sin(θ)-sqrt(3)cos(θ)=2

{ "query": { "display": "$$\\sin\\left(θ\\right)-\\sqrt{3}\\cos\\left(θ\\right)=2$$", "symbolab_question": "EQUATION#\\sin(θ)-\\sqrt{3}\\cos(θ)=2" }, "solution": { "level": "PERFORMED", "subject": "Trigonometry", "topic": "Trig Equations", "subTopic": "Trig Equations", "default": "θ=\\frac{5π}{6}+2πn", "degrees": "θ=150^{\\circ }+360^{\\circ }n", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\sin\\left(θ\\right)-\\sqrt{3}\\cos\\left(θ\\right)=2{\\quad:\\quad}θ=\\frac{5π}{6}+2πn$$", "input": "\\sin\\left(θ\\right)-\\sqrt{3}\\cos\\left(θ\\right)=2", "steps": [ { "type": "step", "primary": "Add $$\\sqrt{3}\\cos\\left(θ\\right)$$ to both sides", "result": "\\sin\\left(θ\\right)=2+\\sqrt{3}\\cos\\left(θ\\right)" }, { "type": "step", "primary": "Square both sides", "result": "\\sin^{2}\\left(θ\\right)=\\left(2+\\sqrt{3}\\cos\\left(θ\\right)\\right)^{2}" }, { "type": "step", "primary": "Subtract $$\\left(2+\\sqrt{3}\\cos\\left(θ\\right)\\right)^{2}$$ from both sides", "result": "\\sin^{2}\\left(θ\\right)-4-4\\sqrt{3}\\cos\\left(θ\\right)-3\\cos^{2}\\left(θ\\right)=0" }, { "type": "interim", "title": "Rewrite using trig identities", "input": "-4+\\sin^{2}\\left(θ\\right)-3\\cos^{2}\\left(θ\\right)-4\\cos\\left(θ\\right)\\sqrt{3}", "result": "-3-4\\cos^{2}\\left(θ\\right)-4\\cos\\left(θ\\right)\\sqrt{3}=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(θ\\right)-3\\cos^{2}\\left(θ\\right)-4\\cos\\left(θ\\right)\\sqrt{3}" }, { "type": "interim", "title": "Simplify $$-4+1-\\cos^{2}\\left(θ\\right)-3\\cos^{2}\\left(θ\\right)-4\\cos\\left(θ\\right)\\sqrt{3}:{\\quad}-4\\cos^{2}\\left(θ\\right)-4\\sqrt{3}\\cos\\left(θ\\right)-3$$", "input": "-4+1-\\cos^{2}\\left(θ\\right)-3\\cos^{2}\\left(θ\\right)-4\\cos\\left(θ\\right)\\sqrt{3}", "result": "=-4\\cos^{2}\\left(θ\\right)-4\\sqrt{3}\\cos\\left(θ\\right)-3", "steps": [ { "type": "step", "primary": "Add similar elements: $$-\\cos^{2}\\left(θ\\right)-3\\cos^{2}\\left(θ\\right)=-4\\cos^{2}\\left(θ\\right)$$", "result": "=-4+1-4\\cos^{2}\\left(θ\\right)-4\\sqrt{3}\\cos\\left(θ\\right)" }, { "type": "step", "primary": "Add/Subtract the numbers: $$-4+1=-3$$", "result": "=-4\\cos^{2}\\left(θ\\right)-4\\sqrt{3}\\cos\\left(θ\\right)-3" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7T06shGQOeEftvEmzESFae/pWKbkIIJtzjRon000y7xnd+G7JIwOwCJjkBU9qPWCXUpePku2bIkpxpx/sZ1eiUQlAlm5MBjmOz6iqN1PySyXSNOYTjJcTVwfGeGti+Q55n/ziqMrJOwT0L9FRSjIX/O57Suk6r5rx0JR94nMrHwTWwPs1+Gw97t4MeuaNjSYTVeH60jNPIHBoeIeWKi52fhEH2XvHLdCvDXp2zn8AhXQsWPBujxfT+mPs+i/Q1yvZ8QCgvGuvlEPSq2MhjtjxoompXFf3SOUx+H18qfp3MLg=" } } ], "meta": { "interimType": "Trig Rewrite Using Trig identities 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7oWWoY5ZFl206HiWStjmRL+mg83AxypoMUhHOqnNaPW0c3/5vq4dO9BGFFC2QPC8clqClbHbWusutXJm2QiyEXwUxLQbQeCNRfBzdFA6af0FjOct0gFocPiJsGcQgslMNdQdbWYTlRBRMjC5B3pqD7KU7z9dJNGGpEQezwyiNvBVp91KsI33ns5s4zO7JGAXGVqAHAalJtMlHZvkZFo8YL8rDeZ0Uy0k88mjvA1+2U1SIP2TH9VLmoizBQwdUXcYEMwlmQv/qPmO1VQ4glIE4qqXFidbElO7XH8+PPiEMftA=" } }, { "type": "interim", "title": "Solve by substitution", "input": "-3-4\\cos^{2}\\left(θ\\right)-4\\cos\\left(θ\\right)\\sqrt{3}=0", "result": "\\cos\\left(θ\\right)=-\\frac{\\sqrt{3}}{2}", "steps": [ { "type": "step", "primary": "Let: $$\\cos\\left(θ\\right)=u$$", "result": "-3-4u^{2}-4u\\sqrt{3}=0" }, { "type": "interim", "title": "$$-3-4u^{2}-4u\\sqrt{3}=0{\\quad:\\quad}u=-\\frac{\\sqrt{3}}{2}$$", "input": "-3-4u^{2}-4u\\sqrt{3}=0", "steps": [ { "type": "step", "primary": "Write in the standard form $$ax^{2}+bx+c=0$$", "result": "-4u^{2}-4\\sqrt{3}u-3=0" }, { "type": "interim", "title": "Solve with the quadratic formula", "input": "-4u^{2}-4\\sqrt{3}u-3=0", "result": "{u}_{1,\\:2}=\\frac{-\\left(-4\\sqrt{3}\\right)\\pm\\:\\sqrt{\\left(-4\\sqrt{3}\\right)^{2}-4\\left(-4\\right)\\left(-3\\right)}}{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 $${\\quad}a=-4,\\:b=-4\\sqrt{3},\\:c=-3$$", "result": "{u}_{1,\\:2}=\\frac{-\\left(-4\\sqrt{3}\\right)\\pm\\:\\sqrt{\\left(-4\\sqrt{3}\\right)^{2}-4\\left(-4\\right)\\left(-3\\right)}}{2\\left(-4\\right)}" } ], "meta": { "interimType": "Solving The Quadratic Equation With Quadratic Formula Definition 0Eq", "gptData": "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" } }, { "type": "interim", "title": "$$\\left(-4\\sqrt{3}\\right)^{2}-4\\left(-4\\right)\\left(-3\\right)=0$$", "input": "\\left(-4\\sqrt{3}\\right)^{2}-4\\left(-4\\right)\\left(-3\\right)", "result": "{u}_{1,\\:2}=\\frac{-\\left(-4\\sqrt{3}\\right)\\pm\\:\\sqrt{0}}{2\\left(-4\\right)}", "steps": [ { "type": "step", "primary": "Apply rule $$-\\left(-a\\right)=a$$", "result": "=\\left(-4\\sqrt{3}\\right)^{2}-4\\cdot\\:4\\cdot\\:3" }, { "type": "interim", "title": "$$\\left(-4\\sqrt{3}\\right)^{2}=4^{2}\\cdot\\:3$$", "input": "\\left(-4\\sqrt{3}\\right)^{2}", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$\\left(-a\\right)^{n}=a^{n},\\:$$if $$n$$ is even", "secondary": [ "$$\\left(-4\\sqrt{3}\\right)^{2}=\\left(4\\sqrt{3}\\right)^{2}$$" ], "result": "=\\left(4\\sqrt{3}\\right)^{2}" }, { "type": "step", "primary": "Apply exponent rule: $$\\left(a\\cdot\\:b\\right)^{n}=a^{n}b^{n}$$", "result": "=4^{2}\\left(\\sqrt{3}\\right)^{2}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "interim", "title": "$$\\left(\\sqrt{3}\\right)^{2}:{\\quad}3$$", "steps": [ { "type": "step", "primary": "Apply radical rule: $$\\sqrt{a}=a^{\\frac{1}{2}}$$", "result": "=\\left(3^{\\frac{1}{2}}\\right)^{2}", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } }, { "type": "step", "primary": "Apply exponent rule: $$\\left(a^{b}\\right)^{c}=a^{bc}$$", "result": "=3^{\\frac{1}{2}\\cdot\\:2}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "interim", "title": "$$\\frac{1}{2}\\cdot\\:2=1$$", "input": "\\frac{1}{2}\\cdot\\:2", "result": "=3", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{1\\cdot\\:2}{2}" }, { "type": "step", "primary": "Cancel the common factor: $$2$$", "result": "=1" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l3vTdf410Ywhq1vZ0kzF8e30Fwl9QKPJxyO/TFRCb5Grju+5Z51e/ZZSD3gRHwjBE9/03SOiEv+BIHutWLr6nUfz18ijmoplMAomfJM9x8W1GdKgiNs+PolKvTuWzYk/" } } ], "meta": { "interimType": "N/A" } }, { "type": "step", "result": "=4^{2}\\cdot\\:3" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s71xvHyH5eNNgs8yW3fa2CKydVBn2NNCFZqg4ZoVh6UwqjkVi15I8rBefLi4Iyt2wr4/14o8RLtSK0WJRLt3AWMbu7qyZDwFK9jWGpMmc0pr4LfOxeCwWHeBw2KfnPes1vMdGh0b4sYpbe5HFlR6NPJA==" } }, { "type": "interim", "title": "$$4\\cdot\\:4\\cdot\\:3=48$$", "input": "4\\cdot\\:4\\cdot\\:3", "steps": [ { "type": "step", "primary": "Multiply the numbers: $$4\\cdot\\:4\\cdot\\:3=48$$", "result": "=48" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7aqTaUg1qT9nDhx1eUoutzxUbjr1up91GPxQqAij33CujkVi15I8rBefLi4Iyt2wrCklV65q5uyJBBcrfcaVZq7xcSXgbJgwa5HuBBkPlckn++yrn+9cNysRon8HIu7bT" } }, { "type": "step", "result": "=4^{2}\\cdot\\:3-48" }, { "type": "interim", "title": "$$4^{2}\\cdot\\:3=48$$", "input": "4^{2}\\cdot\\:3", "steps": [ { "type": "step", "primary": "$$4^{2}=16$$", "result": "=16\\cdot\\:3" }, { "type": "step", "primary": "Multiply the numbers: $$16\\cdot\\:3=48$$", "result": "=48" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7VPQv5NmETWojg73v8B9EnC061ljBSPJeENOw2efoSWumgEsDh9eg4me0entNzllgwUz9JjPZK5WSZB53PR+nvQW+99ohgnhk2vrkHIRsmBUkt3WiGR7ZaCaXvz77bMjS" } }, { "type": "step", "result": "=48-48" }, { "type": "step", "primary": "Subtract the numbers: $$48-48=0$$", "result": "=0" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s71xvHyH5eNNgs8yW3fa2CK565HV5Etvy5npqB8iJxZzHehkKrn0era9rz8TlL+x/vZuJKdCFsPJy1+5gBMEc9dkt+ykOJBuJSRvF8I9Rrs53zTI/i1xF2GOqdQtBTT+c8wT7KbPrmvDecvDbmS27gbQ==" } }, { "type": "step", "result": "u=\\frac{-\\left(-4\\sqrt{3}\\right)}{2\\left(-4\\right)}" }, { "type": "interim", "title": "$$\\frac{-\\left(-4\\sqrt{3}\\right)}{2\\left(-4\\right)}=-\\frac{\\sqrt{3}}{2}$$", "input": "\\frac{-\\left(-4\\sqrt{3}\\right)}{2\\left(-4\\right)}", "result": "u=-\\frac{\\sqrt{3}}{2}", "steps": [ { "type": "step", "primary": "Remove parentheses: $$\\left(-a\\right)=-a,\\:-\\left(-a\\right)=a$$", "result": "=\\frac{4\\sqrt{3}}{-2\\cdot\\:4}" }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:4=8$$", "result": "=\\frac{4\\sqrt{3}}{-8}" }, { "type": "step", "primary": "Apply the fraction rule: $$\\frac{a}{-b}=-\\frac{a}{b}$$", "result": "=-\\frac{4\\sqrt{3}}{8}" }, { "type": "step", "primary": "Cancel the common factor: $$4$$", "result": "=-\\frac{\\sqrt{3}}{2}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7BaRGlw68yflRXEFZzbCaTGInUDCP7b00dXLyj7VdDNctOtZYwUjyXhDTsNnn6ElryOsg4xTbsj8PJfnagYu7Qycg0t9KufQ5/Ecxx5MB743/P/+v51eTuH2/F4MIu5mqVkJ7mOgTTTE7+LV6kzrc+jdepsiTxBpWU6vmWa+XxBom/e8+HIkQc4F7XJ2JfHIpbqwnNL+vkns16SbvknXX+A==" } }, { "type": "step", "primary": "The solution to the quadratic equation is:", "result": "u=-\\frac{\\sqrt{3}}{2}" } ], "meta": { "solvingClass": "Equations", "interimType": "Equations" } }, { "type": "step", "primary": "Substitute back $$u=\\cos\\left(θ\\right)$$", "result": "\\cos\\left(θ\\right)=-\\frac{\\sqrt{3}}{2}" } ], "meta": { "interimType": "Substitution Method 0Eq" } }, { "type": "interim", "title": "$$\\cos\\left(θ\\right)=-\\frac{\\sqrt{3}}{2}{\\quad:\\quad}θ=\\frac{5π}{6}+2πn,\\:θ=\\frac{7π}{6}+2πn$$", "input": "\\cos\\left(θ\\right)=-\\frac{\\sqrt{3}}{2}", "steps": [ { "type": "interim", "title": "General solutions for $$\\cos\\left(θ\\right)=-\\frac{\\sqrt{3}}{2}$$", "result": "θ=\\frac{5π}{6}+2πn,\\:θ=\\frac{7π}{6}+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": "θ=\\frac{5π}{6}+2πn,\\:θ=\\frac{7π}{6}+2πn" } ], "meta": { "interimType": "Trig General Solutions cos 1Eq" } } ], "meta": { "interimType": "N/A" } }, { "type": "step", "primary": "Combine all the solutions", "result": "θ=\\frac{5π}{6}+2πn,\\:θ=\\frac{7π}{6}+2πn" }, { "type": "interim", "title": "Verify solutions by plugging them into the original equation", "steps": [ { "type": "step", "primary": "Check the solutions by plugging them into $$\\sin\\left(θ\\right)-\\sqrt{3}\\cos\\left(θ\\right)=2$$<br/>Remove the ones that don't agree with the equation." }, { "type": "interim", "title": "Check the solution $$\\frac{5π}{6}+2πn:{\\quad}$$True", "input": "\\frac{5π}{6}+2πn", "steps": [ { "type": "step", "primary": "Plug in $$n=1$$", "result": "\\frac{5π}{6}+2π1" }, { "type": "step", "primary": "For $$\\sin\\left(θ\\right)-\\sqrt{3}\\cos\\left(θ\\right)=2{\\quad}$$plug in$${\\quad}θ=\\frac{5π}{6}+2π1$$", "result": "\\sin\\left(\\frac{5π}{6}+2π1\\right)-\\sqrt{3}\\cos\\left(\\frac{5π}{6}+2π1\\right)=2" }, { "type": "step", "primary": "Refine", "result": "2=2" }, { "type": "step", "result": "\\Rightarrow\\:\\mathrm{True}" } ], "meta": { "interimType": "Check One Solution 1Eq" } }, { "type": "interim", "title": "Check the solution $$\\frac{7π}{6}+2πn:{\\quad}$$False", "input": "\\frac{7π}{6}+2πn", "steps": [ { "type": "step", "primary": "Plug in $$n=1$$", "result": "\\frac{7π}{6}+2π1" }, { "type": "step", "primary": "For $$\\sin\\left(θ\\right)-\\sqrt{3}\\cos\\left(θ\\right)=2{\\quad}$$plug in$${\\quad}θ=\\frac{7π}{6}+2π1$$", "result": "\\sin\\left(\\frac{7π}{6}+2π1\\right)-\\sqrt{3}\\cos\\left(\\frac{7π}{6}+2π1\\right)=2" }, { "type": "step", "primary": "Refine", "result": "1=2" }, { "type": "step", "result": "\\Rightarrow\\:\\mathrm{False}" } ], "meta": { "interimType": "Check One Solution 1Eq" } } ], "meta": { "interimType": "Check Solutions Plug Preface 1Eq" } }, { "type": "step", "result": "θ=\\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": "θ", "plotRequest": "\\sin(θ)-\\sqrt{3}\\cos(θ)-2" }, "showViewLarger": true } }, "meta": { "showVerify": true } }