2=4cos(x)

{ "query": { "display": "$$2=4\\cos\\left(x\\right)$$", "symbolab_question": "EQUATION#2=4\\cos(x)" }, "solution": { "level": "PERFORMED", "subject": "Trigonometry", "topic": "Trig Equations", "subTopic": "Trig Equations", "default": "x=\\frac{π}{3}+2πn,x=\\frac{5π}{3}+2πn", "degrees": "x=60^{\\circ }+360^{\\circ }n,x=300^{\\circ }+360^{\\circ }n", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$2=4\\cos\\left(x\\right){\\quad:\\quad}x=\\frac{π}{3}+2πn,\\:x=\\frac{5π}{3}+2πn$$", "input": "2=4\\cos\\left(x\\right)", "steps": [ { "type": "step", "primary": "Switch sides", "result": "4\\cos\\left(x\\right)=2" }, { "type": "interim", "title": "Divide both sides by $$4$$", "input": "4\\cos\\left(x\\right)=2", "result": "\\cos\\left(x\\right)=\\frac{1}{2}", "steps": [ { "type": "step", "primary": "Divide both sides by $$4$$", "result": "\\frac{4\\cos\\left(x\\right)}{4}=\\frac{2}{4}" }, { "type": "step", "primary": "Simplify", "result": "\\cos\\left(x\\right)=\\frac{1}{2}" } ], "meta": { "interimType": "Divide Both Sides Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s78xk2pWtXOxpmo58sRKEQmrYQkMNjndEipufh+Wqx31NPnie/F7TDTqTxkucfXTe/anfB9LIhO2yo4oZlMuz5YG6DDJHWnEbFgS5QJ7Yus3MjkYhWaYooxcLQLbyMdQYurTH6CH2AfoyKEr0+/Uxe+cZoxkl7XKCNfz0BLTb8L1fdFpY8+TJQURh2lEt0g23416V4XDAqdH7arA5dzvyiHKJ4EOUSN7omJaP3ec2BE5cVk8t2Twjh+zjp9dlYTco/GgwSbIcd5JjP/d1KuCbW94zDSf2vh/i+3s4cO/oKwZy1rKw+72V41LgxeCOecKXOfIvtRFItCUfO72DblJDvr6ag359nJICSNxR1sdkiJYDwt9LEn7QCBUukJKctfSJK0WcS2OQ+KOcgbSBFF8tALeNj26EEByMJ+Did4E29hUw=" } }, { "type": "interim", "title": "General solutions for $$\\cos\\left(x\\right)=\\frac{1}{2}$$", "result": "x=\\frac{π}{3}+2πn,\\:x=\\frac{5π}{3}+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}{3}+2πn,\\:x=\\frac{5π}{3}+2πn" } ], "meta": { "interimType": "Trig General Solutions cos 1Eq" } } ], "meta": { "solvingClass": "Trig Equations", "practiceLink": "/practice/trigonometry-practice#area=main&subtopic=Trig%20Equations", "practiceTopic": "Trig Equations" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "2-4\\cos(x)" }, "showViewLarger": true } }, "meta": { "showVerify": true } }