sin(x)= 5/(sqrt(29))

{ "query": { "display": "$$\\sin\\left(x\\right)=\\frac{5}{\\sqrt{29}}$$", "symbolab_question": "EQUATION#\\sin(x)=\\frac{5}{\\sqrt{29}}" }, "solution": { "level": "PERFORMED", "subject": "Trigonometry", "topic": "Trig Equations", "subTopic": "Trig Equations", "default": "x=1.19028…+2πn,x=π-1.19028…+2πn", "radians": "x=1.19028…+6.28318…n,x=π-1.19028…+6.28318…n", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\sin\\left(x\\right)=\\frac{5}{\\sqrt{29}}{\\quad:\\quad}x=1.19028…+2πn,\\:x=π-1.19028…+2πn$$", "input": "\\sin\\left(x\\right)=\\frac{5}{\\sqrt{29}}", "steps": [ { "type": "interim", "title": "Simplify $$\\frac{5}{\\sqrt{29}}:{\\quad}\\frac{5\\sqrt{29}}{29}$$", "input": "\\frac{5}{\\sqrt{29}}", "steps": [ { "type": "step", "primary": "Multiply by the conjugate $$\\frac{\\sqrt{29}}{\\sqrt{29}}$$", "result": "=\\frac{5\\sqrt{29}}{\\sqrt{29}\\sqrt{29}}", "meta": { "title": { "extension": "To rationalize the denominator, multiply numerator and denominator by the conjugate of the radical $$\\sqrt{29}$$" } } }, { "type": "interim", "title": "$$\\sqrt{29}\\sqrt{29}=29$$", "input": "\\sqrt{29}\\sqrt{29}", "result": "=\\frac{5\\sqrt{29}}{29}", "steps": [ { "type": "step", "primary": "Apply radical rule: $$\\sqrt{a}\\sqrt{a}=a$$", "secondary": [ "$$\\sqrt{29}\\sqrt{29}=29$$" ], "result": "=29", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7HIUz0ZrCw0IcSX7ELREfng3T8l0DLehSgjG8HuZqSOurju+5Z51e/ZZSD3gRHwjBczg8josUq56WkH8jc8dl8Vau/z08uT3+ptxweKstxOUakf2lbi2o+kE8ueSo7Iw7" } }, { "type": "step", "result": "\\sin\\left(x\\right)=\\frac{5\\sqrt{29}}{29}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s77Si2C3qEpH3IWqnVcEkiz1H2zKkfByZqxn8Hez08UlMJQJZuTAY5js+oqjdT8ksl6plhBSPl+zW+sGaAL6SjNTH1Q3wwsSlKfhjxILmh6ZOBBTEk/JQ2cZ9WKuRzClU7QG+dQXHhPmuaaYlTGyzk2YdvBBpvpzIdjx034jlPcHm10PVfxMZZ26cDb5wZR/k9" } }, { "type": "interim", "title": "Apply trig inverse properties", "input": "\\sin\\left(x\\right)=\\frac{5\\sqrt{29}}{29}", "result": "x=\\arcsin\\left(\\frac{5\\sqrt{29}}{29}\\right)+2πn,\\:x=π-\\arcsin\\left(\\frac{5\\sqrt{29}}{29}\\right)+2πn", "steps": [ { "type": "step", "primary": "General solutions for $$\\sin\\left(x\\right)=\\frac{5\\sqrt{29}}{29}$$", "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{5\\sqrt{29}}{29}\\right)+2πn,\\:x=π-\\arcsin\\left(\\frac{5\\sqrt{29}}{29}\\right)+2πn" } ], "meta": { "interimType": "Trig Apply Inverse Props 0Eq" } }, { "type": "step", "primary": "Show solutions in decimal form", "result": "x=1.19028…+2πn,\\:x=π-1.19028…+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(x)-\\frac{5}{\\sqrt{29}}" }, "showViewLarger": true } }, "meta": { "showVerify": true } }