sin^2(x)=((4m-9))/8

{ "query": { "display": "$$\\sin^{2}\\left(x\\right)=\\frac{\\left(4m-9\\right)}{8}$$", "symbolab_question": "EQUATION#\\sin^{2}(x)=\\frac{(4m-9)}{8}" }, "solution": { "level": "PERFORMED", "subject": "Trigonometry", "topic": "Trig Equations", "subTopic": "Trig Equations", "default": "x=\\arcsin(\\frac{\\sqrt{2}\\sqrt{4m-9}}{4})+2πn,x=π+\\arcsin(-\\frac{\\sqrt{2}\\sqrt{4m-9}}{4})+2πn,x=\\arcsin(-\\frac{\\sqrt{2}\\sqrt{4m-9}}{4})+2πn,x=π+\\arcsin(\\frac{\\sqrt{2}\\sqrt{4m-9}}{4})+2πn", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\sin^{2}\\left(x\\right)=\\frac{\\left(4m-9\\right)}{8}{\\quad:\\quad}x=\\arcsin\\left(\\frac{\\sqrt{2}\\sqrt{4m-9}}{4}\\right)+2πn,\\:x=π+\\arcsin\\left(-\\frac{\\sqrt{2}\\sqrt{4m-9}}{4}\\right)+2πn,\\:x=\\arcsin\\left(-\\frac{\\sqrt{2}\\sqrt{4m-9}}{4}\\right)+2πn,\\:x=π+\\arcsin\\left(\\frac{\\sqrt{2}\\sqrt{4m-9}}{4}\\right)+2πn$$", "input": "\\sin^{2}\\left(x\\right)=\\frac{\\left(4m-9\\right)}{8}", "steps": [ { "type": "interim", "title": "Solve by substitution", "input": "\\sin^{2}\\left(x\\right)=\\frac{4m-9}{8}", "result": "\\sin\\left(x\\right)=\\frac{\\sqrt{2}\\sqrt{4m-9}}{4},\\:\\sin\\left(x\\right)=-\\frac{\\sqrt{2}\\sqrt{4m-9}}{4}", "steps": [ { "type": "step", "primary": "Let: $$\\sin\\left(x\\right)=u$$", "result": "u^{2}=\\frac{4m-9}{8}" }, { "type": "interim", "title": "$$u^{2}=\\frac{4m-9}{8}{\\quad:\\quad}u=\\frac{\\sqrt{2}\\sqrt{4m-9}}{4},\\:u=-\\frac{\\sqrt{2}\\sqrt{4m-9}}{4}$$", "input": "u^{2}=\\frac{4m-9}{8}", "steps": [ { "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{4m-9}{8}},\\:u=-\\sqrt{\\frac{4m-9}{8}}" }, { "type": "interim", "title": "Simplify $$\\sqrt{\\frac{4m-9}{8}}:{\\quad}\\frac{\\sqrt{2}\\sqrt{4m-9}}{4}$$", "input": "\\sqrt{\\frac{4m-9}{8}}", "steps": [ { "type": "step", "primary": "Apply radical rule: $$\\sqrt[n]{\\frac{a}{b}}=\\frac{\\sqrt[n]{a}}{\\sqrt[n]{b}},\\:\\quad$$ assuming $$a\\ge0,\\:b\\ge0$$", "result": "=\\frac{\\sqrt{4m-9}}{\\sqrt{8}}", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } }, { "type": "interim", "title": "$$\\sqrt{8}=2\\sqrt{2}$$", "input": "\\sqrt{8}", "result": "=\\frac{\\sqrt{4m-9}}{2\\sqrt{2}}", "steps": [ { "type": "interim", "title": "Prime factorization of $$8:{\\quad}2^{3}$$", "input": "8", "result": "=\\sqrt{2^{3}}", "steps": [ { "type": "step", "primary": "$$8\\:$$divides by $$2\\quad\\:8=4\\cdot\\:2$$", "result": "=2\\cdot\\:4" }, { "type": "step", "primary": "$$4\\:$$divides by $$2\\quad\\:4=2\\cdot\\:2$$", "result": "=2\\cdot\\:2\\cdot\\:2" }, { "type": "step", "primary": "$$2$$ is a prime number, therefore no further factorization is possible", "result": "=2\\cdot\\:2\\cdot\\:2" }, { "type": "step", "result": "=2^{3}" } ], "meta": { "solvingClass": "Composite Integer", "interimType": "Prime Fac 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xyoU2cWyPLDgE1QLLHeauuc3PHQdChPJ2JhfqHT+ZU0OMrfn8NOj0LUzuzje6xTyxRtXnBik8vDPXVw+nKWp28DI/y9DKGIPglJ+qMi9xDu2KE1OovxZAaXg7BtrFPk4UcCzRnGgMN6CYRfod7Mq0dp1RcgsS082tQWmOBW6FvhEw" } }, { "type": "step", "primary": "Apply exponent rule: $$a^{b+c}=a^b\\cdot\\:a^c$$", "result": "=\\sqrt{2^{2}\\cdot\\:2}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Apply radical rule: $$\\sqrt[n]{ab}=\\sqrt[n]{a}\\sqrt[n]{b}$$", "result": "=\\sqrt{2}\\sqrt{2^{2}}", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } }, { "type": "step", "primary": "Apply radical rule: $$\\sqrt[n]{a^n}=a$$", "secondary": [ "$$\\sqrt{2^{2}}=2$$" ], "result": "=2\\sqrt{2}", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } } ], "meta": { "interimType": "N/A" } }, { "type": "interim", "title": "Rationalize $$\\frac{\\sqrt{4m-9}}{2\\sqrt{2}}:{\\quad}\\frac{\\sqrt{2}\\sqrt{4m-9}}{4}$$", "input": "\\frac{\\sqrt{4m-9}}{2\\sqrt{2}}", "result": "=\\frac{\\sqrt{2}\\sqrt{4m-9}}{4}", "steps": [ { "type": "step", "primary": "Multiply by the conjugate $$\\frac{\\sqrt{2}}{\\sqrt{2}}$$", "result": "=\\frac{\\sqrt{4m-9}\\sqrt{2}}{2\\sqrt{2}\\sqrt{2}}", "meta": { "title": { "extension": "To rationalize the denominator, multiply numerator and denominator by the conjugate of the radical $$\\sqrt{2}$$" } } }, { "type": "interim", "title": "$$2\\sqrt{2}\\sqrt{2}=4$$", "input": "2\\sqrt{2}\\sqrt{2}", "result": "=\\frac{\\sqrt{2}\\sqrt{4m-9}}{4}", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$2\\sqrt{2}\\sqrt{2}=\\:2\\cdot\\:2^{\\frac{1}{2}}\\cdot\\:2^{\\frac{1}{2}}=\\:2^{1+\\frac{1}{2}+\\frac{1}{2}}$$" ], "result": "=2^{1+\\frac{1}{2}+\\frac{1}{2}}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add similar elements: $$\\frac{1}{2}+\\frac{1}{2}=2\\cdot\\:\\frac{1}{2}$$", "result": "=2^{1+2\\cdot\\:\\frac{1}{2}}" }, { "type": "interim", "title": "$$2\\cdot\\:\\frac{1}{2}=1$$", "input": "2\\cdot\\:\\frac{1}{2}", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{1\\cdot\\:2}{2}" }, { "type": "step", "primary": "Cancel the common factor: $$2$$", "result": "=1" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7/OsC643lXZbU+VEjF1qviunYaDZZLRLMVMbjt+IYzOarju+5Z51e/ZZSD3gRHwjBE9/03SOiEv+BIHutWLr6nTQhqRfFhpwnpyuz2TS2M3xMz+u325qtellRtF+nqNeo" } }, { "type": "step", "result": "=2^{1+1}" }, { "type": "step", "primary": "Refine", "result": "=4" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7mGTgoVo05Sts7XaxvZWn37c/1aV0UaN4psdFpdDCCJpwkKGJWEPFPk38sdJMsyPIpxDcd3ce1jrro8deQTJg9Dy5FZVhty+dm32gdTi9c/N3RwQhNxCnUEv+iF15JJUV" } } ], "meta": { "interimType": "Rationalize Title 1Eq" } } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "interim", "title": "Simplify $$-\\sqrt{\\frac{4m-9}{8}}:{\\quad}-\\frac{\\sqrt{2}\\sqrt{4m-9}}{4}$$", "input": "-\\sqrt{\\frac{4m-9}{8}}", "steps": [ { "type": "interim", "title": "Simplify $$\\sqrt{\\frac{4m-9}{8}}:{\\quad}\\frac{\\sqrt{4m-9}}{2\\sqrt{2}}$$", "input": "\\sqrt{\\frac{4m-9}{8}}", "result": "=-\\frac{\\sqrt{4m-9}}{2\\sqrt{2}}", "steps": [ { "type": "step", "primary": "Apply radical rule: $$\\sqrt[n]{\\frac{a}{b}}=\\frac{\\sqrt[n]{a}}{\\sqrt[n]{b}},\\:\\quad$$ assuming $$a\\ge0,\\:b\\ge0$$", "result": "=\\frac{\\sqrt{4m-9}}{\\sqrt{8}}", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } }, { "type": "interim", "title": "$$\\sqrt{8}=2\\sqrt{2}$$", "input": "\\sqrt{8}", "result": "=\\frac{\\sqrt{4m-9}}{2\\sqrt{2}}", "steps": [ { "type": "interim", "title": "Prime factorization of $$8:{\\quad}2^{3}$$", "input": "8", "result": "=\\sqrt{2^{3}}", "steps": [ { "type": "step", "primary": "$$8\\:$$divides by $$2\\quad\\:8=4\\cdot\\:2$$", "result": "=2\\cdot\\:4" }, { "type": "step", "primary": "$$4\\:$$divides by $$2\\quad\\:4=2\\cdot\\:2$$", "result": "=2\\cdot\\:2\\cdot\\:2" }, { "type": "step", "primary": "$$2$$ is a prime number, therefore no further factorization is possible", "result": "=2\\cdot\\:2\\cdot\\:2" }, { "type": "step", "result": "=2^{3}" } ], "meta": { "solvingClass": "Composite Integer", "interimType": "Prime Fac 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xyoU2cWyPLDgE1QLLHeauuc3PHQdChPJ2JhfqHT+ZU0OMrfn8NOj0LUzuzje6xTyxRtXnBik8vDPXVw+nKWp28DI/y9DKGIPglJ+qMi9xDu2KE1OovxZAaXg7BtrFPk4UcCzRnGgMN6CYRfod7Mq0dp1RcgsS082tQWmOBW6FvhEw" } }, { "type": "step", "primary": "Apply exponent rule: $$a^{b+c}=a^b\\cdot\\:a^c$$", "result": "=\\sqrt{2^{2}\\cdot\\:2}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Apply radical rule: $$\\sqrt[n]{ab}=\\sqrt[n]{a}\\sqrt[n]{b}$$", "result": "=\\sqrt{2}\\sqrt{2^{2}}", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } }, { "type": "step", "primary": "Apply radical rule: $$\\sqrt[n]{a^n}=a$$", "secondary": [ "$$\\sqrt{2^{2}}=2$$" ], "result": "=2\\sqrt{2}", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } } ], "meta": { "interimType": "N/A" } } ], "meta": { "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "interim", "title": "Rationalize $$-\\frac{\\sqrt{4m-9}}{2\\sqrt{2}}:{\\quad}-\\frac{\\sqrt{2}\\sqrt{4m-9}}{4}$$", "input": "-\\frac{\\sqrt{4m-9}}{2\\sqrt{2}}", "result": "=-\\frac{\\sqrt{2}\\sqrt{4m-9}}{4}", "steps": [ { "type": "step", "primary": "Multiply by the conjugate $$\\frac{\\sqrt{2}}{\\sqrt{2}}$$", "result": "=-\\frac{\\sqrt{4m-9}\\sqrt{2}}{2\\sqrt{2}\\sqrt{2}}", "meta": { "title": { "extension": "To rationalize the denominator, multiply numerator and denominator by the conjugate of the radical $$\\sqrt{2}$$" } } }, { "type": "interim", "title": "$$2\\sqrt{2}\\sqrt{2}=4$$", "input": "2\\sqrt{2}\\sqrt{2}", "result": "=-\\frac{\\sqrt{2}\\sqrt{4m-9}}{4}", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$2\\sqrt{2}\\sqrt{2}=\\:2\\cdot\\:2^{\\frac{1}{2}}\\cdot\\:2^{\\frac{1}{2}}=\\:2^{1+\\frac{1}{2}+\\frac{1}{2}}$$" ], "result": "=2^{1+\\frac{1}{2}+\\frac{1}{2}}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add similar elements: $$\\frac{1}{2}+\\frac{1}{2}=2\\cdot\\:\\frac{1}{2}$$", "result": "=2^{1+2\\cdot\\:\\frac{1}{2}}" }, { "type": "interim", "title": "$$2\\cdot\\:\\frac{1}{2}=1$$", "input": "2\\cdot\\:\\frac{1}{2}", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{1\\cdot\\:2}{2}" }, { "type": "step", "primary": "Cancel the common factor: $$2$$", "result": "=1" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7/OsC643lXZbU+VEjF1qviunYaDZZLRLMVMbjt+IYzOarju+5Z51e/ZZSD3gRHwjBE9/03SOiEv+BIHutWLr6nTQhqRfFhpwnpyuz2TS2M3xMz+u325qtellRtF+nqNeo" } }, { "type": "step", "result": "=2^{1+1}" }, { "type": "step", "primary": "Refine", "result": "=4" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7mGTgoVo05Sts7XaxvZWn37c/1aV0UaN4psdFpdDCCJpwkKGJWEPFPk38sdJMsyPIpxDcd3ce1jrro8deQTJg9Dy5FZVhty+dm32gdTi9c/N3RwQhNxCnUEv+iF15JJUV" } } ], "meta": { "interimType": "Rationalize Title 1Eq" } } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "result": "u=\\frac{\\sqrt{2}\\sqrt{4m-9}}{4},\\:u=-\\frac{\\sqrt{2}\\sqrt{4m-9}}{4}" } ], "meta": { "solvingClass": "Equations", "interimType": "Equations" } }, { "type": "step", "primary": "Substitute back $$u=\\sin\\left(x\\right)$$", "result": "\\sin\\left(x\\right)=\\frac{\\sqrt{2}\\sqrt{4m-9}}{4},\\:\\sin\\left(x\\right)=-\\frac{\\sqrt{2}\\sqrt{4m-9}}{4}" } ], "meta": { "interimType": "Substitution Method 0Eq" } }, { "type": "interim", "title": "$$\\sin\\left(x\\right)=\\frac{\\sqrt{2}\\sqrt{4m-9}}{4}{\\quad:\\quad}x=\\arcsin\\left(\\frac{\\sqrt{2}\\sqrt{4m-9}}{4}\\right)+2πn,\\:x=π+\\arcsin\\left(-\\frac{\\sqrt{2}\\sqrt{4m-9}}{4}\\right)+2πn$$", "input": "\\sin\\left(x\\right)=\\frac{\\sqrt{2}\\sqrt{4m-9}}{4}", "steps": [ { "type": "interim", "title": "Apply trig inverse properties", "input": "\\sin\\left(x\\right)=\\frac{\\sqrt{2}\\sqrt{4m-9}}{4}", "result": "x=\\arcsin\\left(\\frac{\\sqrt{2}\\sqrt{4m-9}}{4}\\right)+2πn,\\:x=π+\\arcsin\\left(-\\frac{\\sqrt{2}\\sqrt{4m-9}}{4}\\right)+2πn", "steps": [ { "type": "step", "primary": "General solutions for $$\\sin\\left(x\\right)=\\frac{\\sqrt{2}\\sqrt{4m-9}}{4}$$", "secondary": [ "$$\\sin\\left(x\\right)=a\\quad\\Rightarrow\\quad\\:x=\\arcsin\\left(a\\right)+2πn,\\:\\quad\\:x=π+\\arcsin\\left(a\\right)+2πn$$" ], "result": "x=\\arcsin\\left(\\frac{\\sqrt{2}\\sqrt{4m-9}}{4}\\right)+2πn,\\:x=π+\\arcsin\\left(-\\frac{\\sqrt{2}\\sqrt{4m-9}}{4}\\right)+2πn" } ], "meta": { "interimType": "Trig Apply Inverse Props 0Eq" } } ], "meta": { "interimType": "N/A" } }, { "type": "interim", "title": "$$\\sin\\left(x\\right)=-\\frac{\\sqrt{2}\\sqrt{4m-9}}{4}{\\quad:\\quad}x=\\arcsin\\left(-\\frac{\\sqrt{2}\\sqrt{4m-9}}{4}\\right)+2πn,\\:x=π+\\arcsin\\left(\\frac{\\sqrt{2}\\sqrt{4m-9}}{4}\\right)+2πn$$", "input": "\\sin\\left(x\\right)=-\\frac{\\sqrt{2}\\sqrt{4m-9}}{4}", "steps": [ { "type": "interim", "title": "Apply trig inverse properties", "input": "\\sin\\left(x\\right)=-\\frac{\\sqrt{2}\\sqrt{4m-9}}{4}", "result": "x=\\arcsin\\left(-\\frac{\\sqrt{2}\\sqrt{4m-9}}{4}\\right)+2πn,\\:x=π+\\arcsin\\left(\\frac{\\sqrt{2}\\sqrt{4m-9}}{4}\\right)+2πn", "steps": [ { "type": "step", "primary": "General solutions for $$\\sin\\left(x\\right)=-\\frac{\\sqrt{2}\\sqrt{4m-9}}{4}$$", "secondary": [ "$$\\sin\\left(x\\right)=a\\quad\\Rightarrow\\quad\\:x=\\arcsin\\left(a\\right)+2πn,\\:\\quad\\:x=π+\\arcsin\\left(a\\right)+2πn$$" ], "result": "x=\\arcsin\\left(-\\frac{\\sqrt{2}\\sqrt{4m-9}}{4}\\right)+2πn,\\:x=π+\\arcsin\\left(\\frac{\\sqrt{2}\\sqrt{4m-9}}{4}\\right)+2πn" } ], "meta": { "interimType": "Trig Apply Inverse Props 0Eq" } } ], "meta": { "interimType": "N/A" } }, { "type": "step", "primary": "Combine all the solutions", "result": "x=\\arcsin\\left(\\frac{\\sqrt{2}\\sqrt{4m-9}}{4}\\right)+2πn,\\:x=π+\\arcsin\\left(-\\frac{\\sqrt{2}\\sqrt{4m-9}}{4}\\right)+2πn,\\:x=\\arcsin\\left(-\\frac{\\sqrt{2}\\sqrt{4m-9}}{4}\\right)+2πn,\\:x=π+\\arcsin\\left(\\frac{\\sqrt{2}\\sqrt{4m-9}}{4}\\right)+2πn" } ], "meta": { "solvingClass": "Trig Equations", "practiceLink": "/practice/trigonometry-practice#area=main&subtopic=Trig%20Equations", "practiceTopic": "Trig Equations" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "\\sin^{2}(x)=\\frac{(4m-9)}{8}" }, "showViewLarger": true } }, "meta": { "showVerify": true } }