\\right)}{5}", "steps": [ { "type": "step", "primary": "Divide both sides by $$15$$", "result": "\\frac{15\\sin\\left(b\\right)}{15}=\\frac{\\sin\\left(32^{\\circ\\:}\\right)\\cdot\\:12}{15}" }, { "type": "step", "primary": "Simplify", "result": "\\sin\\left(b\\right)=\\frac{4\\sin\\left(32^{\\circ\\:}\\right)}{5}" } ], "meta": { "interimType": "Divide Both Sides Specific 1Eq", "gptData": "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" } }, { "type": "step", "primary": "Verify Solutions" }, { "type": "interim", "title": "Find undefined (singularity) points:$${\\quad}\\sin\\left(b\\right)=0$$", "steps": [ { "type": "step", "primary": "Take the denominator(s) of $$\\frac{12}{\\sin\\left(b\\right)}$$ and compare to zero" }, { "type": "step", "result": "\\sin\\left(b\\right)=0" }, { "type": "step", "primary": "The following points are undefined", "result": "\\sin\\left(b\\right)=0" } ], "meta": { "interimType": "Undefined Points 0Eq" } }, { "type": "step", "primary": "Combine undefined points with solutions:" }, { "type": "step", "result": "\\sin\\left(b\\right)=\\frac{4\\sin\\left(32^{\\circ\\:}\\right)}{5}" }, { "type": "interim", "title": "Apply trig inverse properties", "input": "\\sin\\left(b\\right)=\\frac{4\\sin\\left(32^{\\circ\\:}\\right)}{5}", "result": "b=\\arcsin\\left(\\frac{4\\sin\\left(32^{\\circ\\:}\\right)}{5}\\right)+360^{\\circ\\:}n,\\:b=180^{\\circ\\:}-\\arcsin\\left(\\frac{4\\sin\\left(32^{\\circ\\:}\\right)}{5}\\right)+360^{\\circ\\:}n", "steps": [ { "type": "step", "primary": "General solutions for $$\\sin\\left(b\\right)=\\frac{4\\sin\\left(32^{\\circ\\:}\\right)}{5}$$", "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": "b=\\arcsin\\left(\\frac{4\\sin\\left(32^{\\circ\\:}\\right)}{5}\\right)+360^{\\circ\\:}n,\\:b=180^{\\circ\\:}-\\arcsin\\left(\\frac{4\\sin\\left(32^{\\circ\\:}\\right)}{5}\\right)+360^{\\circ\\:}n" } ], "meta": { "interimType": "Trig Apply Inverse Props 0Eq" } }, { "type": "step", "primary": "Show solutions in decimal form", "result": "b=0.43778…+360^{\\circ\\:}n,\\:b=180^{\\circ\\:}-0.43778…+360^{\\circ\\:}n" } ], "meta": { "solvingClass": "Trig Equations", "practiceLink": "/practice/trigonometry-practice#area=main&subtopic=Trig%20Equations", "practiceTopic": "Trig Equations" } }, "plot_output": { "meta": { "plotInfo": { "variable": "b", "plotRequest": "\\frac{15}{\\sin(32^{\\circ })}-\\frac{12}{\\sin(b)}" }, "showViewLarger": true } }, "meta": { "showVerify": true } }