3csc^2(x)+cot^2(x)-4=0

{ "query": { "display": "$$3\\csc^{2}\\left(x\\right)+\\cot^{2}\\left(x\\right)-4=0$$", "symbolab_question": "EQUATION#3\\csc^{2}(x)+\\cot^{2}(x)-4=0" }, "solution": { "level": "PERFORMED", "subject": "Trigonometry", "topic": "Trig Equations", "subTopic": "Trig Equations", "default": "x=1.10714…+2πn,x=π-1.10714…+2πn,x=-1.10714…+2πn,x=π+1.10714…+2πn", "degrees": "x=63.43494…^{\\circ }+360^{\\circ }n,x=116.56505…^{\\circ }+360^{\\circ }n,x=-63.43494…^{\\circ }+360^{\\circ }n,x=243.43494…^{\\circ }+360^{\\circ }n", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$3\\csc^{2}\\left(x\\right)+\\cot^{2}\\left(x\\right)-4=0{\\quad:\\quad}x=1.10714…+2πn,\\:x=π-1.10714…+2πn,\\:x=-1.10714…+2πn,\\:x=π+1.10714…+2πn$$", "input": "3\\csc^{2}\\left(x\\right)+\\cot^{2}\\left(x\\right)-4=0", "steps": [ { "type": "interim", "title": "Rewrite using trig identities", "input": "-4+\\cot^{2}\\left(x\\right)+3\\csc^{2}\\left(x\\right)", "result": "-5+4\\csc^{2}\\left(x\\right)=0", "steps": [ { "type": "step", "primary": "Use the Pythagorean identity: $$1+\\cot^{2}\\left(x\\right)=\\csc^{2}\\left(x\\right)$$", "secondary": [ "$$\\cot^{2}\\left(x\\right)=\\csc^{2}\\left(x\\right)-1$$" ], "result": "=-4+\\csc^{2}\\left(x\\right)-1+3\\csc^{2}\\left(x\\right)" }, { "type": "interim", "title": "Simplify $$-4+\\csc^{2}\\left(x\\right)-1+3\\csc^{2}\\left(x\\right):{\\quad}4\\csc^{2}\\left(x\\right)-5$$", "input": "-4+\\csc^{2}\\left(x\\right)-1+3\\csc^{2}\\left(x\\right)", "result": "=4\\csc^{2}\\left(x\\right)-5", "steps": [ { "type": "step", "primary": "Group like terms", "result": "=\\csc^{2}\\left(x\\right)+3\\csc^{2}\\left(x\\right)-4-1" }, { "type": "step", "primary": "Add similar elements: $$\\csc^{2}\\left(x\\right)+3\\csc^{2}\\left(x\\right)=4\\csc^{2}\\left(x\\right)$$", "result": "=4\\csc^{2}\\left(x\\right)-4-1" }, { "type": "step", "primary": "Subtract the numbers: $$-4-1=-5$$", "result": "=4\\csc^{2}\\left(x\\right)-5" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s72EY6ivuWcmx2LPQ9bPCM2SWDBAb9YlOLrgm2VLr7cIrTLx8mOdHYVzxX643JqKFIvTZkYD4gUeSoFrhGGhxq92JsNf7vbpFZFGLmSqyJ3FhFKk3fejFkyiOiq9iG9IkAIuSmJUDtLxRPyJ57Nkq7qsVno+6g9Xt721n1DXKlQZpCxvE6+nyyc/2wAGs8Ng3BvzIPeEtDfcHv/z8uls8Teg==" } } ], "meta": { "interimType": "Trig Rewrite Using Trig identities 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7L3hjgZFKTWfrMkuwjPw7a57tpnS6XdAsu/VTdHkmRyNBjGZ+c+wIWHxP5VQdJDji8wQyZ2Br8agbY8qkLUt6WPm0tu7qq7Iur/j8bUReEEOq5I4KKx/r65u7kTSvPypsTAGfLXhvDxHSrKO+Eewjma9DUyifL3JKhVgQZPPaAlarve3E7cDlwD8G9VYfu6duTZGH8SqHtUWuYa2dpw0bulhsd89TDzuC3BUug+Pj7eQ=" } }, { "type": "interim", "title": "Solve by substitution", "input": "-5+4\\csc^{2}\\left(x\\right)=0", "result": "\\csc\\left(x\\right)=\\frac{\\sqrt{5}}{2},\\:\\csc\\left(x\\right)=-\\frac{\\sqrt{5}}{2}", "steps": [ { "type": "step", "primary": "Let: $$\\csc\\left(x\\right)=u$$", "result": "-5+4u^{2}=0" }, { "type": "interim", "title": "$$-5+4u^{2}=0{\\quad:\\quad}u=\\frac{\\sqrt{5}}{2},\\:u=-\\frac{\\sqrt{5}}{2}$$", "input": "-5+4u^{2}=0", "steps": [ { "type": "interim", "title": "Move $$5\\:$$to the right side", "input": "-5+4u^{2}=0", "result": "4u^{2}=5", "steps": [ { "type": "step", "primary": "Add $$5$$ to both sides", "result": "-5+4u^{2}+5=0+5" }, { "type": "step", "primary": "Simplify", "result": "4u^{2}=5" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": "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" } }, { "type": "interim", "title": "Divide both sides by $$4$$", "input": "4u^{2}=5", "result": "u^{2}=\\frac{5}{4}", "steps": [ { "type": "step", "primary": "Divide both sides by $$4$$", "result": "\\frac{4u^{2}}{4}=\\frac{5}{4}" }, { "type": "step", "primary": "Simplify", "result": "u^{2}=\\frac{5}{4}" } ], "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{5}{4}},\\:u=-\\sqrt{\\frac{5}{4}}" }, { "type": "interim", "title": "$$\\sqrt{\\frac{5}{4}}=\\frac{\\sqrt{5}}{2}$$", "input": "\\sqrt{\\frac{5}{4}}", "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{5}}{\\sqrt{4}}", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } }, { "type": "interim", "title": "$$\\sqrt{4}=2$$", "input": "\\sqrt{4}", "result": "=\\frac{\\sqrt{5}}{2}", "steps": [ { "type": "step", "primary": "Factor the number: $$4=2^{2}$$", "result": "=\\sqrt{2^{2}}" }, { "type": "step", "primary": "Apply radical rule: $$\\sqrt[n]{a^n}=a$$", "secondary": [ "$$\\sqrt{2^{2}}=2$$" ], "result": "=2", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } } ], "meta": { "interimType": "N/A" } } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7FJIkkmi1CWuhEmDQGlA0N2ac2b92HIU7JT2fammBuwCrju+5Z51e/ZZSD3gRHwjBnvjDY21b5XBQ44AG3rKeoG/SCkwwUCGKnKbtgoSHij2Ky+4oc4WkS69iOkw+tuw7LWZ4Xe/1PBsqIb6VlQjpP53u2ZL8N7AJVbuPKwxBqwuwiNrEngO+NNvZ9sqNu+2V" } }, { "type": "interim", "title": "$$-\\sqrt{\\frac{5}{4}}=-\\frac{\\sqrt{5}}{2}$$", "input": "-\\sqrt{\\frac{5}{4}}", "steps": [ { "type": "interim", "title": "Simplify $$\\sqrt{\\frac{5}{4}}:{\\quad}\\frac{\\sqrt{5}}{2}$$", "input": "\\sqrt{\\frac{5}{4}}", "result": "=-\\frac{\\sqrt{5}}{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{5}}{\\sqrt{4}}", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } }, { "type": "interim", "title": "$$\\sqrt{4}=2$$", "input": "\\sqrt{4}", "result": "=\\frac{\\sqrt{5}}{2}", "steps": [ { "type": "step", "primary": "Factor the number: $$4=2^{2}$$", "result": "=\\sqrt{2^{2}}" }, { "type": "step", "primary": "Apply radical rule: $$\\sqrt[n]{a^n}=a$$", "secondary": [ "$$\\sqrt{2^{2}}=2$$" ], "result": "=2", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } } ], "meta": { "interimType": "N/A" } } ], "meta": { "interimType": "Algebraic Manipulation Simplify Title 1Eq" } } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7xVDfakA4g1ZczTHh+Arxm9QeV72FIQzMkoGy2TWAl2gJQJZuTAY5js+oqjdT8kslDa1hUvtzgBcwD7zey8pjXxhv0DD+Guy21dttppVeU0BrXh3OSOXhogkv14GjeoU6U+eSMWjs3p8KfHPIAu9XC+3Qn1PQ3Sk29dmk8nD+rYFurCc0v6+SezXpJu+Sddf4" } }, { "type": "step", "result": "u=\\frac{\\sqrt{5}}{2},\\:u=-\\frac{\\sqrt{5}}{2}" } ], "meta": { "solvingClass": "Equations", "interimType": "Equations" } }, { "type": "step", "primary": "Substitute back $$u=\\csc\\left(x\\right)$$", "result": "\\csc\\left(x\\right)=\\frac{\\sqrt{5}}{2},\\:\\csc\\left(x\\right)=-\\frac{\\sqrt{5}}{2}" } ], "meta": { "interimType": "Substitution Method 0Eq" } }, { "type": "interim", "title": "$$\\csc\\left(x\\right)=\\frac{\\sqrt{5}}{2}{\\quad:\\quad}x=\\arccsc\\left(\\frac{\\sqrt{5}}{2}\\right)+2πn,\\:x=π-\\arccsc\\left(\\frac{\\sqrt{5}}{2}\\right)+2πn$$", "input": "\\csc\\left(x\\right)=\\frac{\\sqrt{5}}{2}", "steps": [ { "type": "interim", "title": "Apply trig inverse properties", "input": "\\csc\\left(x\\right)=\\frac{\\sqrt{5}}{2}", "result": "x=\\arccsc\\left(\\frac{\\sqrt{5}}{2}\\right)+2πn,\\:x=π-\\arccsc\\left(\\frac{\\sqrt{5}}{2}\\right)+2πn", "steps": [ { "type": "step", "primary": "General solutions for $$\\csc\\left(x\\right)=\\frac{\\sqrt{5}}{2}$$", "secondary": [ "$$\\csc\\left(x\\right)=a\\quad\\Rightarrow\\quad\\:x=\\arccsc\\left(a\\right)+2πn,\\:\\quad\\:x=π-\\arccsc\\left(a\\right)+2πn$$" ], "result": "x=\\arccsc\\left(\\frac{\\sqrt{5}}{2}\\right)+2πn,\\:x=π-\\arccsc\\left(\\frac{\\sqrt{5}}{2}\\right)+2πn" } ], "meta": { "interimType": "Trig Apply Inverse Props 0Eq" } } ], "meta": { "interimType": "N/A" } }, { "type": "interim", "title": "$$\\csc\\left(x\\right)=-\\frac{\\sqrt{5}}{2}{\\quad:\\quad}x=\\arccsc\\left(-\\frac{\\sqrt{5}}{2}\\right)+2πn,\\:x=π+\\arccsc\\left(\\frac{\\sqrt{5}}{2}\\right)+2πn$$", "input": "\\csc\\left(x\\right)=-\\frac{\\sqrt{5}}{2}", "steps": [ { "type": "interim", "title": "Apply trig inverse properties", "input": "\\csc\\left(x\\right)=-\\frac{\\sqrt{5}}{2}", "result": "x=\\arccsc\\left(-\\frac{\\sqrt{5}}{2}\\right)+2πn,\\:x=π+\\arccsc\\left(\\frac{\\sqrt{5}}{2}\\right)+2πn", "steps": [ { "type": "step", "primary": "General solutions for $$\\csc\\left(x\\right)=-\\frac{\\sqrt{5}}{2}$$", "secondary": [ "$$\\csc\\left(x\\right)=-a\\quad\\Rightarrow\\quad\\:x=\\arccsc\\left(-a\\right)+2πn,\\:\\quad\\:x=π+\\arccsc\\left(a\\right)+2πn$$" ], "result": "x=\\arccsc\\left(-\\frac{\\sqrt{5}}{2}\\right)+2πn,\\:x=π+\\arccsc\\left(\\frac{\\sqrt{5}}{2}\\right)+2πn" } ], "meta": { "interimType": "Trig Apply Inverse Props 0Eq" } } ], "meta": { "interimType": "N/A" } }, { "type": "step", "primary": "Combine all the solutions", "result": "x=\\arccsc\\left(\\frac{\\sqrt{5}}{2}\\right)+2πn,\\:x=π-\\arccsc\\left(\\frac{\\sqrt{5}}{2}\\right)+2πn,\\:x=\\arccsc\\left(-\\frac{\\sqrt{5}}{2}\\right)+2πn,\\:x=π+\\arccsc\\left(\\frac{\\sqrt{5}}{2}\\right)+2πn" }, { "type": "step", "primary": "Show solutions in decimal form", "result": "x=1.10714…+2πn,\\:x=π-1.10714…+2πn,\\:x=-1.10714…+2πn,\\:x=π+1.10714…+2πn" } ], "meta": { "solvingClass": "Trig Equations", "practiceLink": "/practice/trigonometry-practice#area=main&subtopic=Trig%20Equations", "practiceTopic": "Trig Equations" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "3\\csc^{2}(x)+\\cot^{2}(x)-4" }, "showViewLarger": true } }, "meta": { "showVerify": true } }