2sec^2(a)+tan^2(a)=3

{ "query": { "display": "$$2\\sec^{2}\\left(a\\right)+\\tan^{2}\\left(a\\right)=3$$", "symbolab_question": "EQUATION#2\\sec^{2}(a)+\\tan^{2}(a)=3" }, "solution": { "level": "PERFORMED", "subject": "Trigonometry", "topic": "Trig Equations", "subTopic": "Trig Equations", "default": "a=\\frac{π}{6}+2πn,a=\\frac{11π}{6}+2πn,a=\\frac{5π}{6}+2πn,a=\\frac{7π}{6}+2πn", "degrees": "a=30^{\\circ }+360^{\\circ }n,a=330^{\\circ }+360^{\\circ }n,a=150^{\\circ }+360^{\\circ }n,a=210^{\\circ }+360^{\\circ }n", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$2\\sec^{2}\\left(a\\right)+\\tan^{2}\\left(a\\right)=3{\\quad:\\quad}a=\\frac{π}{6}+2πn,\\:a=\\frac{11π}{6}+2πn,\\:a=\\frac{5π}{6}+2πn,\\:a=\\frac{7π}{6}+2πn$$", "input": "2\\sec^{2}\\left(a\\right)+\\tan^{2}\\left(a\\right)=3", "steps": [ { "type": "step", "primary": "Subtract $$3$$ from both sides", "result": "2\\sec^{2}\\left(a\\right)+\\tan^{2}\\left(a\\right)-3=0" }, { "type": "interim", "title": "Rewrite using trig identities", "input": "-3+\\tan^{2}\\left(a\\right)+2\\sec^{2}\\left(a\\right)", "result": "-4+3\\sec^{2}\\left(a\\right)=0", "steps": [ { "type": "step", "primary": "Use the Pythagorean identity: $$\\tan^{2}\\left(x\\right)+1=\\sec^{2}\\left(x\\right)$$", "secondary": [ "$$\\tan^{2}\\left(x\\right)=\\sec^{2}\\left(x\\right)-1$$" ], "result": "=-3+\\sec^{2}\\left(a\\right)-1+2\\sec^{2}\\left(a\\right)" }, { "type": "interim", "title": "Simplify $$-3+\\sec^{2}\\left(a\\right)-1+2\\sec^{2}\\left(a\\right):{\\quad}3\\sec^{2}\\left(a\\right)-4$$", "input": "-3+\\sec^{2}\\left(a\\right)-1+2\\sec^{2}\\left(a\\right)", "result": "=3\\sec^{2}\\left(a\\right)-4", "steps": [ { "type": "step", "primary": "Group like terms", "result": "=\\sec^{2}\\left(a\\right)+2\\sec^{2}\\left(a\\right)-3-1" }, { "type": "step", "primary": "Add similar elements: $$\\sec^{2}\\left(a\\right)+2\\sec^{2}\\left(a\\right)=3\\sec^{2}\\left(a\\right)$$", "result": "=3\\sec^{2}\\left(a\\right)-3-1" }, { "type": "step", "primary": "Subtract the numbers: $$-3-1=-4$$", "result": "=3\\sec^{2}\\left(a\\right)-4" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7UgklGu6MHENMdvR89Sn2jOA76o+2Fsiksz3EnU2UUKzTLx8mOdHYVzxX643JqKFIxQEDtqOROE0LMgYioHJKEz+8CXcMzyL3NKDr/Vsa6phFKk3fejFkyiOiq9iG9IkAIuSmJUDtLxRPyJ57Nkq7qqFWb29NMgjN+IrYTYzlmJ7FjvPquMWChWq/mbRQnkdAvzIPeEtDfcHv/z8uls8Teg==" } } ], "meta": { "interimType": "Trig Rewrite Using Trig identities 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7A2ndqzl6s8qYyDCrnag1bRM8wUMPc+HnPfm8PEnw/jrKrZyZu9mb9/mRlogZMpBC8wQyZ2Br8agbY8qkLUt6WPm0tu7qq7Iur/j8bUReEEOq5I4KKx/r65u7kTSvPypspi+a+CkvtcH9io/Zp3P8unNYIbtIWdPep0QwdS7Fvsurve3E7cDlwD8G9VYfu6duTZGH8SqHtUWuYa2dpw0bulhsd89TDzuC3BUug+Pj7eQ=" } }, { "type": "interim", "title": "Solve by substitution", "input": "-4+3\\sec^{2}\\left(a\\right)=0", "result": "\\sec\\left(a\\right)=\\frac{2\\sqrt{3}}{3},\\:\\sec\\left(a\\right)=-\\frac{2\\sqrt{3}}{3}", "steps": [ { "type": "step", "primary": "Let: $$\\sec\\left(a\\right)=u$$", "result": "-4+3u^{2}=0" }, { "type": "interim", "title": "$$-4+3u^{2}=0{\\quad:\\quad}u=\\frac{2\\sqrt{3}}{3},\\:u=-\\frac{2\\sqrt{3}}{3}$$", "input": "-4+3u^{2}=0", "steps": [ { "type": "interim", "title": "Move $$4\\:$$to the right side", "input": "-4+3u^{2}=0", "result": "3u^{2}=4", "steps": [ { "type": "step", "primary": "Add $$4$$ to both sides", "result": "-4+3u^{2}+4=0+4" }, { "type": "step", "primary": "Simplify", "result": "3u^{2}=4" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": "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" } }, { "type": "interim", "title": "Divide both sides by $$3$$", "input": "3u^{2}=4", "result": "u^{2}=\\frac{4}{3}", "steps": [ { "type": "step", "primary": "Divide both sides by $$3$$", "result": "\\frac{3u^{2}}{3}=\\frac{4}{3}" }, { "type": "step", "primary": "Simplify", "result": "u^{2}=\\frac{4}{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{4}{3}},\\:u=-\\sqrt{\\frac{4}{3}}" }, { "type": "interim", "title": "$$\\sqrt{\\frac{4}{3}}=\\frac{2\\sqrt{3}}{3}$$", "input": "\\sqrt{\\frac{4}{3}}", "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{4}}{\\sqrt{3}}", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } }, { "type": "interim", "title": "$$\\sqrt{4}=2$$", "input": "\\sqrt{4}", "result": "=\\frac{2}{\\sqrt{3}}", "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" } }, { "type": "interim", "title": "Rationalize $$\\frac{2}{\\sqrt{3}}:{\\quad}\\frac{2\\sqrt{3}}{3}$$", "input": "\\frac{2}{\\sqrt{3}}", "result": "=\\frac{2\\sqrt{3}}{3}", "steps": [ { "type": "step", "primary": "Multiply by the conjugate $$\\frac{\\sqrt{3}}{\\sqrt{3}}$$", "result": "=\\frac{2\\sqrt{3}}{\\sqrt{3}\\sqrt{3}}", "meta": { "title": { "extension": "To rationalize the denominator, multiply numerator and denominator by the conjugate of the radical $$\\sqrt{3}$$" } } }, { "type": "interim", "title": "$$\\sqrt{3}\\sqrt{3}=3$$", "input": "\\sqrt{3}\\sqrt{3}", "result": "=\\frac{2\\sqrt{3}}{3}", "steps": [ { "type": "step", "primary": "Apply radical rule: $$\\sqrt{a}\\sqrt{a}=a$$", "secondary": [ "$$\\sqrt{3}\\sqrt{3}=3$$" ], "result": "=3", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7KTpmXttz9tzhFROsvdfdpl9t0rGzXfqIeCX9GJLyBFHMwViaLUXkeD+JukROhWdjqqRb86vK1jAPxwJmTBc7jxuwBfr4jqpPqXmFXXvxZ1uqe0B1E2GM2M0eTcXoyACf" } } ], "meta": { "interimType": "Rationalize Title 1Eq" } } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7FJIkkmi1CWuhEmDQGlA0N4P4i7rn7kQdPkXBtoGBetGrju+5Z51e/ZZSD3gRHwjBFwUvfj7MBkZSw9utD6uBwCqRWAnthMgJzhBiWt2FFSYwwBGXNBWz4bN2v1kzlVrruZ41Mpl4cVRd9z4etrBgHNONRoG/W96wFGMG7eB2COBXZeNU0R6SHwm0bdlcOVNB" } }, { "type": "interim", "title": "$$-\\sqrt{\\frac{4}{3}}=-\\frac{2\\sqrt{3}}{3}$$", "input": "-\\sqrt{\\frac{4}{3}}", "steps": [ { "type": "interim", "title": "Simplify $$\\sqrt{\\frac{4}{3}}:{\\quad}\\frac{2}{\\sqrt{3}}$$", "input": "\\sqrt{\\frac{4}{3}}", "result": "=-\\frac{2}{\\sqrt{3}}", "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{4}}{\\sqrt{3}}", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } }, { "type": "interim", "title": "$$\\sqrt{4}=2$$", "input": "\\sqrt{4}", "result": "=\\frac{2}{\\sqrt{3}}", "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" } }, { "type": "interim", "title": "Rationalize $$-\\frac{2}{\\sqrt{3}}:{\\quad}-\\frac{2\\sqrt{3}}{3}$$", "input": "-\\frac{2}{\\sqrt{3}}", "result": "=-\\frac{2\\sqrt{3}}{3}", "steps": [ { "type": "step", "primary": "Multiply by the conjugate $$\\frac{\\sqrt{3}}{\\sqrt{3}}$$", "result": "=-\\frac{2\\sqrt{3}}{\\sqrt{3}\\sqrt{3}}", "meta": { "title": { "extension": "To rationalize the denominator, multiply numerator and denominator by the conjugate of the radical $$\\sqrt{3}$$" } } }, { "type": "interim", "title": "$$\\sqrt{3}\\sqrt{3}=3$$", "input": "\\sqrt{3}\\sqrt{3}", "result": "=-\\frac{2\\sqrt{3}}{3}", "steps": [ { "type": "step", "primary": "Apply radical rule: $$\\sqrt{a}\\sqrt{a}=a$$", "secondary": [ "$$\\sqrt{3}\\sqrt{3}=3$$" ], "result": "=3", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7KTpmXttz9tzhFROsvdfdpl9t0rGzXfqIeCX9GJLyBFHMwViaLUXkeD+JukROhWdjqqRb86vK1jAPxwJmTBc7jxuwBfr4jqpPqXmFXXvxZ1uqe0B1E2GM2M0eTcXoyACf" } } ], "meta": { "interimType": "Rationalize Title 1Eq" } } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7xVDfakA4g1ZczTHh+Arxm3d5ib5iZU80LXX/Sy5aO94JQJZuTAY5js+oqjdT8ksl3SW91Kyp3H5B/zOUV5Tbk34APy6SRyb9uqPXtO6MrHZknSRozsjPuvLfbiOO4vVlZmFe0Q+dO6iAefQwiP4W4xyIib/o5w2zV927TVIlN9fMFpomwc7sL5G+bzGnxA1h" } }, { "type": "step", "result": "u=\\frac{2\\sqrt{3}}{3},\\:u=-\\frac{2\\sqrt{3}}{3}" } ], "meta": { "solvingClass": "Equations", "interimType": "Equations" } }, { "type": "step", "primary": "Substitute back $$u=\\sec\\left(a\\right)$$", "result": "\\sec\\left(a\\right)=\\frac{2\\sqrt{3}}{3},\\:\\sec\\left(a\\right)=-\\frac{2\\sqrt{3}}{3}" } ], "meta": { "interimType": "Substitution Method 0Eq" } }, { "type": "interim", "title": "$$\\sec\\left(a\\right)=\\frac{2\\sqrt{3}}{3}{\\quad:\\quad}a=\\frac{π}{6}+2πn,\\:a=\\frac{11π}{6}+2πn$$", "input": "\\sec\\left(a\\right)=\\frac{2\\sqrt{3}}{3}", "steps": [ { "type": "interim", "title": "General solutions for $$\\sec\\left(a\\right)=\\frac{2\\sqrt{3}}{3}$$", "result": "a=\\frac{π}{6}+2πn,\\:a=\\frac{11π}{6}+2πn", "steps": [ { "type": "step", "primary": "$$\\sec\\left(x\\right)$$ periodicity table with $$2πn$$ cycle:<br/>$$\\begin{array}{|c|c|c|c|}\\hline x&\\sec(x)&x&\\sec(x)\\\\\\hline 0&1&π&-1\\\\\\hline \\frac{π}{6}&\\frac{2\\sqrt{3}}{3}&\\frac{7π}{6}&-\\frac{2\\sqrt{3}}{3}\\\\\\hline \\frac{π}{4}&\\sqrt{2}&\\frac{5π}{4}&-\\sqrt{2}\\\\\\hline \\frac{π}{3}&2&\\frac{4π}{3}&-2\\\\\\hline \\frac{π}{2}&\\mathrm{Undefined}&\\frac{3π}{2}&\\mathrm{Undefined}\\\\\\hline \\frac{2π}{3}&-2&\\frac{5π}{3}&2\\\\\\hline \\frac{3π}{4}&-\\sqrt{2}&\\frac{7π}{4}&\\sqrt{2}\\\\\\hline \\frac{5π}{6}&-\\frac{2\\sqrt{3}}{3}&\\frac{11π}{6}&\\frac{2\\sqrt{3}}{3}\\\\\\hline \\end{array}$$" }, { "type": "step", "result": "a=\\frac{π}{6}+2πn,\\:a=\\frac{11π}{6}+2πn" } ], "meta": { "interimType": "Trig General Solutions sec 1Eq" } } ], "meta": { "interimType": "N/A" } }, { "type": "interim", "title": "$$\\sec\\left(a\\right)=-\\frac{2\\sqrt{3}}{3}{\\quad:\\quad}a=\\frac{5π}{6}+2πn,\\:a=\\frac{7π}{6}+2πn$$", "input": "\\sec\\left(a\\right)=-\\frac{2\\sqrt{3}}{3}", "steps": [ { "type": "interim", "title": "General solutions for $$\\sec\\left(a\\right)=-\\frac{2\\sqrt{3}}{3}$$", "result": "a=\\frac{5π}{6}+2πn,\\:a=\\frac{7π}{6}+2πn", "steps": [ { "type": "step", "primary": "$$\\sec\\left(x\\right)$$ periodicity table with $$2πn$$ cycle:<br/>$$\\begin{array}{|c|c|c|c|}\\hline x&\\sec(x)&x&\\sec(x)\\\\\\hline 0&1&π&-1\\\\\\hline \\frac{π}{6}&\\frac{2\\sqrt{3}}{3}&\\frac{7π}{6}&-\\frac{2\\sqrt{3}}{3}\\\\\\hline \\frac{π}{4}&\\sqrt{2}&\\frac{5π}{4}&-\\sqrt{2}\\\\\\hline \\frac{π}{3}&2&\\frac{4π}{3}&-2\\\\\\hline \\frac{π}{2}&\\mathrm{Undefined}&\\frac{3π}{2}&\\mathrm{Undefined}\\\\\\hline \\frac{2π}{3}&-2&\\frac{5π}{3}&2\\\\\\hline \\frac{3π}{4}&-\\sqrt{2}&\\frac{7π}{4}&\\sqrt{2}\\\\\\hline \\frac{5π}{6}&-\\frac{2\\sqrt{3}}{3}&\\frac{11π}{6}&\\frac{2\\sqrt{3}}{3}\\\\\\hline \\end{array}$$" }, { "type": "step", "result": "a=\\frac{5π}{6}+2πn,\\:a=\\frac{7π}{6}+2πn" } ], "meta": { "interimType": "Trig General Solutions sec 1Eq" } } ], "meta": { "interimType": "N/A" } }, { "type": "step", "primary": "Combine all the solutions", "result": "a=\\frac{π}{6}+2πn,\\:a=\\frac{11π}{6}+2πn,\\:a=\\frac{5π}{6}+2πn,\\:a=\\frac{7π}{6}+2πn" } ], "meta": { "solvingClass": "Trig Equations", "practiceLink": "/practice/trigonometry-practice#area=main&subtopic=Trig%20Equations", "practiceTopic": "Trig Equations" } }, "plot_output": { "meta": { "plotInfo": { "variable": "a", "plotRequest": "2\\sec^{2}(a)+\\tan^{2}(a)-3" }, "showViewLarger": true } }, "meta": { "showVerify": true } }