2cos(2x)=4cos(x)

{ "query": { "display": "$$2\\cos\\left(2x\\right)=4\\cos\\left(x\\right)$$", "symbolab_question": "EQUATION#2\\cos(2x)=4\\cos(x)" }, "solution": { "level": "PERFORMED", "subject": "Trigonometry", "topic": "Trig Equations", "subTopic": "Trig Equations", "default": "x=1.94553…+2πn,x=-1.94553…+2πn", "degrees": "x=111.47070…^{\\circ }+360^{\\circ }n,x=-111.47070…^{\\circ }+360^{\\circ }n", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$2\\cos\\left(2x\\right)=4\\cos\\left(x\\right){\\quad:\\quad}x=1.94553…+2πn,\\:x=-1.94553…+2πn$$", "input": "2\\cos\\left(2x\\right)=4\\cos\\left(x\\right)", "steps": [ { "type": "step", "primary": "Subtract $$4\\cos\\left(x\\right)$$ from both sides", "result": "2\\cos\\left(2x\\right)-4\\cos\\left(x\\right)=0" }, { "type": "interim", "title": "Rewrite using trig identities", "input": "2\\cos\\left(2x\\right)-4\\cos\\left(x\\right)", "result": "\\left(-1+2\\cos^{2}\\left(x\\right)\\right)\\cdot\\:2-4\\cos\\left(x\\right)=0", "steps": [ { "type": "step", "primary": "Use the Double Angle identity: $$\\cos\\left(2x\\right)=2\\cos^{2}\\left(x\\right)-1$$", "result": "=2\\left(2\\cos^{2}\\left(x\\right)-1\\right)-4\\cos\\left(x\\right)" } ], "meta": { "interimType": "Trig Rewrite Using Trig identities 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s791AXYceO9Pc4CRAMY99YBbETrgVsOFKD+6sr52dgCjr1fWutlCU4zUzgkFchUVlKOeWgsE4Mk40prEDZkKQ8Xcq64+b8YguXf4qCtKW9b7LmGkXjFCrtLFIxDvils2UNTQhGpKpnv7rV1P8sJbv5+98cd17+D2uaxTvKn+KCe0nWwPs1+Gw97t4MeuaNjSYTRvemj3GBE2iIDcXU+cR6iI+gxrQ1tCXeKlYIJ1n6NLewiNrEngO+NNvZ9sqNu+2V" } }, { "type": "interim", "title": "Solve by substitution", "input": "\\left(-1+2\\cos^{2}\\left(x\\right)\\right)\\cdot\\:2-4\\cos\\left(x\\right)=0", "result": "\\cos\\left(x\\right)=\\frac{1+\\sqrt{3}}{2},\\:\\cos\\left(x\\right)=\\frac{1-\\sqrt{3}}{2}", "steps": [ { "type": "step", "primary": "Let: $$\\cos\\left(x\\right)=u$$", "result": "\\left(-1+2u^{2}\\right)\\cdot\\:2-4u=0" }, { "type": "interim", "title": "$$\\left(-1+2u^{2}\\right)\\cdot\\:2-4u=0{\\quad:\\quad}u=\\frac{1+\\sqrt{3}}{2},\\:u=\\frac{1-\\sqrt{3}}{2}$$", "input": "\\left(-1+2u^{2}\\right)\\cdot\\:2-4u=0", "steps": [ { "type": "interim", "title": "Expand $$\\left(-1+2u^{2}\\right)\\cdot\\:2-4u:{\\quad}-2+4u^{2}-4u$$", "input": "\\left(-1+2u^{2}\\right)\\cdot\\:2-4u", "steps": [ { "type": "step", "result": "=2\\left(-1+2u^{2}\\right)-4u" }, { "type": "interim", "title": "Expand $$2\\left(-1+2u^{2}\\right):{\\quad}-2+4u^{2}$$", "input": "2\\left(-1+2u^{2}\\right)", "result": "=-2+4u^{2}-4u", "steps": [ { "type": "step", "primary": "Apply the distributive law: $$a\\left(b+c\\right)=ab+ac$$", "secondary": [ "$$a=2,\\:b=-1,\\:c=2u^{2}$$" ], "result": "=2\\left(-1\\right)+2\\cdot\\:2u^{2}", "meta": { "practiceLink": "/practice/expansion-practice", "practiceTopic": "Expand Rules" } }, { "type": "step", "primary": "Apply minus-plus rules", "secondary": [ "$$+\\left(-a\\right)=-a$$" ], "result": "=-2\\cdot\\:1+2\\cdot\\:2u^{2}" }, { "type": "interim", "title": "Simplify $$-2\\cdot\\:1+2\\cdot\\:2u^{2}:{\\quad}-2+4u^{2}$$", "input": "-2\\cdot\\:1+2\\cdot\\:2u^{2}", "result": "=-2+4u^{2}", "steps": [ { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:1=2$$", "result": "=-2+2\\cdot\\:2u^{2}" }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:2=4$$", "result": "=-2+4u^{2}" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } } ], "meta": { "interimType": "Algebraic Manipulation Expand Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s72q98Xh8AJIfTC60rpA5YJ1XTSum/z5kLpMzXS1UJIezaPA+z2Rjc0XxxVqYgQm7ph0R4kHw1LArZyd/YFzVYTR7MWKURN+43KCOzRc+RXaH6GyAXaxp1KFgmyoPwh9I84gBJl4WMO1rA0a30/bUYlg==" } } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Expand Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7b69jmYnUdXcFDF2vLCuj3+U1Q9cotxZWfn6O0z/hhTULAZlDhoAdFQF6AF4pPagvRKAKbro8EavzLN8GuIDU/PWVUjtAdd6EqSvsyX6r+5AezFilETfuNygjs0XPkV2hVA2+5AcEPm4qNY1/pGFl6x4DXf76rZP1+6PID4YXjRA=" } }, { "type": "step", "result": "-2+4u^{2}-4u=0" }, { "type": "step", "primary": "Write in the standard form $$ax^{2}+bx+c=0$$", "result": "4u^{2}-4u-2=0" }, { "type": "interim", "title": "Solve with the quadratic formula", "input": "4u^{2}-4u-2=0", "result": "{u}_{1,\\:2}=\\frac{-\\left(-4\\right)\\pm\\:\\sqrt{\\left(-4\\right)^{2}-4\\cdot\\:4\\left(-2\\right)}}{2\\cdot\\:4}", "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 $${\\quad}a=4,\\:b=-4,\\:c=-2$$", "result": "{u}_{1,\\:2}=\\frac{-\\left(-4\\right)\\pm\\:\\sqrt{\\left(-4\\right)^{2}-4\\cdot\\:4\\left(-2\\right)}}{2\\cdot\\:4}" } ], "meta": { "interimType": "Solving The Quadratic Equation With Quadratic Formula Definition 0Eq", "gptData": "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" } }, { "type": "interim", "title": "$$\\sqrt{\\left(-4\\right)^{2}-4\\cdot\\:4\\left(-2\\right)}=4\\sqrt{3}$$", "input": "\\sqrt{\\left(-4\\right)^{2}-4\\cdot\\:4\\left(-2\\right)}", "result": "{u}_{1,\\:2}=\\frac{-\\left(-4\\right)\\pm\\:4\\sqrt{3}}{2\\cdot\\:4}", "steps": [ { "type": "step", "primary": "Apply rule $$-\\left(-a\\right)=a$$", "result": "=\\sqrt{\\left(-4\\right)^{2}+4\\cdot\\:4\\cdot\\:2}" }, { "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\\:2}" }, { "type": "step", "primary": "Multiply the numbers: $$4\\cdot\\:4\\cdot\\:2=32$$", "result": "=\\sqrt{4^{2}+32}" }, { "type": "step", "primary": "$$4^{2}=16$$", "result": "=\\sqrt{16+32}" }, { "type": "step", "primary": "Add the numbers: $$16+32=48$$", "result": "=\\sqrt{48}" }, { "type": "interim", "title": "Prime factorization of $$48:{\\quad}2^{4}\\cdot\\:3$$", "input": "48", "result": "=\\sqrt{2^{4}\\cdot\\:3}", "steps": [ { "type": "step", "primary": "$$48\\:$$divides by $$2\\quad\\:48=24\\cdot\\:2$$", "result": "=2\\cdot\\:24" }, { "type": "step", "primary": "$$24\\:$$divides by $$2\\quad\\:24=12\\cdot\\:2$$", "result": "=2\\cdot\\:2\\cdot\\:12" }, { "type": "step", "primary": "$$12\\:$$divides by $$2\\quad\\:12=6\\cdot\\:2$$", "result": "=2\\cdot\\:2\\cdot\\:2\\cdot\\:6" }, { "type": "step", "primary": "$$6\\:$$divides by $$2\\quad\\:6=3\\cdot\\:2$$", "result": "=2\\cdot\\:2\\cdot\\:2\\cdot\\:2\\cdot\\:3" }, { "type": "step", "primary": "$$2,\\:3$$ are all prime numbers, therefore no further factorization is possible", "result": "=2\\cdot\\:2\\cdot\\:2\\cdot\\:2\\cdot\\:3" }, { "type": "step", "result": "=2^{4}\\cdot\\:3" } ], "meta": { "solvingClass": "Composite Integer", "interimType": "Prime Fac 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xyoU2cWyPLDgE1QLLHeauuc3PHQdChPJ2JhfqHT+ZU0OMrfn8NOj0LUzuzje6xTyxRsG/uC0ndYtZpJL4uAxK7FKuEeNBgSa3LrIvx33A/jwUB4gitN/2ICkrV6ivfiR3BLFRzd4QlsM8ugKm4vxBIEDxzy+DdBKC0kEoj+60KMUV" } }, { "type": "step", "primary": "Apply radical rule: $$\\sqrt[n]{ab}=\\sqrt[n]{a}\\sqrt[n]{b}$$", "result": "=\\sqrt{3}\\sqrt{2^{4}}", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } }, { "type": "step", "primary": "Apply radical rule: $$\\sqrt[n]{a^m}=a^{\\frac{m}{n}}$$", "secondary": [ "$$\\sqrt{2^{4}}=2^{\\frac{4}{2}}=2^{2}$$" ], "result": "=2^{2}\\sqrt{3}", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } }, { "type": "step", "primary": "Refine", "result": "=4\\sqrt{3}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7UJ8pwdbiCoYgDDkgm+KlvcxHTuOb6t4zxqqdK8Vq2s0AlilG71elit3w1IBbYN0PcN8MCunLJ8HtC1DgqImf/SELuC9uG3tq7Z3/2V23T2X93Ul/TpNwtO5K6HdO9r7roQ1PEEL1PoUpfwJ2ZQEu67A3kT/KrjOSt98KefkjT2mwiNrEngO+NNvZ9sqNu+2V" } }, { "type": "step", "primary": "Separate the solutions", "result": "{u}_{1}=\\frac{-\\left(-4\\right)+4\\sqrt{3}}{2\\cdot\\:4},\\:{u}_{2}=\\frac{-\\left(-4\\right)-4\\sqrt{3}}{2\\cdot\\:4}" }, { "type": "interim", "title": "$$u=\\frac{-\\left(-4\\right)+4\\sqrt{3}}{2\\cdot\\:4}:{\\quad}\\frac{1+\\sqrt{3}}{2}$$", "input": "\\frac{-\\left(-4\\right)+4\\sqrt{3}}{2\\cdot\\:4}", "steps": [ { "type": "step", "primary": "Apply rule $$-\\left(-a\\right)=a$$", "result": "=\\frac{4+4\\sqrt{3}}{2\\cdot\\:4}" }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:4=8$$", "result": "=\\frac{4+4\\sqrt{3}}{8}" }, { "type": "interim", "title": "Factor $$4+4\\sqrt{3}:{\\quad}4\\left(1+\\sqrt{3}\\right)$$", "input": "4+4\\sqrt{3}", "result": "=\\frac{4\\left(1+\\sqrt{3}\\right)}{8}", "steps": [ { "type": "step", "primary": "Rewrite as", "result": "=4\\cdot\\:1+4\\sqrt{3}" }, { "type": "step", "primary": "Factor out common term $$4$$", "result": "=4\\left(1+\\sqrt{3}\\right)", "meta": { "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } } ], "meta": { "interimType": "Algebraic Manipulation Factor Title 1Eq" } }, { "type": "step", "primary": "Cancel the common factor: $$4$$", "result": "=\\frac{1+\\sqrt{3}}{2}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7eti+L3MUqrrXPEzllVGmSa325RVX3gvSk/0tevtLX29OXjDqgZceJmQvXbNh57qqcJChiVhDxT5N/LHSTLMjyGXRwVFt3MpkAxqPC419n7etCZA9IvYdgMxbjpdehNrsP8B5gQhfgjXUFrdjPTeFIEoITjE39ygFXmV413UTIbXr1zn/SlH8Lh2NTs7GejAb" } }, { "type": "interim", "title": "$$u=\\frac{-\\left(-4\\right)-4\\sqrt{3}}{2\\cdot\\:4}:{\\quad}\\frac{1-\\sqrt{3}}{2}$$", "input": "\\frac{-\\left(-4\\right)-4\\sqrt{3}}{2\\cdot\\:4}", "steps": [ { "type": "step", "primary": "Apply rule $$-\\left(-a\\right)=a$$", "result": "=\\frac{4-4\\sqrt{3}}{2\\cdot\\:4}" }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:4=8$$", "result": "=\\frac{4-4\\sqrt{3}}{8}" }, { "type": "interim", "title": "Factor $$4-4\\sqrt{3}:{\\quad}4\\left(1-\\sqrt{3}\\right)$$", "input": "4-4\\sqrt{3}", "result": "=\\frac{4\\left(1-\\sqrt{3}\\right)}{8}", "steps": [ { "type": "step", "primary": "Rewrite as", "result": "=4\\cdot\\:1-4\\sqrt{3}" }, { "type": "step", "primary": "Factor out common term $$4$$", "result": "=4\\left(1-\\sqrt{3}\\right)", "meta": { "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } } ], "meta": { "interimType": "Algebraic Manipulation Factor Title 1Eq" } }, { "type": "step", "primary": "Cancel the common factor: $$4$$", "result": "=\\frac{1-\\sqrt{3}}{2}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7aldFBKdPNptfOboJzW+FCa325RVX3gvSk/0tevtLX29OXjDqgZceJmQvXbNh57qqcJChiVhDxT5N/LHSTLMjyJXjKiI2jP70puVkz73DPSitCZA9IvYdgMxbjpdehNrsP8B5gQhfgjXUFrdjPTeFIKaEdTKuXctYcU+K9p46uA7r1zn/SlH8Lh2NTs7GejAb" } }, { "type": "step", "primary": "The solutions to the quadratic equation are:", "result": "u=\\frac{1+\\sqrt{3}}{2},\\:u=\\frac{1-\\sqrt{3}}{2}" } ], "meta": { "solvingClass": "Equations", "interimType": "Equations" } }, { "type": "step", "primary": "Substitute back $$u=\\cos\\left(x\\right)$$", "result": "\\cos\\left(x\\right)=\\frac{1+\\sqrt{3}}{2},\\:\\cos\\left(x\\right)=\\frac{1-\\sqrt{3}}{2}" } ], "meta": { "interimType": "Substitution Method 0Eq" } }, { "type": "interim", "title": "$$\\cos\\left(x\\right)=\\frac{1+\\sqrt{3}}{2}{\\quad:\\quad}$$No Solution", "input": "\\cos\\left(x\\right)=\\frac{1+\\sqrt{3}}{2}", "steps": [ { "type": "step", "primary": "$$-1\\le\\cos\\left(x\\right)\\le1$$", "result": "\\mathrm{No\\:Solution}" } ], "meta": { "interimType": "N/A" } }, { "type": "interim", "title": "$$\\cos\\left(x\\right)=\\frac{1-\\sqrt{3}}{2}{\\quad:\\quad}x=\\arccos\\left(\\frac{1-\\sqrt{3}}{2}\\right)+2πn,\\:x=-\\arccos\\left(\\frac{1-\\sqrt{3}}{2}\\right)+2πn$$", "input": "\\cos\\left(x\\right)=\\frac{1-\\sqrt{3}}{2}", "steps": [ { "type": "interim", "title": "Apply trig inverse properties", "input": "\\cos\\left(x\\right)=\\frac{1-\\sqrt{3}}{2}", "result": "x=\\arccos\\left(\\frac{1-\\sqrt{3}}{2}\\right)+2πn,\\:x=-\\arccos\\left(\\frac{1-\\sqrt{3}}{2}\\right)+2πn", "steps": [ { "type": "step", "primary": "General solutions for $$\\cos\\left(x\\right)=\\frac{1-\\sqrt{3}}{2}$$", "secondary": [ "$$\\cos\\left(x\\right)=-a\\quad\\Rightarrow\\quad\\:x=\\arccos\\left(-a\\right)+2πn,\\:\\quad\\:x=-\\arccos\\left(-a\\right)+2πn$$" ], "result": "x=\\arccos\\left(\\frac{1-\\sqrt{3}}{2}\\right)+2πn,\\:x=-\\arccos\\left(\\frac{1-\\sqrt{3}}{2}\\right)+2πn" } ], "meta": { "interimType": "Trig Apply Inverse Props 0Eq" } } ], "meta": { "interimType": "N/A" } }, { "type": "step", "primary": "Combine all the solutions", "result": "x=\\arccos\\left(\\frac{1-\\sqrt{3}}{2}\\right)+2πn,\\:x=-\\arccos\\left(\\frac{1-\\sqrt{3}}{2}\\right)+2πn" }, { "type": "step", "primary": "Show solutions in decimal form", "result": "x=1.94553…+2πn,\\:x=-1.94553…+2πn" } ], "meta": { "solvingClass": "Trig Equations", "practiceLink": "/practice/trigonometry-practice#area=main&subtopic=Trig%20Equations", "practiceTopic": "Trig Equations" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "2\\cos(2x)-4\\cos(x)" }, "showViewLarger": true } }, "meta": { "showVerify": true } }