0.08=0.12cos^2(3.922t)

{ "query": { "display": "$$0.08=0.12\\cos^{2}\\left(3.922t\\right)$$", "symbolab_question": "EQUATION#0.08=0.12\\cos^{2}(3.922t)" }, "solution": { "level": "PERFORMED", "subject": "Trigonometry", "topic": "Trig Equations", "subTopic": "Trig Equations", "default": "t=\\frac{0.61547…}{3.922}+\\frac{2πn}{3.922},t=\\frac{2π}{3.922}-\\frac{0.61547…}{3.922}+\\frac{2πn}{3.922},t=\\frac{2.52611…}{3.922}+\\frac{2πn}{3.922},t=-\\frac{2.52611…}{3.922}+\\frac{2πn}{3.922}", "degrees": "t=8.99143…^{\\circ }+91.78990…^{\\circ }n,t=82.79847…^{\\circ }+91.78990…^{\\circ }n,t=36.90352…^{\\circ }+91.78990…^{\\circ }n,t=-36.90352…^{\\circ }+91.78990…^{\\circ }n", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$0.08=0.12\\cos^{2}\\left(3.922t\\right){\\quad:\\quad}t=\\frac{0.61547…}{3.922}+\\frac{2πn}{3.922},\\:t=\\frac{2π}{3.922}-\\frac{0.61547…}{3.922}+\\frac{2πn}{3.922},\\:t=\\frac{2.52611…}{3.922}+\\frac{2πn}{3.922},\\:t=-\\frac{2.52611…}{3.922}+\\frac{2πn}{3.922}$$", "input": "0.08=0.12\\cos^{2}\\left(3.922t\\right)", "steps": [ { "type": "step", "primary": "Switch sides", "result": "0.12\\cos^{2}\\left(3.922t\\right)=0.08" }, { "type": "interim", "title": "Solve by substitution", "input": "0.12\\cos^{2}\\left(3.922t\\right)=0.08", "result": "\\cos\\left(3.922t\\right)=\\sqrt{\\frac{2}{3}},\\:\\cos\\left(3.922t\\right)=-\\sqrt{\\frac{2}{3}}", "steps": [ { "type": "step", "primary": "Let: $$\\cos\\left(3.922t\\right)=u$$", "result": "0.12u^{2}=0.08" }, { "type": "interim", "title": "$$0.12u^{2}=0.08{\\quad:\\quad}u=\\sqrt{\\frac{2}{3}},\\:u=-\\sqrt{\\frac{2}{3}}$$", "input": "0.12u^{2}=0.08", "steps": [ { "type": "interim", "title": "Multiply both sides by $$100$$", "input": "0.12u^{2}=0.08", "result": "12u^{2}=8", "steps": [ { "type": "step", "primary": "To eliminate decimal points, multiply by 10 for every digit after the decimal point", "secondary": [ "There are $$2\\:$$digits to the right of the decimal point, therefore multiply by $$100$$" ], "result": "0.12u^{2}\\cdot\\:100=0.08\\cdot\\:100" }, { "type": "step", "primary": "Refine", "result": "12u^{2}=8" } ], "meta": { "interimType": "Multiply Both Sides Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s735tE+PDK58N12+sc3s9A7K/a1B3IUZpFQ4qD7FDzgYzsFA7GvaMs+fzlDjmlcVRXaCj47GMC73PUEOQxvsSjQGsXpeZ8abTmHsfSyfd6ixNaxX6v7cggjt2xcFpAlvYGsoh1N1bMWdCb5YM9Q4f9ltMyKXqjig1oe8nVG8YCoXV694Ph869lyDoQUdss535rH5VsZEFE5bIye1EkjNkyumZfqkaAoWwZzpVnzOGh6n2U3Ivi6Jw5WzIVkoj67obgo+TLSd/+z47y7tSQmVO2R/+io0XpeCEVkKWH7k/AAgdoX5mCrYP0f9OYSSmutw+IFx8+wilMNmK/TZrUT39yqO3kBdtF9Z7/O54JnPKwuiC6djUi+Gv40tGMhEB/ZddUtBQABXdCU08T73LuSgBVPZZ3p0+zhdnfVu+f0BCo6lc34oLn7c0nhzXM7wbqfHSIWfVV320lmGqcplFsNJbfCTSRaYnlKsEjNdJpJWYt48immmfHn9rGr9hNGoXTD+uKzumcdQ/N+9Ye+ltHemffHRnReJINv015UetB8nD+tqWhCzTT7H2LIFekqKk0KFYOY0KINx7aCCcjHY7WaAbVDIZsRJXFE2L0jxuZjJ92ZjQ=" } }, { "type": "interim", "title": "Divide both sides by $$12$$", "input": "12u^{2}=8", "result": "u^{2}=\\frac{2}{3}", "steps": [ { "type": "step", "primary": "Divide both sides by $$12$$", "result": "\\frac{12u^{2}}{12}=\\frac{8}{12}" }, { "type": "step", "primary": "Simplify", "result": "u^{2}=\\frac{2}{3}" } ], "meta": { "interimType": "Divide Both Sides Specific 1Eq", "gptData": "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" } }, { "type": "step", "primary": "For $$x^{2}=f\\left(a\\right)$$ the solutions are $$x=\\sqrt{f\\left(a\\right)},\\:\\:-\\sqrt{f\\left(a\\right)}$$" }, { "type": "step", "result": "u=\\sqrt{\\frac{2}{3}},\\:u=-\\sqrt{\\frac{2}{3}}" } ], "meta": { "solvingClass": "Equations", "interimType": "Equations" } }, { "type": "step", "primary": "Substitute back $$u=\\cos\\left(3.922t\\right)$$", "result": "\\cos\\left(3.922t\\right)=\\sqrt{\\frac{2}{3}},\\:\\cos\\left(3.922t\\right)=-\\sqrt{\\frac{2}{3}}" } ], "meta": { "interimType": "Substitution Method 0Eq" } }, { "type": "interim", "title": "$$\\cos\\left(3.922t\\right)=\\sqrt{\\frac{2}{3}}{\\quad:\\quad}t=\\frac{\\arccos\\left(\\sqrt{\\frac{2}{3}}\\right)}{3.922}+\\frac{2πn}{3.922},\\:t=\\frac{2π}{3.922}-\\frac{\\arccos\\left(\\sqrt{\\frac{2}{3}}\\right)}{3.922}+\\frac{2πn}{3.922}$$", "input": "\\cos\\left(3.922t\\right)=\\sqrt{\\frac{2}{3}}", "steps": [ { "type": "interim", "title": "Apply trig inverse properties", "input": "\\cos\\left(3.922t\\right)=\\sqrt{\\frac{2}{3}}", "result": "3.922t=\\arccos\\left(\\sqrt{\\frac{2}{3}}\\right)+2πn,\\:3.922t=2π-\\arccos\\left(\\sqrt{\\frac{2}{3}}\\right)+2πn", "steps": [ { "type": "step", "primary": "General solutions for $$\\cos\\left(3.922t\\right)=\\sqrt{\\frac{2}{3}}$$", "secondary": [ "$$\\cos\\left(x\\right)=a\\quad\\Rightarrow\\quad\\:x=\\arccos\\left(a\\right)+2πn,\\:\\quad\\:x=2π-\\arccos\\left(a\\right)+2πn$$" ], "result": "3.922t=\\arccos\\left(\\sqrt{\\frac{2}{3}}\\right)+2πn,\\:3.922t=2π-\\arccos\\left(\\sqrt{\\frac{2}{3}}\\right)+2πn" } ], "meta": { "interimType": "Trig Apply Inverse Props 0Eq" } }, { "type": "interim", "title": "Solve $$3.922t=\\arccos\\left(\\sqrt{\\frac{2}{3}}\\right)+2πn:{\\quad}t=\\frac{\\arccos\\left(\\sqrt{\\frac{2}{3}}\\right)}{3.922}+\\frac{2πn}{3.922}$$", "input": "3.922t=\\arccos\\left(\\sqrt{\\frac{2}{3}}\\right)+2πn", "steps": [ { "type": "interim", "title": "Divide both sides by $$3.922$$", "input": "3.922t=\\arccos\\left(\\sqrt{\\frac{2}{3}}\\right)+2πn", "result": "t=\\frac{\\arccos\\left(\\sqrt{\\frac{2}{3}}\\right)}{3.922}+\\frac{2πn}{3.922}", "steps": [ { "type": "step", "primary": "Divide both sides by $$3.922$$", "result": "\\frac{3.922t}{3.922}=\\frac{\\arccos\\left(\\sqrt{\\frac{2}{3}}\\right)}{3.922}+\\frac{2πn}{3.922}" }, { "type": "step", "primary": "Simplify", "result": "t=\\frac{\\arccos\\left(\\sqrt{\\frac{2}{3}}\\right)}{3.922}+\\frac{2πn}{3.922}" } ], "meta": { "interimType": "Divide Both Sides Specific 1Eq", "gptData": 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$$3.922t=2π-\\arccos\\left(\\sqrt{\\frac{2}{3}}\\right)+2πn:{\\quad}t=\\frac{2π}{3.922}-\\frac{\\arccos\\left(\\sqrt{\\frac{2}{3}}\\right)}{3.922}+\\frac{2πn}{3.922}$$", "input": "3.922t=2π-\\arccos\\left(\\sqrt{\\frac{2}{3}}\\right)+2πn", "steps": [ { "type": "interim", "title": "Divide both sides by $$3.922$$", "input": "3.922t=2π-\\arccos\\left(\\sqrt{\\frac{2}{3}}\\right)+2πn", "result": "t=\\frac{2π}{3.922}-\\frac{\\arccos\\left(\\sqrt{\\frac{2}{3}}\\right)}{3.922}+\\frac{2πn}{3.922}", "steps": [ { "type": "step", "primary": "Divide both sides by $$3.922$$", "result": "\\frac{3.922t}{3.922}=\\frac{2π}{3.922}-\\frac{\\arccos\\left(\\sqrt{\\frac{2}{3}}\\right)}{3.922}+\\frac{2πn}{3.922}" }, { "type": "step", "primary": "Simplify", "result": "t=\\frac{2π}{3.922}-\\frac{\\arccos\\left(\\sqrt{\\frac{2}{3}}\\right)}{3.922}+\\frac{2πn}{3.922}" } ], "meta": { "interimType": "Divide Both Sides Specific 1Eq", "gptData": 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"t=\\frac{\\arccos\\left(\\sqrt{\\frac{2}{3}}\\right)}{3.922}+\\frac{2πn}{3.922},\\:t=\\frac{2π}{3.922}-\\frac{\\arccos\\left(\\sqrt{\\frac{2}{3}}\\right)}{3.922}+\\frac{2πn}{3.922}" } ], "meta": { "interimType": "N/A" } }, { "type": "interim", "title": "$$\\cos\\left(3.922t\\right)=-\\sqrt{\\frac{2}{3}}{\\quad:\\quad}t=\\frac{\\arccos\\left(-\\sqrt{\\frac{2}{3}}\\right)}{3.922}+\\frac{2πn}{3.922},\\:t=-\\frac{\\arccos\\left(-\\sqrt{\\frac{2}{3}}\\right)}{3.922}+\\frac{2πn}{3.922}$$", "input": "\\cos\\left(3.922t\\right)=-\\sqrt{\\frac{2}{3}}", "steps": [ { "type": "interim", "title": "Apply trig inverse properties", "input": "\\cos\\left(3.922t\\right)=-\\sqrt{\\frac{2}{3}}", "result": "3.922t=\\arccos\\left(-\\sqrt{\\frac{2}{3}}\\right)+2πn,\\:3.922t=-\\arccos\\left(-\\sqrt{\\frac{2}{3}}\\right)+2πn", "steps": [ { "type": "step", "primary": "General solutions for $$\\cos\\left(3.922t\\right)=-\\sqrt{\\frac{2}{3}}$$", "secondary": [ "$$\\cos\\left(x\\right)=-a\\quad\\Rightarrow\\quad\\:x=\\arccos\\left(-a\\right)+2πn,\\:\\quad\\:x=-\\arccos\\left(-a\\right)+2πn$$" ], "result": "3.922t=\\arccos\\left(-\\sqrt{\\frac{2}{3}}\\right)+2πn,\\:3.922t=-\\arccos\\left(-\\sqrt{\\frac{2}{3}}\\right)+2πn" } ], "meta": { "interimType": "Trig Apply Inverse Props 0Eq" } }, { "type": "interim", "title": "Solve $$3.922t=\\arccos\\left(-\\sqrt{\\frac{2}{3}}\\right)+2πn:{\\quad}t=\\frac{\\arccos\\left(-\\sqrt{\\frac{2}{3}}\\right)}{3.922}+\\frac{2πn}{3.922}$$", "input": "3.922t=\\arccos\\left(-\\sqrt{\\frac{2}{3}}\\right)+2πn", "steps": [ { "type": "interim", "title": "Divide both sides by $$3.922$$", "input": "3.922t=\\arccos\\left(-\\sqrt{\\frac{2}{3}}\\right)+2πn", "result": "t=\\frac{\\arccos\\left(-\\sqrt{\\frac{2}{3}}\\right)}{3.922}+\\frac{2πn}{3.922}", "steps": [ { "type": "step", "primary": "Divide both sides by $$3.922$$", "result": "\\frac{3.922t}{3.922}=\\frac{\\arccos\\left(-\\sqrt{\\frac{2}{3}}\\right)}{3.922}+\\frac{2πn}{3.922}" }, { "type": "step", "primary": "Simplify", "result": "t=\\frac{\\arccos\\left(-\\sqrt{\\frac{2}{3}}\\right)}{3.922}+\\frac{2πn}{3.922}" } ], "meta": { "interimType": "Divide Both Sides Specific 1Eq", "gptData": 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$$3.922t=-\\arccos\\left(-\\sqrt{\\frac{2}{3}}\\right)+2πn:{\\quad}t=-\\frac{\\arccos\\left(-\\sqrt{\\frac{2}{3}}\\right)}{3.922}+\\frac{2πn}{3.922}$$", "input": "3.922t=-\\arccos\\left(-\\sqrt{\\frac{2}{3}}\\right)+2πn", "steps": [ { "type": "interim", "title": "Divide both sides by $$3.922$$", "input": "3.922t=-\\arccos\\left(-\\sqrt{\\frac{2}{3}}\\right)+2πn", "result": "t=-\\frac{\\arccos\\left(-\\sqrt{\\frac{2}{3}}\\right)}{3.922}+\\frac{2πn}{3.922}", "steps": [ { "type": "step", "primary": "Divide both sides by $$3.922$$", "result": "\\frac{3.922t}{3.922}=-\\frac{\\arccos\\left(-\\sqrt{\\frac{2}{3}}\\right)}{3.922}+\\frac{2πn}{3.922}" }, { "type": "step", "primary": "Simplify", "result": "t=-\\frac{\\arccos\\left(-\\sqrt{\\frac{2}{3}}\\right)}{3.922}+\\frac{2πn}{3.922}" } ], "meta": { "interimType": "Divide Both Sides Specific 1Eq", "gptData": 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"t=\\frac{\\arccos\\left(-\\sqrt{\\frac{2}{3}}\\right)}{3.922}+\\frac{2πn}{3.922},\\:t=-\\frac{\\arccos\\left(-\\sqrt{\\frac{2}{3}}\\right)}{3.922}+\\frac{2πn}{3.922}" } ], "meta": { "interimType": "N/A" } }, { "type": "step", "primary": "Combine all the solutions", "result": "t=\\frac{\\arccos\\left(\\sqrt{\\frac{2}{3}}\\right)}{3.922}+\\frac{2πn}{3.922},\\:t=\\frac{2π}{3.922}-\\frac{\\arccos\\left(\\sqrt{\\frac{2}{3}}\\right)}{3.922}+\\frac{2πn}{3.922},\\:t=\\frac{\\arccos\\left(-\\sqrt{\\frac{2}{3}}\\right)}{3.922}+\\frac{2πn}{3.922},\\:t=-\\frac{\\arccos\\left(-\\sqrt{\\frac{2}{3}}\\right)}{3.922}+\\frac{2πn}{3.922}" }, { "type": "step", "primary": "Show solutions in decimal form", "result": "t=\\frac{0.61547…}{3.922}+\\frac{2πn}{3.922},\\:t=\\frac{2π}{3.922}-\\frac{0.61547…}{3.922}+\\frac{2πn}{3.922},\\:t=\\frac{2.52611…}{3.922}+\\frac{2πn}{3.922},\\:t=-\\frac{2.52611…}{3.922}+\\frac{2πn}{3.922}" } ], "meta": { "solvingClass": "Trig Equations", "practiceLink": "/practice/trigonometry-practice#area=main&subtopic=Trig%20Equations", "practiceTopic": "Trig Equations" } }, "plot_output": { "meta": { "plotInfo": { "variable": "t", "plotRequest": "0.08-0.12\\cos^{2}(3.922t)" }, "showViewLarger": true } }, "meta": { "showVerify": true } }