tan(x)+cos(x)=0

{ "query": { "display": "$$\\tan\\left(x\\right)+\\cos\\left(x\\right)=0$$", "symbolab_question": "EQUATION#\\tan(x)+\\cos(x)=0" }, "solution": { "level": "PERFORMED", "subject": "Trigonometry", "topic": "Trig Equations", "subTopic": "Trig Equations", "default": "x=-0.66623…+2πn,x=π+0.66623…+2πn", "degrees": "x=-38.17270…^{\\circ }+360^{\\circ }n,x=218.17270…^{\\circ }+360^{\\circ }n", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\tan\\left(x\\right)+\\cos\\left(x\\right)=0{\\quad:\\quad}x=-0.66623…+2πn,\\:x=π+0.66623…+2πn$$", "input": "\\tan\\left(x\\right)+\\cos\\left(x\\right)=0", "steps": [ { "type": "interim", "title": "Express with sin, cos", "input": "\\cos\\left(x\\right)+\\tan\\left(x\\right)", "result": "\\frac{\\cos^{2}\\left(x\\right)+\\sin\\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": "=\\cos\\left(x\\right)+\\frac{\\sin\\left(x\\right)}{\\cos\\left(x\\right)}" }, { "type": "interim", "title": "Simplify $$\\cos\\left(x\\right)+\\frac{\\sin\\left(x\\right)}{\\cos\\left(x\\right)}:{\\quad}\\frac{\\cos^{2}\\left(x\\right)+\\sin\\left(x\\right)}{\\cos\\left(x\\right)}$$", "input": "\\cos\\left(x\\right)+\\frac{\\sin\\left(x\\right)}{\\cos\\left(x\\right)}", "result": "=\\frac{\\cos^{2}\\left(x\\right)+\\sin\\left(x\\right)}{\\cos\\left(x\\right)}", "steps": [ { "type": "step", "primary": "Convert element to fraction: $$\\cos\\left(x\\right)=\\frac{\\cos\\left(x\\right)\\cos\\left(x\\right)}{\\cos\\left(x\\right)}$$", "result": "=\\frac{\\cos\\left(x\\right)\\cos\\left(x\\right)}{\\cos\\left(x\\right)}+\\frac{\\sin\\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{\\cos\\left(x\\right)\\cos\\left(x\\right)+\\sin\\left(x\\right)}{\\cos\\left(x\\right)}" }, { "type": "interim", "title": "$$\\cos\\left(x\\right)\\cos\\left(x\\right)+\\sin\\left(x\\right)=\\cos^{2}\\left(x\\right)+\\sin\\left(x\\right)$$", "input": "\\cos\\left(x\\right)\\cos\\left(x\\right)+\\sin\\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": "=\\cos^{2}\\left(x\\right)+\\sin\\left(x\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7OgJyajQjkqczngxvLtluw1iCh4YvB7OYFu29/9OJCg3dd47a0hQ8flDbGsI5To1dfZAUp6oTTB7dj4hFbvM10WRCKxaw3mIX9wdhjyGIEzJsJLgH4PxJhAtZcKZcwgBL0k/N4qMKqA1xd9FUgNUY5xaYKIpfQBJ7oPa7JmsEwhxfzj+LPLyx7lxOvWRG7qca" } }, { "type": "step", "result": "=\\frac{\\cos^{2}\\left(x\\right)+\\sin\\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\\left(x\\right)=0" }, { "type": "interim", "title": "Rewrite using trig identities", "input": "\\cos^{2}\\left(x\\right)+\\sin\\left(x\\right)", "result": "1+\\sin\\left(x\\right)-\\sin^{2}\\left(x\\right)=0", "steps": [ { "type": "step", "primary": "Use the Pythagorean identity: $$\\cos^{2}\\left(x\\right)+\\sin^{2}\\left(x\\right)=1$$", "secondary": [ "$$\\cos^{2}\\left(x\\right)=1-\\sin^{2}\\left(x\\right)$$" ], "result": "=1-\\sin^{2}\\left(x\\right)+\\sin\\left(x\\right)" } ], "meta": { "interimType": "Trig Rewrite Using Trig identities 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7lr/MB5ele2A6D5h4s+0Rx75ft2AtSA6BmT8JgEOBLPI52FciCV6Q/ZuTzBHIPdDy6bFjjx5IuxvIy7/nJMTFRsLgcgrrcmkH6sZs7VXsfRCXglwnmROl9cGKdo0OgfgyqGZUu+CPQE64Rxwg38I8Ladpa3M4SuhKB9tuevFDkRirve3E7cDlwD8G9VYfu6duTZGH8SqHtUWuYa2dpw0bulhsd89TDzuC3BUug+Pj7eQ=" } }, { "type": "interim", "title": "Solve by substitution", "input": "1+\\sin\\left(x\\right)-\\sin^{2}\\left(x\\right)=0", "result": "\\sin\\left(x\\right)=-\\frac{-1+\\sqrt{5}}{2},\\:\\sin\\left(x\\right)=\\frac{1+\\sqrt{5}}{2}", "steps": [ { "type": "step", "primary": "Let: $$\\sin\\left(x\\right)=u$$", "result": "1+u-u^{2}=0" }, { "type": "interim", "title": "$$1+u-u^{2}=0{\\quad:\\quad}u=-\\frac{-1+\\sqrt{5}}{2},\\:u=\\frac{1+\\sqrt{5}}{2}$$", "input": "1+u-u^{2}=0", "steps": [ { "type": "step", "primary": "Write in the standard form $$ax^{2}+bx+c=0$$", "result": "-u^{2}+u+1=0" }, { "type": "interim", "title": "Solve with the quadratic formula", "input": "-u^{2}+u+1=0", "result": "{u}_{1,\\:2}=\\frac{-1\\pm\\:\\sqrt{1^{2}-4\\left(-1\\right)\\cdot\\:1}}{2\\left(-1\\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 $${\\quad}a=-1,\\:b=1,\\:c=1$$", "result": "{u}_{1,\\:2}=\\frac{-1\\pm\\:\\sqrt{1^{2}-4\\left(-1\\right)\\cdot\\:1}}{2\\left(-1\\right)}" } ], "meta": { "interimType": "Solving The Quadratic Equation With Quadratic Formula Definition 0Eq", "gptData": "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" } }, { "type": "interim", "title": "$$\\sqrt{1^{2}-4\\left(-1\\right)\\cdot\\:1}=\\sqrt{5}$$", "input": "\\sqrt{1^{2}-4\\left(-1\\right)\\cdot\\:1}", "result": "{u}_{1,\\:2}=\\frac{-1\\pm\\:\\sqrt{5}}{2\\left(-1\\right)}", "steps": [ { "type": "step", "primary": "Apply rule $$1^{a}=1$$", "secondary": [ "$$1^{2}=1$$" ], "result": "=\\sqrt{1-4\\left(-1\\right)\\cdot\\:1}" }, { "type": "step", "primary": "Apply rule $$-\\left(-a\\right)=a$$", "result": "=\\sqrt{1+4\\cdot\\:1\\cdot\\:1}" }, { "type": "step", "primary": "Multiply the numbers: $$4\\cdot\\:1\\cdot\\:1=4$$", "result": "=\\sqrt{1+4}" }, { "type": "step", "primary": "Add the numbers: $$1+4=5$$", "result": "=\\sqrt{5}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7Uac2nmABaurcWf8xH8OQy33oGRuidDUanDuyywGs3HDehkKrn0era9rz8TlL+x/vBfhxL0S14glIZCZ6ZA0AY2qVOhDc44PEh+mhJiNpBfU1QyDoB0OrWwwVyAC6WXt6eHT6uFFeiceLrTXtMaUUWkLK/49YEbk1ZWCaH6K4ELk=" } }, { "type": "step", "primary": "Separate the solutions", "result": "{u}_{1}=\\frac{-1+\\sqrt{5}}{2\\left(-1\\right)},\\:{u}_{2}=\\frac{-1-\\sqrt{5}}{2\\left(-1\\right)}" }, { "type": "interim", "title": "$$u=\\frac{-1+\\sqrt{5}}{2\\left(-1\\right)}:{\\quad}-\\frac{-1+\\sqrt{5}}{2}$$", "input": "\\frac{-1+\\sqrt{5}}{2\\left(-1\\right)}", "steps": [ { "type": "step", "primary": "Remove parentheses: $$\\left(-a\\right)=-a$$", "result": "=\\frac{-1+\\sqrt{5}}{-2\\cdot\\:1}" }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:1=2$$", "result": "=\\frac{-1+\\sqrt{5}}{-2}" }, { "type": "step", "primary": "Apply the fraction rule: $$\\frac{a}{-b}=-\\frac{a}{b}$$", "result": "=-\\frac{-1+\\sqrt{5}}{2}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7gI32wa8JgqOZw2m0j1OS90QP2BiXFRrqt5sCjolWDPnehkKrn0era9rz8TlL+x/vttdvQxZI3PlVepHWO3+UgshWRHqs0dS9ZhvF6coEeYsRztKE552b3ssyeALQtWRjFKUfisIc5ub5fZa89rjWkM+wUOE0U5cHD20x8B9kEhq/Mg94S0N9we//Py6WzxN6" } }, { "type": "interim", "title": "$$u=\\frac{-1-\\sqrt{5}}{2\\left(-1\\right)}:{\\quad}\\frac{1+\\sqrt{5}}{2}$$", "input": "\\frac{-1-\\sqrt{5}}{2\\left(-1\\right)}", "steps": [ { "type": "step", "primary": "Remove parentheses: $$\\left(-a\\right)=-a$$", "result": "=\\frac{-1-\\sqrt{5}}{-2\\cdot\\:1}" }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:1=2$$", "result": "=\\frac{-1-\\sqrt{5}}{-2}" }, { "type": "step", "primary": "Apply the fraction rule: $$\\frac{-a}{-b}=\\frac{a}{b}$$", "secondary": [ "$$-1-\\sqrt{5}=-\\left(1+\\sqrt{5}\\right)$$" ], "result": "=\\frac{1+\\sqrt{5}}{2}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7LcON8lIS+hibufdR4EzXnEQP2BiXFRrqt5sCjolWDPnehkKrn0era9rz8TlL+x/vBVZ9vx5jzfo/n1rSDQAgpiHiAmUkic6yAdOR6gEgyxE/y9DKGIPglJ+qMi9xDu2K49eAGHKP0Q/cW9+uQGTAeFBxkPFXXco7wtYZkFzyIdY=" } }, { "type": "step", "primary": "The solutions to the quadratic equation are:", "result": "u=-\\frac{-1+\\sqrt{5}}{2},\\:u=\\frac{1+\\sqrt{5}}{2}" } ], "meta": { "solvingClass": "Equations", "interimType": "Equations" } }, { "type": "step", "primary": "Substitute back $$u=\\sin\\left(x\\right)$$", "result": "\\sin\\left(x\\right)=-\\frac{-1+\\sqrt{5}}{2},\\:\\sin\\left(x\\right)=\\frac{1+\\sqrt{5}}{2}" } ], "meta": { "interimType": "Substitution Method 0Eq" } }, { "type": "interim", "title": "$$\\sin\\left(x\\right)=-\\frac{-1+\\sqrt{5}}{2}{\\quad:\\quad}x=\\arcsin\\left(-\\frac{-1+\\sqrt{5}}{2}\\right)+2πn,\\:x=π+\\arcsin\\left(\\frac{-1+\\sqrt{5}}{2}\\right)+2πn$$", "input": "\\sin\\left(x\\right)=-\\frac{-1+\\sqrt{5}}{2}", "steps": [ { "type": "interim", "title": "Apply trig inverse properties", "input": "\\sin\\left(x\\right)=-\\frac{-1+\\sqrt{5}}{2}", "result": "x=\\arcsin\\left(-\\frac{-1+\\sqrt{5}}{2}\\right)+2πn,\\:x=π+\\arcsin\\left(\\frac{-1+\\sqrt{5}}{2}\\right)+2πn", "steps": [ { "type": "step", "primary": "General solutions for $$\\sin\\left(x\\right)=-\\frac{-1+\\sqrt{5}}{2}$$", "secondary": [ "$$\\sin\\left(x\\right)=-a\\quad\\Rightarrow\\quad\\:x=\\arcsin\\left(-a\\right)+2πn,\\:\\quad\\:x=π+\\arcsin\\left(a\\right)+2πn$$" ], "result": "x=\\arcsin\\left(-\\frac{-1+\\sqrt{5}}{2}\\right)+2πn,\\:x=π+\\arcsin\\left(\\frac{-1+\\sqrt{5}}{2}\\right)+2πn" } ], "meta": { "interimType": "Trig Apply Inverse Props 0Eq" } } ], "meta": { "interimType": "N/A" } }, { "type": "interim", "title": "$$\\sin\\left(x\\right)=\\frac{1+\\sqrt{5}}{2}{\\quad:\\quad}$$No Solution", "input": "\\sin\\left(x\\right)=\\frac{1+\\sqrt{5}}{2}", "steps": [ { "type": "step", "primary": "$$-1\\le\\sin\\left(x\\right)\\le1$$", "result": "\\mathrm{No\\:Solution}" } ], "meta": { "interimType": "N/A" } }, { "type": "step", "primary": "Combine all the solutions", "result": "x=\\arcsin\\left(-\\frac{-1+\\sqrt{5}}{2}\\right)+2πn,\\:x=π+\\arcsin\\left(\\frac{-1+\\sqrt{5}}{2}\\right)+2πn" }, { "type": "step", "primary": "Show solutions in decimal form", "result": "x=-0.66623…+2πn,\\:x=π+0.66623…+2πn" } ], "meta": { "solvingClass": "Trig Equations", "practiceLink": "/practice/trigonometry-practice#area=main&subtopic=Trig%20Equations", "practiceTopic": "Trig Equations" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "\\tan(x)+\\cos(x)" }, "showViewLarger": true } }, "meta": { "showVerify": true } }