cos(3x)=-3cos(x)

{ "query": { "display": "$$\\cos\\left(3x\\right)=-3\\cos\\left(x\\right)$$", "symbolab_question": "EQUATION#\\cos(3x)=-3\\cos(x)" }, "solution": { "level": "PERFORMED", "subject": "Trigonometry", "topic": "Trig Equations", "subTopic": "Trig Equations", "default": "x=\\frac{π}{2}+2πn,x=\\frac{3π}{2}+2πn", "degrees": "x=90^{\\circ }+360^{\\circ }n,x=270^{\\circ }+360^{\\circ }n", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\cos\\left(3x\\right)=-3\\cos\\left(x\\right){\\quad:\\quad}x=\\frac{π}{2}+2πn,\\:x=\\frac{3π}{2}+2πn$$", "input": "\\cos\\left(3x\\right)=-3\\cos\\left(x\\right)", "steps": [ { "type": "step", "primary": "Subtract $$-3\\cos\\left(x\\right)$$ from both sides", "result": "\\cos\\left(3x\\right)+3\\cos\\left(x\\right)=0" }, { "type": "interim", "title": "Rewrite using trig identities", "input": "\\cos\\left(3x\\right)+3\\cos\\left(x\\right)", "result": "4\\cos^{3}\\left(x\\right)=0", "steps": [ { "type": "interim", "title": "$$\\cos\\left(3x\\right)=4\\cos^{3}\\left(x\\right)-3\\cos\\left(x\\right)$$", "input": "\\cos\\left(3x\\right)", "steps": [ { "type": "interim", "title": "Rewrite using trig identities", "input": "\\cos\\left(3x\\right)", "result": "=\\cos\\left(x\\right)\\cos\\left(2x\\right)-2\\sin^{2}\\left(x\\right)\\cos\\left(x\\right)", "steps": [ { "type": "step", "primary": "Rewrite as", "result": "=\\cos\\left(2x+x\\right)" }, { "type": "step", "primary": "Use the Angle Sum identity: $$\\cos\\left(s+t\\right)=\\cos\\left(s\\right)\\cos\\left(t\\right)-\\sin\\left(s\\right)\\sin\\left(t\\right)$$", "result": "=\\cos\\left(2x\\right)\\cos\\left(x\\right)-\\sin\\left(2x\\right)\\sin\\left(x\\right)" }, { "type": "step", "primary": "Use the Double Angle identity: $$\\sin\\left(2x\\right)=2\\sin\\left(x\\right)\\cos\\left(x\\right)$$", "result": "=\\cos\\left(2x\\right)\\cos\\left(x\\right)-2\\sin\\left(x\\right)\\cos\\left(x\\right)\\sin\\left(x\\right)" }, { "type": "interim", "title": "Simplify $$\\cos\\left(2x\\right)\\cos\\left(x\\right)-2\\sin\\left(x\\right)\\cos\\left(x\\right)\\sin\\left(x\\right):{\\quad}\\cos\\left(x\\right)\\cos\\left(2x\\right)-2\\sin^{2}\\left(x\\right)\\cos\\left(x\\right)$$", "input": "\\cos\\left(2x\\right)\\cos\\left(x\\right)-2\\sin\\left(x\\right)\\cos\\left(x\\right)\\sin\\left(x\\right)", "result": "=\\cos\\left(x\\right)\\cos\\left(2x\\right)-2\\sin^{2}\\left(x\\right)\\cos\\left(x\\right)", "steps": [ { "type": "interim", "title": "$$2\\sin\\left(x\\right)\\cos\\left(x\\right)\\sin\\left(x\\right)=2\\sin^{2}\\left(x\\right)\\cos\\left(x\\right)$$", "input": "2\\sin\\left(x\\right)\\cos\\left(x\\right)\\sin\\left(x\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\sin\\left(x\\right)\\sin\\left(x\\right)=\\:\\sin^{1+1}\\left(x\\right)$$" ], "result": "=2\\cos\\left(x\\right)\\sin^{1+1}\\left(x\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$1+1=2$$", "result": "=2\\cos\\left(x\\right)\\sin^{2}\\left(x\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7dCLXYy2Yt9p1+SBnDXNf7O9drm6SkPRRfRCRWaenDXXdd47a0hQ8flDbGsI5To1dp4ERY5rl3RmA4Q2oLBCOcmg5s5mL5kdzgH1S4tWlBNh04xTNlEc8iWnVHxPamSnbpepZqlZAYNxOypikTE7Zg8DY5rcbbXm63hZvX3JEjZKU8y0ZqgCmUFkzmDfKYnYn" } }, { "type": "step", "result": "=\\cos\\left(x\\right)\\cos\\left(2x\\right)-2\\sin^{2}\\left(x\\right)\\cos\\left(x\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } } ], "meta": { "interimType": "Trig Rewrite Using Trig identities 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7lRPp6QndqgNT+L6IZutAbBuW+T7VWHANzzVbDcPRRXoIvVZ5+IpoLiKFp9zdXqo3RiJG90FtsO8uDfN0UpLW3P5FLqcqe8V8QfGJihEOR6DyI9ef8OnL+4RiuMPvaKALCSoj8WxzNkU1xYqYJngqhaFnx3zs9U5EV+00OjB0LWPWwPs1+Gw97t4MeuaNjSYTRvemj3GBE2iIDcXU+cR6iI+gxrQ1tCXeKlYIJ1n6NLewiNrEngO+NNvZ9sqNu+2V" } }, { "type": "step", "primary": "Use the Double Angle identity: $$\\cos\\left(2x\\right)=2\\cos^{2}\\left(x\\right)-1$$", "result": "=\\left(2\\cos^{2}\\left(x\\right)-1\\right)\\cos\\left(x\\right)-2\\sin^{2}\\left(x\\right)\\cos\\left(x\\right)" }, { "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": "=\\left(2\\cos^{2}\\left(x\\right)-1\\right)\\cos\\left(x\\right)-2\\left(1-\\cos^{2}\\left(x\\right)\\right)\\cos\\left(x\\right)" }, { "type": "interim", "title": "Expand $$\\left(2\\cos^{2}\\left(x\\right)-1\\right)\\cos\\left(x\\right)-2\\left(1-\\cos^{2}\\left(x\\right)\\right)\\cos\\left(x\\right):{\\quad}4\\cos^{3}\\left(x\\right)-3\\cos\\left(x\\right)$$", "input": "\\left(2\\cos^{2}\\left(x\\right)-1\\right)\\cos\\left(x\\right)-2\\left(1-\\cos^{2}\\left(x\\right)\\right)\\cos\\left(x\\right)", "result": "=4\\cos^{3}\\left(x\\right)-3\\cos\\left(x\\right)", "steps": [ { "type": "step", "result": "=\\cos\\left(x\\right)\\left(2\\cos^{2}\\left(x\\right)-1\\right)-2\\cos\\left(x\\right)\\left(1-\\cos^{2}\\left(x\\right)\\right)" }, { "type": "interim", "title": "Expand $$\\cos\\left(x\\right)\\left(2\\cos^{2}\\left(x\\right)-1\\right):{\\quad}2\\cos^{3}\\left(x\\right)-\\cos\\left(x\\right)$$", "input": "\\cos\\left(x\\right)\\left(2\\cos^{2}\\left(x\\right)-1\\right)", "result": "=2\\cos^{3}\\left(x\\right)-\\cos\\left(x\\right)-2\\left(1-\\cos^{2}\\left(x\\right)\\right)\\cos\\left(x\\right)", "steps": [ { "type": "step", "primary": "Apply the distributive law: $$a\\left(b-c\\right)=ab-ac$$", "secondary": [ "$$a=\\cos\\left(x\\right),\\:b=2\\cos^{2}\\left(x\\right),\\:c=1$$" ], "result": "=\\cos\\left(x\\right)2\\cos^{2}\\left(x\\right)-\\cos\\left(x\\right)1", "meta": { "practiceLink": "/practice/expansion-practice", "practiceTopic": "Expand Rules" } }, { "type": "step", "result": "=2\\cos^{2}\\left(x\\right)\\cos\\left(x\\right)-1\\cos\\left(x\\right)" }, { "type": "interim", "title": "Simplify $$2\\cos^{2}\\left(x\\right)\\cos\\left(x\\right)-1\\cdot\\:\\cos\\left(x\\right):{\\quad}2\\cos^{3}\\left(x\\right)-\\cos\\left(x\\right)$$", "input": "2\\cos^{2}\\left(x\\right)\\cos\\left(x\\right)-1\\cos\\left(x\\right)", "result": "=2\\cos^{3}\\left(x\\right)-\\cos\\left(x\\right)", "steps": [ { "type": "interim", "title": "$$2\\cos^{2}\\left(x\\right)\\cos\\left(x\\right)=2\\cos^{3}\\left(x\\right)$$", "input": "2\\cos^{2}\\left(x\\right)\\cos\\left(x\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\cos^{2}\\left(x\\right)\\cos\\left(x\\right)=\\:\\cos^{2+1}\\left(x\\right)$$" ], "result": "=2\\cos^{2+1}\\left(x\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$2+1=3$$", "result": "=2\\cos^{3}\\left(x\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7FMDrC3J/ZVrenAVm+Grw5tL3+aeFexLDgCRLUCdS8UcJQJZuTAY5js+oqjdT8kslZT2T5Oe/rGefEnvU1621pu5byrQDQVCXUD0vH/fvOdybuDtfuUsd7wBmEwIunWRKbtdkfpC4az4mQsTNEyAfAIlHwIJiU7AeWULaq8SsF+M=" } }, { "type": "interim", "title": "$$1\\cdot\\:\\cos\\left(x\\right)=\\cos\\left(x\\right)$$", "input": "1\\cos\\left(x\\right)", "steps": [ { "type": "step", "primary": "Multiply: $$1\\cdot\\:\\cos\\left(x\\right)=\\cos\\left(x\\right)$$", "result": "=\\cos\\left(x\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7RRhblr7eOi0Q5jpt8sVUBt13jtrSFDx+UNsawjlOjV3Y54gBruGA/XAJJviwUWdhP8vQyhiD4JSfqjIvcQ7til0y8Cqq4EyFS+F2d9rFxX5ewHvtinFuxmQY3pL50/lu" } }, { "type": "step", "result": "=2\\cos^{3}\\left(x\\right)-\\cos\\left(x\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } } ], "meta": { "interimType": "Algebraic Manipulation Expand Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7NciHuLB6v3lGOOFG9cOx5MADykWcpxGFaMDJmVs16cHdd47a0hQ8flDbGsI5To1dJW4YOR/tooxVjdJm2Ug49x/w0d8sok0hbrog7Ppo9T+jeh7+jKEzLb7VNCEMF3Z/TzUpvge51uP/vAYJhDDKq22a7NflAm5Sxhjk4yIsfPHWjHXRgXCaj7ppu4kBUW6S" } }, { "type": "interim", "title": "Expand $$-2\\cos\\left(x\\right)\\left(1-\\cos^{2}\\left(x\\right)\\right):{\\quad}-2\\cos\\left(x\\right)+2\\cos^{3}\\left(x\\right)$$", "input": "-2\\cos\\left(x\\right)\\left(1-\\cos^{2}\\left(x\\right)\\right)", "result": "=2\\cos^{3}\\left(x\\right)-\\cos\\left(x\\right)-2\\cos\\left(x\\right)+2\\cos^{3}\\left(x\\right)", "steps": [ { "type": "step", "primary": "Apply the distributive law: $$a\\left(b-c\\right)=ab-ac$$", "secondary": [ "$$a=-2\\cos\\left(x\\right),\\:b=1,\\:c=\\cos^{2}\\left(x\\right)$$" ], "result": "=-2\\cos\\left(x\\right)1-\\left(-2\\cos\\left(x\\right)\\right)\\cos^{2}\\left(x\\right)", "meta": { "practiceLink": "/practice/expansion-practice", "practiceTopic": "Expand Rules" } }, { "type": "step", "primary": "Apply minus-plus rules", "secondary": [ "$$-\\left(-a\\right)=a$$" ], "result": "=-2\\cdot\\:1\\cos\\left(x\\right)+2\\cos^{2}\\left(x\\right)\\cos\\left(x\\right)" }, { "type": "interim", "title": "Simplify $$-2\\cdot\\:1\\cdot\\:\\cos\\left(x\\right)+2\\cos^{2}\\left(x\\right)\\cos\\left(x\\right):{\\quad}-2\\cos\\left(x\\right)+2\\cos^{3}\\left(x\\right)$$", "input": "-2\\cdot\\:1\\cos\\left(x\\right)+2\\cos^{2}\\left(x\\right)\\cos\\left(x\\right)", "result": "=-2\\cos\\left(x\\right)+2\\cos^{3}\\left(x\\right)", "steps": [ { "type": "interim", "title": "$$2\\cdot\\:1\\cdot\\:\\cos\\left(x\\right)=2\\cos\\left(x\\right)$$", "input": "2\\cdot\\:1\\cos\\left(x\\right)", "steps": [ { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:1=2$$", "result": "=2\\cos\\left(x\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7hA94Ei8IRXRSFT5crEz4sNp/E4XcAYTeZX9p+FhOStmjkVi15I8rBefLi4Iyt2wrlpro1S6+PbItwi0Q1DKAMrIhZBS2NlAqsVIN9Vo5i+gP+VRiluAkE88OTs2qaaGN4j/daA44VYx1XGd0VviNZA==" } }, { "type": "interim", "title": "$$2\\cos^{2}\\left(x\\right)\\cos\\left(x\\right)=2\\cos^{3}\\left(x\\right)$$", "input": "2\\cos^{2}\\left(x\\right)\\cos\\left(x\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\cos^{2}\\left(x\\right)\\cos\\left(x\\right)=\\:\\cos^{2+1}\\left(x\\right)$$" ], "result": "=2\\cos^{2+1}\\left(x\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$2+1=3$$", "result": "=2\\cos^{3}\\left(x\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7FMDrC3J/ZVrenAVm+Grw5tL3+aeFexLDgCRLUCdS8UcJQJZuTAY5js+oqjdT8kslZT2T5Oe/rGefEnvU1621pu5byrQDQVCXUD0vH/fvOdybuDtfuUsd7wBmEwIunWRKbtdkfpC4az4mQsTNEyAfAIlHwIJiU7AeWULaq8SsF+M=" } }, { "type": "step", "result": "=-2\\cos\\left(x\\right)+2\\cos^{3}\\left(x\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } } ], "meta": { "interimType": "Algebraic Manipulation Expand Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7sgIZW2WpqowUNh9y1fjDlPdUSbTeEtZo7vQx32xfcrsgJ/ZZA32ZInFBpDtxBfiK2KjlCKtfbs9sbhPov8godLXVlvzE7FVH141qHYId7wbuW8q0A0FQl1A9Lx/37znc/G2ANG+ldkUkzmezrPTTgyye6BEMegnzDzRvrMYrgrV53Kc9eyWImx9IGLQEQMMC" } }, { "type": "interim", "title": "Simplify $$2\\cos^{3}\\left(x\\right)-\\cos\\left(x\\right)-2\\cos\\left(x\\right)+2\\cos^{3}\\left(x\\right):{\\quad}4\\cos^{3}\\left(x\\right)-3\\cos\\left(x\\right)$$", "input": "2\\cos^{3}\\left(x\\right)-\\cos\\left(x\\right)-2\\cos\\left(x\\right)+2\\cos^{3}\\left(x\\right)", "result": "=4\\cos^{3}\\left(x\\right)-3\\cos\\left(x\\right)", "steps": [ { "type": "step", "primary": "Group like terms", "result": "=2\\cos^{3}\\left(x\\right)+2\\cos^{3}\\left(x\\right)-\\cos\\left(x\\right)-2\\cos\\left(x\\right)" }, { "type": "step", "primary": "Add similar elements: $$2\\cos^{3}\\left(x\\right)+2\\cos^{3}\\left(x\\right)=4\\cos^{3}\\left(x\\right)$$", "result": "=4\\cos^{3}\\left(x\\right)-\\cos\\left(x\\right)-2\\cos\\left(x\\right)" }, { "type": "step", "primary": "Add similar elements: $$-\\cos\\left(x\\right)-2\\cos\\left(x\\right)=-3\\cos\\left(x\\right)$$", "result": "=4\\cos^{3}\\left(x\\right)-3\\cos\\left(x\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Expand Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7XbvZgYkIRAGf8zO/3nYgVpq1Ul52PosA5ijcMacz4s+zK8JXsifpOfwStcnE4e6Mq6A2SQZIC0Cr/n5hXYn8pnCQoYlYQ8U+Tfyx0kyzI8jGYZ4KRYtRdJHLRCZEUHDue6HC37hAG55YBlhqI5UXttbA+zX4bD3u3gx65o2NJhOrJQTKrJDuojmio8lQinig0JArzKFCEDRvHdBDUHAoVywMvIgAO1bGqE/1MFJROyHk2l8LUWR5lD/KhDaP3wmw" } } ], "meta": { "interimType": "N/A" } }, { "type": "step", "result": "=4\\cos^{3}\\left(x\\right)-3\\cos\\left(x\\right)+3\\cos\\left(x\\right)" }, { "type": "step", "primary": "Simplify", "result": "=4\\cos^{3}\\left(x\\right)", "meta": { "solvingClass": "Solver" } } ], "meta": { "interimType": "Trig Rewrite Using Trig identities 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7lRPp6QndqgNT+L6IZutAbGRAM84mcck05DWWwOr4TE0FMS0G0HgjUXwc3RQOmn9BYznLdIBaHD4ibBnEILJTDXUHW1mE5UQUTIwuQd6ag+y5lt/xMT1lphMa2xH182VxybLESYjUbv3u1rZQ68TMItbA+zX4bD3u3gx65o2NJhNG96aPcYETaIgNxdT5xHqIj6DGtDW0Jd4qVggnWfo0t7CI2sSeA74029n2yo277ZU=" } }, { "type": "interim", "title": "Divide both sides by $$4$$", "input": "4\\cos^{3}\\left(x\\right)=0", "steps": [ { "type": "interim", "title": "Divide both sides by $$4$$", "input": "4\\cos^{3}\\left(x\\right)=0", "result": "\\cos^{3}\\left(x\\right)=0", "steps": [ { "type": "step", "primary": "Divide both sides by $$4$$", "result": "\\frac{4\\cos^{3}\\left(x\\right)}{4}=\\frac{0}{4}" }, { "type": "step", "primary": "Simplify", "result": "\\cos^{3}\\left(x\\right)=0" } ], "meta": { "interimType": "Divide Both Sides Specific 1Eq", "gptData": "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" } }, { "type": "step", "primary": "Apply rule $$x^n=0\\quad\\Rightarrow\\quad\\:x=0$$" } ], "meta": { "interimType": "Divide Both Sides Specific 1Eq" } }, { "type": "step", "result": "\\cos\\left(x\\right)=0" }, { "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": { "solvingClass": "Trig Equations", "practiceLink": "/practice/trigonometry-practice#area=main&subtopic=Trig%20Equations", "practiceTopic": "Trig Equations" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "\\cos(3x)+3\\cos(x)" }, "showViewLarger": true } }, "meta": { "showVerify": true } }