4cos^2(x)-4sin(x)-1=0

{ "query": { "display": "$$4\\cos^{2}\\left(x\\right)-4\\sin\\left(x\\right)-1=0$$", "symbolab_question": "EQUATION#4\\cos^{2}(x)-4\\sin(x)-1=0" }, "solution": { "level": "PERFORMED", "subject": "Trigonometry", "topic": "Trig Equations", "subTopic": "Trig Equations", "default": "x=\\frac{π}{6}+2πn,x=\\frac{5π}{6}+2πn", "degrees": "x=30^{\\circ }+360^{\\circ }n,x=150^{\\circ }+360^{\\circ }n", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$4\\cos^{2}\\left(x\\right)-4\\sin\\left(x\\right)-1=0{\\quad:\\quad}x=\\frac{π}{6}+2πn,\\:x=\\frac{5π}{6}+2πn$$", "input": "4\\cos^{2}\\left(x\\right)-4\\sin\\left(x\\right)-1=0", "steps": [ { "type": "interim", "title": "Rewrite using trig identities", "input": "-1+4\\cos^{2}\\left(x\\right)-4\\sin\\left(x\\right)", "result": "3-4\\sin\\left(x\\right)-4\\sin^{2}\\left(x\\right)=0", "steps": [ { "type": "step", "primary": "Use the Pythagorean identity: $$\\cos^{2}\\left(x\\right)+\\sin^{2}\\left(x\\right)=1$$", "secondary": [ "$$\\cos^{2}\\left(x\\right)=1-\\sin^{2}\\left(x\\right)$$" ], "result": "=-1+4\\left(1-\\sin^{2}\\left(x\\right)\\right)-4\\sin\\left(x\\right)" }, { "type": "interim", "title": "Simplify $$-1+4\\left(1-\\sin^{2}\\left(x\\right)\\right)-4\\sin\\left(x\\right):{\\quad}-4\\sin^{2}\\left(x\\right)-4\\sin\\left(x\\right)+3$$", "input": "-1+4\\left(1-\\sin^{2}\\left(x\\right)\\right)-4\\sin\\left(x\\right)", "result": "=-4\\sin^{2}\\left(x\\right)-4\\sin\\left(x\\right)+3", "steps": [ { "type": "interim", "title": "Expand $$4\\left(1-\\sin^{2}\\left(x\\right)\\right):{\\quad}4-4\\sin^{2}\\left(x\\right)$$", "input": "4\\left(1-\\sin^{2}\\left(x\\right)\\right)", "result": "=-1+4-4\\sin^{2}\\left(x\\right)-4\\sin\\left(x\\right)", "steps": [ { "type": "step", "primary": "Apply the distributive law: $$a\\left(b-c\\right)=ab-ac$$", "secondary": [ "$$a=4,\\:b=1,\\:c=\\sin^{2}\\left(x\\right)$$" ], "result": "=4\\cdot\\:1-4\\sin^{2}\\left(x\\right)", "meta": { "practiceLink": "/practice/expansion-practice", "practiceTopic": "Expand Rules" } }, { "type": "step", "primary": "Multiply the numbers: $$4\\cdot\\:1=4$$", "result": "=4-4\\sin^{2}\\left(x\\right)" } ], "meta": { "interimType": "Algebraic Manipulation Expand Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s76wPhBQ1UvQOJ+DvF6Odr4UPKZBnILZTwiutqYLWXIDCjkVi15I8rBefLi4Iyt2wropSd6E8O/fJ9cnjzAiGMD2RLd2VwIqlBNByF6663syRU6H0nS++8kDqP632fuVHP3vg36HY8Vu4HxmYY39j9LeIASZeFjDtawNGt9P21GJY=" } }, { "type": "step", "primary": "Add/Subtract the numbers: $$-1+4=3$$", "result": "=-4\\sin^{2}\\left(x\\right)-4\\sin\\left(x\\right)+3" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7s8ZFO0pIFfVQON1nWDpvem+xZLA3M9KnFJKoHwa2VY4AlilG71elit3w1IBbYN0PjH2dQRUYYRp0H/atvIPZyphJQ975Sr7MyB3h2BLkA0Najs0t+f1Rm8a256v3B9Y9TeQKHeh69S6dnv9vSoUoFG0rNtapEMuBdYie0CW5TKtsLKY2VkLs8nX1K4bVbOF1Yw6enetreyrE67exYQOWfA==" } } ], "meta": { "interimType": "Trig Rewrite Using Trig identities 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7CV+5dEVi7RNyMmssdQxp8UId6JXTpd2NMUiq8L3OG1TRD5wscKPBrtJdNEvBXrn5EsLZPw0Vm9MD6l9/kB1c91f2VE3iaVir/MjR2DERcZ0+92eGLSzeVn4MnusSJN6pqSR/zUnvXGWWts+f0cGKEYnPlA5RNiF1ltGSTEL2W/pFKk3fejFkyiOiq9iG9IkAO/B7UtNBfw19bDRpX2AlzDEZOhzOLFFCWOVe0TAs9Ps/tVtg7k21KzJIFYRgfW7H" } }, { "type": "interim", "title": "Solve by substitution", "input": "3-4\\sin\\left(x\\right)-4\\sin^{2}\\left(x\\right)=0", "result": "\\sin\\left(x\\right)=-\\frac{3}{2},\\:\\sin\\left(x\\right)=\\frac{1}{2}", "steps": [ { "type": "step", "primary": "Let: $$\\sin\\left(x\\right)=u$$", "result": "3-4u-4u^{2}=0" }, { "type": "interim", "title": "$$3-4u-4u^{2}=0{\\quad:\\quad}u=-\\frac{3}{2},\\:u=\\frac{1}{2}$$", "input": "3-4u-4u^{2}=0", "steps": [ { "type": "step", "primary": "Write in the standard form $$ax^{2}+bx+c=0$$", "result": "-4u^{2}-4u+3=0" }, { "type": "interim", "title": "Solve with the quadratic formula", "input": "-4u^{2}-4u+3=0", "result": "{u}_{1,\\:2}=\\frac{-\\left(-4\\right)\\pm\\:\\sqrt{\\left(-4\\right)^{2}-4\\left(-4\\right)\\cdot\\:3}}{2\\left(-4\\right)}", "steps": [ { "type": "definition", "title": "Quadratic Equation Formula:", "text": "For a quadratic equation of the form $$ax^2+bx+c=0$$ the solutions are <br/>$${\\quad}x_{1,\\:2}=\\frac{-b\\pm\\sqrt{b^2-4ac}}{2a}$$" }, { "type": "step", "primary": "For $${\\quad}a=-4,\\:b=-4,\\:c=3$$", "result": "{u}_{1,\\:2}=\\frac{-\\left(-4\\right)\\pm\\:\\sqrt{\\left(-4\\right)^{2}-4\\left(-4\\right)\\cdot\\:3}}{2\\left(-4\\right)}" } ], "meta": { "interimType": "Solving The Quadratic Equation With Quadratic Formula Definition 0Eq", "gptData": "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" } }, { "type": "interim", "title": "$$\\sqrt{\\left(-4\\right)^{2}-4\\left(-4\\right)\\cdot\\:3}=8$$", "input": "\\sqrt{\\left(-4\\right)^{2}-4\\left(-4\\right)\\cdot\\:3}", "result": "{u}_{1,\\:2}=\\frac{-\\left(-4\\right)\\pm\\:8}{2\\left(-4\\right)}", "steps": [ { "type": "step", "primary": "Apply rule $$-\\left(-a\\right)=a$$", "result": "=\\sqrt{\\left(-4\\right)^{2}+4\\cdot\\:4\\cdot\\:3}" }, { "type": "step", "primary": "Apply exponent rule: $$\\left(-a\\right)^{n}=a^{n},\\:$$if $$n$$ is even", "secondary": [ "$$\\left(-4\\right)^{2}=4^{2}$$" ], "result": "=\\sqrt{4^{2}+4\\cdot\\:4\\cdot\\:3}" }, { "type": "step", "primary": "Multiply the numbers: $$4\\cdot\\:4\\cdot\\:3=48$$", "result": "=\\sqrt{4^{2}+48}" }, { "type": "step", "primary": "$$4^{2}=16$$", "result": "=\\sqrt{16+48}" }, { "type": "step", "primary": "Add the numbers: $$16+48=64$$", "result": "=\\sqrt{64}" }, { "type": "step", "primary": "Factor the number: $$64=8^{2}$$", "result": "=\\sqrt{8^{2}}" }, { "type": "step", "primary": "Apply radical rule: $$\\sqrt[n]{a^n}=a$$", "secondary": [ "$$\\sqrt{8^{2}}=8$$" ], "result": "=8", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7UJ8pwdbiCoYgDDkgm+KlvZWeDC8aEadl+125ksNZx+EAlilG71elit3w1IBbYN0Ps8vL5gcdXJR+EyRTzCE2LaN6Hv6MoTMtvtU0IQwXdn+pjCNmDjDgZv21ivxFfJgtbT6QeI68EHvYdIB3XiFX+iS3daIZHtloJpe/PvtsyNI=" } }, { "type": "step", "primary": "Separate the solutions", "result": "{u}_{1}=\\frac{-\\left(-4\\right)+8}{2\\left(-4\\right)},\\:{u}_{2}=\\frac{-\\left(-4\\right)-8}{2\\left(-4\\right)}" }, { "type": "interim", "title": "$$u=\\frac{-\\left(-4\\right)+8}{2\\left(-4\\right)}:{\\quad}-\\frac{3}{2}$$", "input": "\\frac{-\\left(-4\\right)+8}{2\\left(-4\\right)}", "steps": [ { "type": "step", "primary": "Remove parentheses: $$\\left(-a\\right)=-a,\\:-\\left(-a\\right)=a$$", "result": "=\\frac{4+8}{-2\\cdot\\:4}" }, { "type": "step", "primary": "Add the numbers: $$4+8=12$$", "result": "=\\frac{12}{-2\\cdot\\:4}" }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:4=8$$", "result": "=\\frac{12}{-8}" }, { "type": "step", "primary": "Apply the fraction rule: $$\\frac{a}{-b}=-\\frac{a}{b}$$", "result": "=-\\frac{12}{8}" }, { "type": "step", "primary": "Cancel the common factor: $$4$$", "result": "=-\\frac{3}{2}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s78ThL1w/DrSsEmb1fi6yyfLZ6h36XqoYM77larUc89WF1g99dC9fj9sg0EHzBIRDRANMyDRWlcO8ZgVjaxiUreRHO0oTnnZveyzJ4AtC1ZGPKObtHdExvWGiGIPjPMFnLbQ98ZwH/6N0R+7P9gXMIVg==" } }, { "type": "interim", "title": "$$u=\\frac{-\\left(-4\\right)-8}{2\\left(-4\\right)}:{\\quad}\\frac{1}{2}$$", "input": "\\frac{-\\left(-4\\right)-8}{2\\left(-4\\right)}", "steps": [ { "type": "step", "primary": "Remove parentheses: $$\\left(-a\\right)=-a,\\:-\\left(-a\\right)=a$$", "result": "=\\frac{4-8}{-2\\cdot\\:4}" }, { "type": "step", "primary": "Subtract the numbers: $$4-8=-4$$", "result": "=\\frac{-4}{-2\\cdot\\:4}" }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:4=8$$", "result": "=\\frac{-4}{-8}" }, { "type": "step", "primary": "Apply the fraction rule: $$\\frac{-a}{-b}=\\frac{a}{b}$$", "result": "=\\frac{4}{8}" }, { "type": "step", "primary": "Cancel the common factor: $$4$$", "result": "=\\frac{1}{2}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7teXmohnKGHxQdjHLifG7drZ6h36XqoYM77larUc89WF1g99dC9fj9sg0EHzBIRDRbnWmTWTmSt0muJWVWz+hO/8//6/nV5O4fb8Xgwi7mapdcleqcBeeyG5M+O18HFvNx7LhuKAvb8WoIF40TEr/QA==" } }, { "type": "step", "primary": "The solutions to the quadratic equation are:", "result": "u=-\\frac{3}{2},\\:u=\\frac{1}{2}" } ], "meta": { "solvingClass": "Equations", "interimType": "Equations" } }, { "type": "step", "primary": "Substitute back $$u=\\sin\\left(x\\right)$$", "result": "\\sin\\left(x\\right)=-\\frac{3}{2},\\:\\sin\\left(x\\right)=\\frac{1}{2}" } ], "meta": { "interimType": "Substitution Method 0Eq" } }, { "type": "interim", "title": "$$\\sin\\left(x\\right)=-\\frac{3}{2}{\\quad:\\quad}$$No Solution", "input": "\\sin\\left(x\\right)=-\\frac{3}{2}", "steps": [ { "type": "step", "primary": "$$-1\\le\\sin\\left(x\\right)\\le1$$", "result": "\\mathrm{No\\:Solution}" } ], "meta": { "interimType": "N/A" } }, { "type": "interim", "title": "$$\\sin\\left(x\\right)=\\frac{1}{2}{\\quad:\\quad}x=\\frac{π}{6}+2πn,\\:x=\\frac{5π}{6}+2πn$$", "input": "\\sin\\left(x\\right)=\\frac{1}{2}", "steps": [ { "type": "interim", "title": "General solutions for $$\\sin\\left(x\\right)=\\frac{1}{2}$$", "result": "x=\\frac{π}{6}+2πn,\\:x=\\frac{5π}{6}+2πn", "steps": [ { "type": "step", "primary": "$$\\sin\\left(x\\right)$$ periodicity table with $$2πn$$ cycle:<br/>$$\\begin{array}{|c|c|c|c|}\\hline x&\\sin(x)&x&\\sin(x)\\\\\\hline 0&0&π&0\\\\\\hline \\frac{π}{6}&\\frac{1}{2}&\\frac{7π}{6}&-\\frac{1}{2}\\\\\\hline \\frac{π}{4}&\\frac{\\sqrt{2}}{2}&\\frac{5π}{4}&-\\frac{\\sqrt{2}}{2}\\\\\\hline \\frac{π}{3}&\\frac{\\sqrt{3}}{2}&\\frac{4π}{3}&-\\frac{\\sqrt{3}}{2}\\\\\\hline \\frac{π}{2}&1&\\frac{3π}{2}&-1\\\\\\hline \\frac{2π}{3}&\\frac{\\sqrt{3}}{2}&\\frac{5π}{3}&-\\frac{\\sqrt{3}}{2}\\\\\\hline \\frac{3π}{4}&\\frac{\\sqrt{2}}{2}&\\frac{7π}{4}&-\\frac{\\sqrt{2}}{2}\\\\\\hline \\frac{5π}{6}&\\frac{1}{2}&\\frac{11π}{6}&-\\frac{1}{2}\\\\\\hline \\end{array}$$" }, { "type": "step", "result": "x=\\frac{π}{6}+2πn,\\:x=\\frac{5π}{6}+2πn" } ], "meta": { "interimType": "Trig General Solutions sin 1Eq" } } ], "meta": { "interimType": "N/A" } }, { "type": "step", "primary": "Combine all the solutions", "result": "x=\\frac{π}{6}+2πn,\\:x=\\frac{5π}{6}+2πn" } ], "meta": { "solvingClass": "Trig Equations", "practiceLink": "/practice/trigonometry-practice#area=main&subtopic=Trig%20Equations", "practiceTopic": "Trig Equations" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "4\\cos^{2}(x)-4\\sin(x)-1" }, "showViewLarger": true } }, "meta": { "showVerify": true } }