sin^2(x)=2cos^2(x/2)

{ "query": { "display": "$$\\sin^{2}\\left(x\\right)=2\\cos^{2}\\left(\\frac{x}{2}\\right)$$", "symbolab_question": "EQUATION#\\sin^{2}(x)=2\\cos^{2}(\\frac{x}{2})" }, "solution": { "level": "PERFORMED", "subject": "Trigonometry", "topic": "Trig Equations", "subTopic": "Trig Equations", "default": "x=π+2πn,x=\\frac{π}{2}+2πn,x=\\frac{3π}{2}+2πn", "degrees": "x=180^{\\circ }+360^{\\circ }n,x=90^{\\circ }+360^{\\circ }n,x=270^{\\circ }+360^{\\circ }n", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\sin^{2}\\left(x\\right)=2\\cos^{2}\\left(\\frac{x}{2}\\right){\\quad:\\quad}x=π+2πn,\\:x=\\frac{π}{2}+2πn,\\:x=\\frac{3π}{2}+2πn$$", "input": "\\sin^{2}\\left(x\\right)=2\\cos^{2}\\left(\\frac{x}{2}\\right)", "steps": [ { "type": "step", "primary": "Subtract $$2\\cos^{2}\\left(\\frac{x}{2}\\right)$$ from both sides", "result": "\\sin^{2}\\left(x\\right)-2\\cos^{2}\\left(\\frac{x}{2}\\right)=0" }, { "type": "interim", "title": "Rewrite using trig identities", "input": "\\sin^{2}\\left(x\\right)-2\\cos^{2}\\left(\\frac{x}{2}\\right)", "result": "-\\cos\\left(x\\right)-\\cos^{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": [ "$$\\sin^{2}\\left(x\\right)=1-\\cos^{2}\\left(x\\right)$$" ], "result": "=1-\\cos^{2}\\left(x\\right)-2\\cos^{2}\\left(\\frac{x}{2}\\right)" }, { "type": "step", "primary": "Use the Double Angle identity: $$2\\cos^{2}\\left(x\\right)-1=\\cos\\left(2x\\right)$$", "secondary": [ "$$-2\\cos^{2}\\left(x\\right)+1=-\\cos\\left(2x\\right)$$" ], "result": "=-\\cos^{2}\\left(x\\right)-\\cos\\left(2\\cdot\\:\\frac{x}{2}\\right)" }, { "type": "interim", "title": "Multiply $$2\\cdot\\:\\frac{x}{2}\\::{\\quad}x$$", "input": "2\\cdot\\:\\frac{x}{2}", "result": "=-\\cos^{2}\\left(x\\right)-\\cos\\left(x\\right)", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{x\\cdot\\:2}{2}" }, { "type": "step", "primary": "Cancel the common factor: $$2$$", "result": "=x" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Multiply Title 1Eq" } } ], "meta": { "interimType": "Trig Rewrite Using Trig identities 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7AlY6PORzP4fzAhs5rb1NzzbCG/YaJPKod6wvrQvIvYnhKh08I6E8qNXj3gVnCa/UOdhXIglekP2bk8wRyD3Q8umxY48eSLsbyMu/5yTExUbC4HIK63JpB+rGbO1V7H0Q8qdONSO1daObxnlCUSb6mXIghiDMV4B1hPDQEDo4wTNsA/M4pBZhoUid3n1B4vre23Xi3IvT385kAYFK2CQOxBkJGB3SwIAzffMMCti9BhrWM+R/5vzPLegxuhGAmoA+" } }, { "type": "interim", "title": "Solve by substitution", "input": "-\\cos\\left(x\\right)-\\cos^{2}\\left(x\\right)=0", "result": "\\cos\\left(x\\right)=-1,\\:\\cos\\left(x\\right)=0", "steps": [ { "type": "step", "primary": "Let: $$\\cos\\left(x\\right)=u$$", "result": "-u-u^{2}=0" }, { "type": "interim", "title": "$$-u-u^{2}=0{\\quad:\\quad}u=-1,\\:u=0$$", "input": "-u-u^{2}=0", "steps": [ { "type": "step", "primary": "Write in the standard form $$ax^{2}+bx+c=0$$", "result": "-u^{2}-u=0" }, { "type": "interim", "title": "Solve with the quadratic formula", "input": "-u^{2}-u=0", "result": "{u}_{1,\\:2}=\\frac{-\\left(-1\\right)\\pm\\:\\sqrt{\\left(-1\\right)^{2}-4\\left(-1\\right)\\cdot\\:0}}{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 $${\\quad}a=-1,\\:b=-1,\\:c=0$$", "result": "{u}_{1,\\:2}=\\frac{-\\left(-1\\right)\\pm\\:\\sqrt{\\left(-1\\right)^{2}-4\\left(-1\\right)\\cdot\\:0}}{2\\left(-1\\right)}" } ], "meta": { "interimType": "Solving The Quadratic Equation With Quadratic Formula Definition 0Eq", "gptData": "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" } }, { "type": "interim", "title": "$$\\sqrt{\\left(-1\\right)^{2}-4\\left(-1\\right)\\cdot\\:0}=1$$", "input": "\\sqrt{\\left(-1\\right)^{2}-4\\left(-1\\right)\\cdot\\:0}", "result": "{u}_{1,\\:2}=\\frac{-\\left(-1\\right)\\pm\\:1}{2\\left(-1\\right)}", "steps": [ { "type": "step", "primary": "Apply rule $$-\\left(-a\\right)=a$$", "result": "=\\sqrt{\\left(-1\\right)^{2}+4\\cdot\\:1\\cdot\\:0}" }, { "type": "interim", "title": "$$\\left(-1\\right)^{2}=1$$", "input": "\\left(-1\\right)^{2}", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$\\left(-a\\right)^{n}=a^{n},\\:$$if $$n$$ is even", "secondary": [ "$$\\left(-1\\right)^{2}=1^{2}$$" ], "result": "=1^{2}" }, { "type": "step", "primary": "Apply rule $$1^{a}=1$$", "result": "=1" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s78E1FVQW6YvXK7raPRxih+c0ag8T1MwTer44+aCS/ZFAdx7pcd1x/bAWpIL8hAintf05A2GsVmPba4FjoW22b4iKyMg44e9p5G7GRfJ2en9g=" } }, { "type": "interim", "title": "$$4\\cdot\\:1\\cdot\\:0=0$$", "input": "4\\cdot\\:1\\cdot\\:0", "steps": [ { "type": "step", "primary": "Apply rule $$0\\cdot\\:a=0$$", "result": "=0" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7xF9rlQkuAOrRzKTf6eQs7SD/swv2EbFFv+X+7iEc3n+jkVi15I8rBefLi4Iyt2wr8D4yaPBYvrqNvcxJbQLVhFxXGJQXf9dTesK2NoC90LlXeV03GYdXLZ3ZnsTzJuK6" } }, { "type": "step", "result": "=\\sqrt{1+0}" }, { "type": "step", "primary": "Add the numbers: $$1+0=1$$", "result": "=\\sqrt{1}" }, { "type": "step", "primary": "Apply rule $$\\sqrt{1}=1$$", "result": "=1" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7gcCSkwHGkSzBJS28+axO6/ozfpat709n1xtF07NKaGIAlilG71elit3w1IBbYN0PfzBJSNoxCO6fFy6iRIjrm6N6Hv6MoTMtvtU0IQwXdn+SVpPUu8d2DohT7uf7kqbJfQtm3YLrBni/HT2B8g+NXiS3daIZHtloJpe/PvtsyNI=" } }, { "type": "step", "primary": "Separate the solutions", "result": "{u}_{1}=\\frac{-\\left(-1\\right)+1}{2\\left(-1\\right)},\\:{u}_{2}=\\frac{-\\left(-1\\right)-1}{2\\left(-1\\right)}" }, { "type": "interim", "title": "$$u=\\frac{-\\left(-1\\right)+1}{2\\left(-1\\right)}:{\\quad}-1$$", "input": "\\frac{-\\left(-1\\right)+1}{2\\left(-1\\right)}", "steps": [ { "type": "step", "primary": "Remove parentheses: $$\\left(-a\\right)=-a,\\:-\\left(-a\\right)=a$$", "result": "=\\frac{1+1}{-2\\cdot\\:1}" }, { "type": "step", "primary": "Add the numbers: $$1+1=2$$", "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": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s73nEuax+m66TuGCt9MmWfwONiDKCNsnmWP0fTjbhukYR1g99dC9fj9sg0EHzBIRDRlcq1iPPbKQKUi0Yqft4tTnXxzR/D3xpyR5yXTZ2YQF0oWmfDawUG2OTsqk0PdSrxvzIPeEtDfcHv/z8uls8Teg==" } }, { "type": "interim", "title": "$$u=\\frac{-\\left(-1\\right)-1}{2\\left(-1\\right)}:{\\quad}0$$", "input": "\\frac{-\\left(-1\\right)-1}{2\\left(-1\\right)}", "steps": [ { "type": "step", "primary": "Remove parentheses: $$\\left(-a\\right)=-a,\\:-\\left(-a\\right)=a$$", "result": "=\\frac{1-1}{-2\\cdot\\:1}" }, { "type": "step", "primary": "Subtract the numbers: $$1-1=0$$", "result": "=\\frac{0}{-2\\cdot\\:1}" }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:1=2$$", "result": "=\\frac{0}{-2}" }, { "type": "step", "primary": "Apply the fraction rule: $$\\frac{a}{-b}=-\\frac{a}{b}$$", "result": "=-\\frac{0}{2}" }, { "type": "step", "primary": "Apply rule $$\\frac{0}{a}=0,\\:a\\ne\\:0$$", "result": "=-0" }, { "type": "step", "result": "=0" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s71Z4tBeCEgm6o48U48XbdAONiDKCNsnmWP0fTjbhukYR1g99dC9fj9sg0EHzBIRDRd79UrkSVT0SCLs80Lgihl8XKhRRe6+fuRKwL9f/rSxRwSdSDkhMgABQT7Jz4EHtrJLd1ohke2Wgml78++2zI0g==" } }, { "type": "step", "primary": "The solutions to the quadratic equation are:", "result": "u=-1,\\:u=0" } ], "meta": { "solvingClass": "Equations", "interimType": "Equations" } }, { "type": "step", "primary": "Substitute back $$u=\\cos\\left(x\\right)$$", "result": "\\cos\\left(x\\right)=-1,\\:\\cos\\left(x\\right)=0" } ], "meta": { "interimType": "Substitution Method 0Eq" } }, { "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": "interim", "title": "$$\\cos\\left(x\\right)=0{\\quad:\\quad}x=\\frac{π}{2}+2πn,\\:x=\\frac{3π}{2}+2πn$$", "input": "\\cos\\left(x\\right)=0", "steps": [ { "type": "interim", "title": "General solutions for $$\\cos\\left(x\\right)=0$$", "result": "x=\\frac{π}{2}+2πn,\\:x=\\frac{3π}{2}+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=\\frac{π}{2}+2πn,\\:x=\\frac{3π}{2}+2πn" } ], "meta": { "interimType": "Trig General Solutions cos 1Eq" } } ], "meta": { "interimType": "N/A" } }, { "type": "step", "primary": "Combine all the solutions", "result": "x=π+2πn,\\:x=\\frac{π}{2}+2πn,\\:x=\\frac{3π}{2}+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^{2}(x)-2\\cos^{2}(\\frac{x}{2})" }, "showViewLarger": true } }, "meta": { "showVerify": true } }