11sin(B)+13=4sin(B)+8

{ "query": { "display": "$$11\\sin\\left(B\\right)+13=4\\sin\\left(B\\right)+8$$", "symbolab_question": "EQUATION#11\\sin(B)+13=4\\sin(B)+8" }, "solution": { "level": "PERFORMED", "subject": "Trigonometry", "topic": "Trig Equations", "subTopic": "Trig Equations", "default": "B=-0.79560…+2πn,B=π+0.79560…+2πn", "degrees": "B=-45.58469…^{\\circ }+360^{\\circ }n,B=225.58469…^{\\circ }+360^{\\circ }n", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$11\\sin\\left(B\\right)+13=4\\sin\\left(B\\right)+8{\\quad:\\quad}B=-0.79560…+2πn,\\:B=π+0.79560…+2πn$$", "input": "11\\sin\\left(B\\right)+13=4\\sin\\left(B\\right)+8", "steps": [ { "type": "interim", "title": "Solve by substitution", "input": "11\\sin\\left(B\\right)+13=4\\sin\\left(B\\right)+8", "result": "\\sin\\left(B\\right)=-\\frac{5}{7}", "steps": [ { "type": "step", "primary": "Let: $$\\sin\\left(B\\right)=u$$", "result": "11u+13=4u+8" }, { "type": "interim", "title": "$$11u+13=4u+8{\\quad:\\quad}u=-\\frac{5}{7}$$", "input": "11u+13=4u+8", "steps": [ { "type": "interim", "title": "Move $$13\\:$$to the right side", "input": "11u+13=4u+8", "result": "11u=4u-5", "steps": [ { "type": "step", "primary": "Subtract $$13$$ from both sides", "result": "11u+13-13=4u+8-13" }, { "type": "step", "primary": "Simplify", "result": "11u=4u-5" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": "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" } }, { "type": "interim", "title": "Move $$4u\\:$$to the left side", "input": "11u=4u-5", "result": "7u=-5", "steps": [ { "type": "step", "primary": "Subtract $$4u$$ from both sides", "result": "11u-4u=4u-5-4u" }, { "type": "step", "primary": "Simplify", "result": "7u=-5" } ], "meta": { "interimType": "Move to the Left Title 1Eq", "gptData": "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" } }, { "type": "interim", "title": "Divide both sides by $$7$$", "input": "7u=-5", "result": "u=-\\frac{5}{7}", "steps": [ { "type": "step", "primary": "Divide both sides by $$7$$", "result": "\\frac{7u}{7}=\\frac{-5}{7}" }, { "type": "step", "primary": "Simplify", "result": "u=-\\frac{5}{7}" } ], "meta": { "interimType": "Divide Both Sides Specific 1Eq", "gptData": "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" } } ], "meta": { "solvingClass": "Equations", "interimType": "Equations" } }, { "type": "step", "primary": "Substitute back $$u=\\sin\\left(B\\right)$$", "result": "\\sin\\left(B\\right)=-\\frac{5}{7}" } ], "meta": { "interimType": "Substitution Method 0Eq" } }, { "type": "interim", "title": "$$\\sin\\left(B\\right)=-\\frac{5}{7}{\\quad:\\quad}B=\\arcsin\\left(-\\frac{5}{7}\\right)+2πn,\\:B=π+\\arcsin\\left(\\frac{5}{7}\\right)+2πn$$", "input": "\\sin\\left(B\\right)=-\\frac{5}{7}", "steps": [ { "type": "interim", "title": "Apply trig inverse properties", "input": "\\sin\\left(B\\right)=-\\frac{5}{7}", "result": "B=\\arcsin\\left(-\\frac{5}{7}\\right)+2πn,\\:B=π+\\arcsin\\left(\\frac{5}{7}\\right)+2πn", "steps": [ { "type": "step", "primary": "General solutions for $$\\sin\\left(B\\right)=-\\frac{5}{7}$$", "secondary": [ "$$\\sin\\left(x\\right)=-a\\quad\\Rightarrow\\quad\\:x=\\arcsin\\left(-a\\right)+2πn,\\:\\quad\\:x=π+\\arcsin\\left(a\\right)+2πn$$" ], "result": "B=\\arcsin\\left(-\\frac{5}{7}\\right)+2πn,\\:B=π+\\arcsin\\left(\\frac{5}{7}\\right)+2πn" } ], "meta": { "interimType": "Trig Apply Inverse Props 0Eq" } } ], "meta": { "interimType": "N/A" } }, { "type": "step", "primary": "Combine all the solutions", "result": "B=\\arcsin\\left(-\\frac{5}{7}\\right)+2πn,\\:B=π+\\arcsin\\left(\\frac{5}{7}\\right)+2πn" }, { "type": "step", "primary": "Show solutions in decimal form", "result": "B=-0.79560…+2πn,\\:B=π+0.79560…+2πn" } ], "meta": { "solvingClass": "Trig Equations", "practiceLink": "/practice/trigonometry-practice#area=main&subtopic=Trig%20Equations", "practiceTopic": "Trig Equations" } }, "plot_output": { "meta": { "plotInfo": { "variable": "B", "plotRequest": "11\\sin(B)+13-4\\sin(B)-8" }, "showViewLarger": true } }, "meta": { "showVerify": true } }