cos(x)+tan(x)sin(x)=2

{ "query": { "display": "$$\\cos\\left(x\\right)+\\tan\\left(x\\right)\\sin\\left(x\\right)=2$$", "symbolab_question": "EQUATION#\\cos(x)+\\tan(x)\\sin(x)=2" }, "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": "$$\\cos\\left(x\\right)+\\tan\\left(x\\right)\\sin\\left(x\\right)=2{\\quad:\\quad}x=\\frac{π}{3}+2πn,\\:x=\\frac{5π}{3}+2πn$$", "input": "\\cos\\left(x\\right)+\\tan\\left(x\\right)\\sin\\left(x\\right)=2", "steps": [ { "type": "step", "primary": "Subtract $$2$$ from both sides", "result": "\\cos\\left(x\\right)+\\tan\\left(x\\right)\\sin\\left(x\\right)-2=0" }, { "type": "interim", "title": "Express with sin, cos", "input": "-2+\\cos\\left(x\\right)+\\sin\\left(x\\right)\\tan\\left(x\\right)", "result": "\\frac{\\cos^{2}\\left(x\\right)+\\sin^{2}\\left(x\\right)-2\\cos\\left(x\\right)}{\\cos\\left(x\\right)}=0", "steps": [ { "type": "step", "primary": "Use the basic trigonometric identity: $$\\tan\\left(x\\right)=\\frac{\\sin\\left(x\\right)}{\\cos\\left(x\\right)}$$", "result": "=-2+\\cos\\left(x\\right)+\\sin\\left(x\\right)\\frac{\\sin\\left(x\\right)}{\\cos\\left(x\\right)}" }, { "type": "interim", "title": "Simplify $$-2+\\cos\\left(x\\right)+\\sin\\left(x\\right)\\frac{\\sin\\left(x\\right)}{\\cos\\left(x\\right)}:{\\quad}\\frac{-2\\cos\\left(x\\right)+\\cos^{2}\\left(x\\right)+\\sin^{2}\\left(x\\right)}{\\cos\\left(x\\right)}$$", "input": "-2+\\cos\\left(x\\right)+\\sin\\left(x\\right)\\frac{\\sin\\left(x\\right)}{\\cos\\left(x\\right)}", "result": "=\\frac{-2\\cos\\left(x\\right)+\\cos^{2}\\left(x\\right)+\\sin^{2}\\left(x\\right)}{\\cos\\left(x\\right)}", "steps": [ { "type": "interim", "title": "$$\\sin\\left(x\\right)\\frac{\\sin\\left(x\\right)}{\\cos\\left(x\\right)}=\\frac{\\sin^{2}\\left(x\\right)}{\\cos\\left(x\\right)}$$", "input": "\\sin\\left(x\\right)\\frac{\\sin\\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{\\sin\\left(x\\right)\\sin\\left(x\\right)}{\\cos\\left(x\\right)}" }, { "type": "interim", "title": "$$\\sin\\left(x\\right)\\sin\\left(x\\right)=\\sin^{2}\\left(x\\right)$$", "input": "\\sin\\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": "=\\sin^{1+1}\\left(x\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$1+1=2$$", "result": "=\\sin^{2}\\left(x\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7Db5kOPGdwaYYCr65H3kEuI7oN3fOm5Kcpc0NdzQFiDj9ovYKijQYhJDCbxu/nAOJh8ihxP+4PwTZMAHoWYe0rYRgj2SDWqhbeg4ibDhNi7JNro/AJAWcGEjut/HzR49zgXIiLmXebpqW8NAeupm/ZQ==" } }, { "type": "step", "result": "=\\frac{\\sin^{2}\\left(x\\right)}{\\cos\\left(x\\right)}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7vDjAnI4+bJMfnz1gQ2685KM5dG3EeXRvnJJvZA7IzMY8h+u5fV4iBAkECPyXf7J5zMFYmi1F5Hg/ibpEToVnYzzxBaKh4A2pz+SncXGp6/b2Q8SaoI7ZIYpDY31RQ4xuZEt3ZXAiqUE0HIXrrrezJNaq5k0eiGSf0z7QripC8KTAcwDR0qqNK1jfOAbxYE0LxPJzd9XLwQ6NF+TMQiN1CO8veVXrIcELR/o8f7R+HhCwiNrEngO+NNvZ9sqNu+2V" } }, { "type": "step", "result": "=-2+\\cos\\left(x\\right)+\\frac{\\sin^{2}\\left(x\\right)}{\\cos\\left(x\\right)}" }, { "type": "step", "primary": "Convert element to fraction: $$2=\\frac{2\\cos\\left(x\\right)}{\\cos\\left(x\\right)},\\:\\cos\\left(x\\right)=\\frac{\\cos\\left(x\\right)\\cos\\left(x\\right)}{\\cos\\left(x\\right)}$$", "result": "=-\\frac{2\\cos\\left(x\\right)}{\\cos\\left(x\\right)}+\\frac{\\cos\\left(x\\right)\\cos\\left(x\\right)}{\\cos\\left(x\\right)}+\\frac{\\sin^{2}\\left(x\\right)}{\\cos\\left(x\\right)}" }, { "type": "step", "primary": "Since the denominators are equal, combine the fractions: $$\\frac{a}{c}\\pm\\frac{b}{c}=\\frac{a\\pm\\:b}{c}$$", "result": "=\\frac{-2\\cos\\left(x\\right)+\\cos\\left(x\\right)\\cos\\left(x\\right)+\\sin^{2}\\left(x\\right)}{\\cos\\left(x\\right)}" }, { "type": "interim", "title": "$$-2\\cos\\left(x\\right)+\\cos\\left(x\\right)\\cos\\left(x\\right)+\\sin^{2}\\left(x\\right)=-2\\cos\\left(x\\right)+\\cos^{2}\\left(x\\right)+\\sin^{2}\\left(x\\right)$$", "input": "-2\\cos\\left(x\\right)+\\cos\\left(x\\right)\\cos\\left(x\\right)+\\sin^{2}\\left(x\\right)", "steps": [ { "type": "interim", "title": "$$\\cos\\left(x\\right)\\cos\\left(x\\right)=\\cos^{2}\\left(x\\right)$$", "input": "\\cos\\left(x\\right)\\cos\\left(x\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\cos\\left(x\\right)\\cos\\left(x\\right)=\\:\\cos^{1+1}\\left(x\\right)$$" ], "result": "=\\cos^{1+1}\\left(x\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$1+1=2$$", "result": "=\\cos^{2}\\left(x\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7OgJyajQjkqczngxvLtluw47oN3fOm5Kcpc0NdzQFiDj9ovYKijQYhJDCbxu/nAOJVxXBxD1gYRAlNp97nQuTZFXRu5R8U1G8Rh9s+llHwfqtic1bCnH3jLV3vr22vWk8gIJE6eFSdaQPkT4FMktmcw==" } }, { "type": "step", "result": "=-2\\cos\\left(x\\right)+\\cos^{2}\\left(x\\right)+\\sin^{2}\\left(x\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7PX0HCNmVyJme9sR5WozT39JPzeKjCqgNcXfRVIDVGOdKSvm3zOo8eGHvXoEovPSvzRqDxPUzBN6vjj5oJL9kUPPKuVEPkyEmJ1a74MqHgcOfElW3eVb7wqxnBKTqlJxOh8ihxP+4PwTZMAHoWYe0rfsL6hFr392h3nkcUCqIWDDw9/zJ4TnfFdyWctQ5b7DIaWSHBWwVSqIvak917VVC0YHlwcodgn2LdoN1sMzDAkMWp32lBij3YO+u0p+hNn+lgXIiLmXebpqW8NAeupm/ZQ==" } }, { "type": "step", "result": "=\\frac{-2\\cos\\left(x\\right)+\\cos^{2}\\left(x\\right)+\\sin^{2}\\left(x\\right)}{\\cos\\left(x\\right)}" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } } ], "meta": { "interimType": "Trig Express Sin Cos 0Eq" } }, { "type": "step", "primary": "$$\\frac{f\\left(x\\right)}{g\\left(x\\right)}=0{\\quad\\Rightarrow\\quad}f\\left(x\\right)=0$$", "result": "\\cos^{2}\\left(x\\right)+\\sin^{2}\\left(x\\right)-2\\cos\\left(x\\right)=0" }, { "type": "interim", "title": "Rewrite using trig identities", "input": "\\cos^{2}\\left(x\\right)+\\sin^{2}\\left(x\\right)-2\\cos\\left(x\\right)", "result": "-2\\cos\\left(x\\right)+1=0", "steps": [ { "type": "step", "primary": "Use the Pythagorean identity: $$\\cos^{2}\\left(x\\right)+\\sin^{2}\\left(x\\right)=1$$", "result": "=-2\\cos\\left(x\\right)+1" } ], "meta": { "interimType": "Trig Rewrite Using Trig identities 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7lr/MB5ele2A6D5h4s+0Rx0x6Ee7qFiY+BtOrucHGoWW8+bFQoS5VVWdKME6m8MgEBTEtBtB4I1F8HN0UDpp/QWM5y3SAWhw+ImwZxCCyUw11B1tZhOVEFEyMLkHemoPszEH7NIGk3xlro3ZeLLYqcdWfoiciINFIkPt89dI9Wjd6pfF1z6umzUJTJvt+ojYZGePNthjlfN9FyWkEQSe2L1EHcM9bz6xhblHaaRGSC5C/Mg94S0N9we//Py6WzxN6" } }, { "type": "interim", "title": "Move $$1\\:$$to the right side", "input": "-2\\cos\\left(x\\right)+1=0", "result": "-2\\cos\\left(x\\right)=-1", "steps": [ { "type": "step", "primary": "Subtract $$1$$ from both sides", "result": "-2\\cos\\left(x\\right)+1-1=0-1" }, { "type": "step", "primary": "Simplify", "result": "-2\\cos\\left(x\\right)=-1" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": "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" } }, { "type": "interim", "title": "Divide both sides by $$-2$$", "input": "-2\\cos\\left(x\\right)=-1", "result": "\\cos\\left(x\\right)=\\frac{1}{2}", "steps": [ { "type": "step", "primary": "Divide both sides by $$-2$$", "result": "\\frac{-2\\cos\\left(x\\right)}{-2}=\\frac{-1}{-2}" }, { "type": "step", "primary": "Simplify", "result": "\\cos\\left(x\\right)=\\frac{1}{2}" } ], "meta": { "interimType": "Divide Both Sides Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7Vyp+7iipLiCmC34kmF1yJWpyKNnVS6286xyIzxDnxfJ32tovqQJUqmDj6vwvJQPSvlgjwPVa21y0zb+fGt7Mh2p7t9VOlxVwXev7QS8K3g/2/yFmPBn9I7Qruh6hzCYyW09jiL5am2bzR3IB671bXnFxn1rssLY3m6zG42uRPCSQw7D74UJEEmspKsfbHPrGfCVeV3Zborci58Dhc7Dmlel5Q9dfYSbH4gy6lXHQaRuXiAEBxoPbme2UIjeXXKLxFzgpezzGAk2IuQnCvRv/k8Vt+7hPAieRjUuMyYe7XLxpDVeK92dMkVb/ckOCbkIaxBVMhfEdue6qsYh7lS/4QsMhSOmsID5cim3F9v8Rs44HgIRL2cbEPZLm634cOSI7G1YrInBT7eibt4cYhvHwDzRQJEdnJEoPNB8oHGc09FO8leuFFBMpmtXRConWh4AZ+97UWQsZjLul80dbuB7w9a1I6yWflsFlt3bztcKjbqc6SvZ4Z3EOq8xsbPTUoN8oZUCPrI3nRtiYfTFzB08Dxg==" } }, { "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": "\\cos(x)+\\tan(x)\\sin(x)-2" }, "showViewLarger": true } }, "meta": { "showVerify": true } }