solvefor x,-4cos(x)=-sin^2(x)+4

{ "query": { "display": "solve for $$x,\\:-4\\cos\\left(x\\right)=-\\sin^{2}\\left(x\\right)+4$$", "symbolab_question": "SOLVE_FOR#solvefor x,-4\\cos(x)=-\\sin^{2}(x)+4" }, "solution": { "level": "PERFORMED", "subject": "Trigonometry", "topic": "Trig Equations", "subTopic": "Trig Equations", "default": "x=π+2πn", "degrees": "x=180^{\\circ }+360^{\\circ }n", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$solve\\:for\\:x,\\:-4\\cos\\left(x\\right)=-\\sin^{2}\\left(x\\right)+4{\\quad:\\quad}x=π+2πn$$", "input": "-4\\cos\\left(x\\right)=-\\sin^{2}\\left(x\\right)+4", "steps": [ { "type": "step", "primary": "Subtract $$-\\sin^{2}\\left(x\\right)+4$$ from both sides", "result": "-4\\cos\\left(x\\right)+\\sin^{2}\\left(x\\right)-4=0" }, { "type": "interim", "title": "Rewrite using trig identities", "input": "-4+\\sin^{2}\\left(x\\right)-4\\cos\\left(x\\right)", "result": "-3-\\cos^{2}\\left(x\\right)-4\\cos\\left(x\\right)=0", "steps": [ { "type": "step", "primary": "Use the Pythagorean identity: $$\\cos^{2}\\left(x\\right)+\\sin^{2}\\left(x\\right)=1$$", "secondary": [ "$$\\sin^{2}\\left(x\\right)=1-\\cos^{2}\\left(x\\right)$$" ], "result": "=-4+1-\\cos^{2}\\left(x\\right)-4\\cos\\left(x\\right)" }, { "type": "step", "primary": "Simplify", "result": "=-\\cos^{2}\\left(x\\right)-4\\cos\\left(x\\right)-3", "meta": { "solvingClass": "Solver" } } ], "meta": { "interimType": "Trig Rewrite Using Trig identities 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7oWWoY5ZFl206HiWStjmRL3J51/0UmnSc/GkZzN8Lpm0pzfAfzSyLRUCFeFU9SuvRCL1WefiKaC4ihafc3V6qN0YiRvdBbbDvLg3zdFKS1tz+RS6nKnvFfEHxiYoRDkegnatgxrm2UZcnj2WgyPW9M3aXZIjcLCc2+X9xGaBGdfSBBTEk/JQ2cZ9WKuRzClU7DcFgmo2GmeG2Kh9EaSPQP8xQp8yk2EdviTvfhz6ruO+9RQa+VFzMcO8uN2y3ZuWk" } }, { "type": "interim", "title": "Solve by substitution", "input": "-3-\\cos^{2}\\left(x\\right)-4\\cos\\left(x\\right)=0", "result": "\\cos\\left(x\\right)=-3,\\:\\cos\\left(x\\right)=-1", "steps": [ { "type": "step", "primary": "Let: $$\\cos\\left(x\\right)=u$$", "result": "-3-u^{2}-4u=0" }, { "type": "interim", "title": "$$-3-u^{2}-4u=0{\\quad:\\quad}u=-3,\\:u=-1$$", "input": "-3-u^{2}-4u=0", "steps": [ { "type": "step", "primary": "Write in the standard form $$ax^{2}+bx+c=0$$", "result": "-u^{2}-4u-3=0" }, { "type": "interim", "title": "Solve with the quadratic formula", "input": "-u^{2}-4u-3=0", "result": "{u}_{1,\\:2}=\\frac{-\\left(-4\\right)\\pm\\:\\sqrt{\\left(-4\\right)^{2}-4\\left(-1\\right)\\left(-3\\right)}}{2\\left(-1\\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 $${\\quad}a=-1,\\:b=-4,\\:c=-3$$", "result": "{u}_{1,\\:2}=\\frac{-\\left(-4\\right)\\pm\\:\\sqrt{\\left(-4\\right)^{2}-4\\left(-1\\right)\\left(-3\\right)}}{2\\left(-1\\right)}" } ], "meta": { "interimType": "Solving The Quadratic Equation With Quadratic Formula Definition 0Eq", "gptData": "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" } }, { "type": "interim", "title": "$$\\sqrt{\\left(-4\\right)^{2}-4\\left(-1\\right)\\left(-3\\right)}=2$$", "input": "\\sqrt{\\left(-4\\right)^{2}-4\\left(-1\\right)\\left(-3\\right)}", "result": "{u}_{1,\\:2}=\\frac{-\\left(-4\\right)\\pm\\:2}{2\\left(-1\\right)}", "steps": [ { "type": "step", "primary": "Apply rule $$-\\left(-a\\right)=a$$", "result": "=\\sqrt{\\left(-4\\right)^{2}-4\\cdot\\:1\\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\\:1\\cdot\\:3}" }, { "type": "step", "primary": "Multiply the numbers: $$4\\cdot\\:1\\cdot\\:3=12$$", "result": "=\\sqrt{4^{2}-12}" }, { "type": "step", "primary": "$$4^{2}=16$$", "result": "=\\sqrt{16-12}" }, { "type": "step", "primary": "Subtract the numbers: $$16-12=4$$", "result": "=\\sqrt{4}" }, { "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": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7UJ8pwdbiCoYgDDkgm+KlvYTJKsrJzofywv72zeCuyjsgJ/ZZA32ZInFBpDtxBfiKRcASOqRpLIeIyUBzgC+nUMxkARoM44WPcMutmKWsbuyEMycS+kOY0O8v//odxcM6aajxn+ha1h9bkOEZl8gjEw==" } }, { "type": "step", "primary": "Separate the solutions", "result": "{u}_{1}=\\frac{-\\left(-4\\right)+2}{2\\left(-1\\right)},\\:{u}_{2}=\\frac{-\\left(-4\\right)-2}{2\\left(-1\\right)}" }, { "type": "interim", "title": "$$u=\\frac{-\\left(-4\\right)+2}{2\\left(-1\\right)}:{\\quad}-3$$", "input": "\\frac{-\\left(-4\\right)+2}{2\\left(-1\\right)}", "steps": [ { "type": "step", "primary": "Remove parentheses: $$\\left(-a\\right)=-a,\\:-\\left(-a\\right)=a$$", "result": "=\\frac{4+2}{-2\\cdot\\:1}" }, { "type": "step", "primary": "Add the numbers: $$4+2=6$$", "result": "=\\frac{6}{-2\\cdot\\:1}" }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:1=2$$", "result": "=\\frac{6}{-2}" }, { "type": "step", "primary": "Apply the fraction rule: $$\\frac{a}{-b}=-\\frac{a}{b}$$", "result": "=-\\frac{6}{2}" }, { "type": "step", "primary": "Divide the numbers: $$\\frac{6}{2}=3$$", "result": "=-3" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7mErZDWWP6MQGJMQ/jBr8p+NiDKCNsnmWP0fTjbhukYR1g99dC9fj9sg0EHzBIRDRDTwfJhbKVRsfj6DjkGblF3XxzR/D3xpyR5yXTZ2YQF2x4aZnjytVNILQaziz42nDvzIPeEtDfcHv/z8uls8Teg==" } }, { "type": "interim", "title": "$$u=\\frac{-\\left(-4\\right)-2}{2\\left(-1\\right)}:{\\quad}-1$$", "input": "\\frac{-\\left(-4\\right)-2}{2\\left(-1\\right)}", "steps": [ { "type": "step", "primary": "Remove parentheses: $$\\left(-a\\right)=-a,\\:-\\left(-a\\right)=a$$", "result": "=\\frac{4-2}{-2\\cdot\\:1}" }, { "type": "step", "primary": "Subtract the numbers: $$4-2=2$$", "result": "=\\frac{2}{-2\\cdot\\:1}" }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:1=2$$", "result": "=\\frac{2}{-2}" }, { "type": "step", "primary": "Apply the fraction rule: $$\\frac{a}{-b}=-\\frac{a}{b}$$", "result": "=-\\frac{2}{2}" }, { "type": "step", "primary": "Apply rule $$\\frac{a}{a}=1$$", "result": "=-1" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7iDVx4wrZkg6rS7/lKSD8zONiDKCNsnmWP0fTjbhukYR1g99dC9fj9sg0EHzBIRDRlcq1iPPbKQKUi0Yqft4tTnXxzR/D3xpyR5yXTZ2YQF2km2EXzUB5QqCoUzpjCcFOvzIPeEtDfcHv/z8uls8Teg==" } }, { "type": "step", "primary": "The solutions to the quadratic equation are:", "result": "u=-3,\\:u=-1" } ], "meta": { "solvingClass": "Equations", "interimType": "Equations" } }, { "type": "step", "primary": "Substitute back $$u=\\cos\\left(x\\right)$$", "result": "\\cos\\left(x\\right)=-3,\\:\\cos\\left(x\\right)=-1" } ], "meta": { "interimType": "Substitution Method 0Eq" } }, { "type": "interim", "title": "$$\\cos\\left(x\\right)=-3{\\quad:\\quad}$$No Solution", "input": "\\cos\\left(x\\right)=-3", "steps": [ { "type": "step", "primary": "$$-1\\le\\cos\\left(x\\right)\\le1$$", "result": "\\mathrm{No\\:Solution}" } ], "meta": { "interimType": "N/A" } }, { "type": "interim", "title": "$$\\cos\\left(x\\right)=-1{\\quad:\\quad}x=π+2πn$$", "input": "\\cos\\left(x\\right)=-1", "steps": [ { "type": "interim", "title": "General solutions for $$\\cos\\left(x\\right)=-1$$", "result": "x=π+2πn", "steps": [ { "type": "step", "primary": "$$\\cos\\left(x\\right)$$ periodicity table with $$2πn$$ cycle:<br/>$$\\begin{array}{|c|c|c|c|}\\hline x&\\cos(x)&x&\\cos(x)\\\\\\hline 0&1&π&-1\\\\\\hline \\frac{π}{6}&\\frac{\\sqrt{3}}{2}&\\frac{7π}{6}&-\\frac{\\sqrt{3}}{2}\\\\\\hline \\frac{π}{4}&\\frac{\\sqrt{2}}{2}&\\frac{5π}{4}&-\\frac{\\sqrt{2}}{2}\\\\\\hline \\frac{π}{3}&\\frac{1}{2}&\\frac{4π}{3}&-\\frac{1}{2}\\\\\\hline \\frac{π}{2}&0&\\frac{3π}{2}&0\\\\\\hline \\frac{2π}{3}&-\\frac{1}{2}&\\frac{5π}{3}&\\frac{1}{2}\\\\\\hline \\frac{3π}{4}&-\\frac{\\sqrt{2}}{2}&\\frac{7π}{4}&\\frac{\\sqrt{2}}{2}\\\\\\hline \\frac{5π}{6}&-\\frac{\\sqrt{3}}{2}&\\frac{11π}{6}&\\frac{\\sqrt{3}}{2}\\\\\\hline \\end{array}$$" }, { "type": "step", "result": "x=π+2πn" } ], "meta": { "interimType": "Trig General Solutions cos 1Eq" } } ], "meta": { "interimType": "N/A" } }, { "type": "step", "primary": "Combine all the solutions", "result": "x=π+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(x)+\\sin^{2}(x)-4" }, "showViewLarger": true } }, "meta": { "showVerify": true } }