4cos^2(x)+17sin(x)=8

{ "query": { "display": "$$4\\cos^{2}\\left(x\\right)+17\\sin\\left(x\\right)=8$$", "symbolab_question": "EQUATION#4\\cos^{2}(x)+17\\sin(x)=8" }, "solution": { "level": "PERFORMED", "subject": "Trigonometry", "topic": "Trig Equations", "subTopic": "Trig Equations", "default": "x=0.25268…+2πn,x=π-0.25268…+2πn", "degrees": "x=14.47751…^{\\circ }+360^{\\circ }n,x=165.52248…^{\\circ }+360^{\\circ }n", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$4\\cos^{2}\\left(x\\right)+17\\sin\\left(x\\right)=8{\\quad:\\quad}x=0.25268…+2πn,\\:x=π-0.25268…+2πn$$", "input": "4\\cos^{2}\\left(x\\right)+17\\sin\\left(x\\right)=8", "steps": [ { "type": "step", "primary": "Subtract $$8$$ from both sides", "result": "4\\cos^{2}\\left(x\\right)+17\\sin\\left(x\\right)-8=0" }, { "type": "interim", "title": "Rewrite using trig identities", "input": "-8+17\\sin\\left(x\\right)+4\\cos^{2}\\left(x\\right)", "result": "-4+17\\sin\\left(x\\right)-4\\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": "=-8+17\\sin\\left(x\\right)+4\\left(1-\\sin^{2}\\left(x\\right)\\right)" }, { "type": "interim", "title": "Simplify $$-8+17\\sin\\left(x\\right)+4\\left(1-\\sin^{2}\\left(x\\right)\\right):{\\quad}17\\sin\\left(x\\right)-4\\sin^{2}\\left(x\\right)-4$$", "input": "-8+17\\sin\\left(x\\right)+4\\left(1-\\sin^{2}\\left(x\\right)\\right)", "result": "=17\\sin\\left(x\\right)-4\\sin^{2}\\left(x\\right)-4", "steps": [ { "type": "interim", "title": "Expand $$4\\left(1-\\sin^{2}\\left(x\\right)\\right):{\\quad}4-4\\sin^{2}\\left(x\\right)$$", "input": "4\\left(1-\\sin^{2}\\left(x\\right)\\right)", "result": "=-8+17\\sin\\left(x\\right)+4-4\\sin^{2}\\left(x\\right)", "steps": [ { "type": "step", "primary": "Apply the distributive law: $$a\\left(b-c\\right)=ab-ac$$", "secondary": [ "$$a=4,\\:b=1,\\:c=\\sin^{2}\\left(x\\right)$$" ], "result": "=4\\cdot\\:1-4\\sin^{2}\\left(x\\right)", "meta": { "practiceLink": "/practice/expansion-practice", "practiceTopic": "Expand Rules" } }, { "type": "step", "primary": "Multiply the numbers: $$4\\cdot\\:1=4$$", "result": "=4-4\\sin^{2}\\left(x\\right)" } ], "meta": { "interimType": "Algebraic Manipulation Expand Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s76wPhBQ1UvQOJ+DvF6Odr4UPKZBnILZTwiutqYLWXIDCjkVi15I8rBefLi4Iyt2wropSd6E8O/fJ9cnjzAiGMD2RLd2VwIqlBNByF6663syRU6H0nS++8kDqP632fuVHP3vg36HY8Vu4HxmYY39j9LeIASZeFjDtawNGt9P21GJY=" } }, { "type": "interim", "title": "Simplify $$-8+17\\sin\\left(x\\right)+4-4\\sin^{2}\\left(x\\right):{\\quad}17\\sin\\left(x\\right)-4\\sin^{2}\\left(x\\right)-4$$", "input": "-8+17\\sin\\left(x\\right)+4-4\\sin^{2}\\left(x\\right)", "result": "=17\\sin\\left(x\\right)-4\\sin^{2}\\left(x\\right)-4", "steps": [ { "type": "step", "primary": "Group like terms", "result": "=17\\sin\\left(x\\right)-4\\sin^{2}\\left(x\\right)-8+4" }, { "type": "step", "primary": "Add/Subtract the numbers: $$-8+4=-4$$", "result": "=17\\sin\\left(x\\right)-4\\sin^{2}\\left(x\\right)-4" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7JlrSL+JrtjUdD2cbK65BOt74N+h2PFbuB8ZmGN/Y/S3TLx8mOdHYVzxX643JqKFIhzztSoslFjqUZ9mat33DIYqZC3r7iKNlMhLzpVmhUBoD+NnDkcRvhRe6ysvqW6NCeqXxdc+rps1CUyb7fqI2Gfe0ABpEQBgtxbwsF8JR4yWPYOjyHivlhJByQCNKVXI18nzzgggzn2lm6poy2q7kiw==" } } ], "meta": { "interimType": "Trig Rewrite Using Trig identities 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s78XwrQ8pMMDMm8Z6rZOjdk+OMxTbTp0V1ktleC+Ke2T7aj2VUeWgC00zqTCXgfBdNgGFMQkUTMNnEnFExdO5KvDLzJz4rkYnjv1/W3glakSm1s1RQOlcS2PC6wMP3eHir86Ch/pv7FH9me467uTfchcwmVRRojVIpmp8Aza6wJ/6jeh7+jKEzLb7VNCEMF3Z/AUodDpEZVCOJY6rkfXcVkqaOQILKf053R9DIJaaWk7Mc6tftTNY2KsUtvVbis5zN" } }, { "type": "interim", "title": "Solve by substitution", "input": "-4+17\\sin\\left(x\\right)-4\\sin^{2}\\left(x\\right)=0", "result": "\\sin\\left(x\\right)=\\frac{1}{4},\\:\\sin\\left(x\\right)=4", "steps": [ { "type": "step", "primary": "Let: $$\\sin\\left(x\\right)=u$$", "result": "-4+17u-4u^{2}=0" }, { "type": "interim", "title": "$$-4+17u-4u^{2}=0{\\quad:\\quad}u=\\frac{1}{4},\\:u=4$$", "input": "-4+17u-4u^{2}=0", "steps": [ { "type": "step", "primary": "Write in the standard form $$ax^{2}+bx+c=0$$", "result": "-4u^{2}+17u-4=0" }, { "type": "interim", "title": "Solve with the quadratic formula", "input": "-4u^{2}+17u-4=0", "result": "{u}_{1,\\:2}=\\frac{-17\\pm\\:\\sqrt{17^{2}-4\\left(-4\\right)\\left(-4\\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 $${\\quad}a=-4,\\:b=17,\\:c=-4$$", "result": "{u}_{1,\\:2}=\\frac{-17\\pm\\:\\sqrt{17^{2}-4\\left(-4\\right)\\left(-4\\right)}}{2\\left(-4\\right)}" } ], "meta": { "interimType": "Solving The Quadratic Equation With Quadratic Formula Definition 0Eq", "gptData": "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" } }, { "type": "interim", "title": "$$\\sqrt{17^{2}-4\\left(-4\\right)\\left(-4\\right)}=15$$", "input": "\\sqrt{17^{2}-4\\left(-4\\right)\\left(-4\\right)}", "result": "{u}_{1,\\:2}=\\frac{-17\\pm\\:15}{2\\left(-4\\right)}", "steps": [ { "type": "step", "primary": "Apply rule $$-\\left(-a\\right)=a$$", "result": "=\\sqrt{17^{2}-4\\cdot\\:4\\cdot\\:4}" }, { "type": "step", "primary": "Multiply the numbers: $$4\\cdot\\:4\\cdot\\:4=64$$", "result": "=\\sqrt{17^{2}-64}" }, { "type": "step", "primary": "$$17^{2}=289$$", "result": "=\\sqrt{289-64}" }, { "type": "step", "primary": "Subtract the numbers: $$289-64=225$$", "result": "=\\sqrt{225}" }, { "type": "step", "primary": "Factor the number: $$225=15^{2}$$", "result": "=\\sqrt{15^{2}}" }, { "type": "step", "primary": "Apply radical rule: $$\\sqrt[n]{a^n}=a$$", "secondary": [ "$$\\sqrt{15^{2}}=15$$" ], "result": "=15", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7VzQB4fe7lT1lCpHARKZ0ZzxnchgraJvI72WWtakX7bnNGoPE9TME3q+OPmgkv2RQixmmSQusUob7WHNEmZm/RXLmwp2GTieS/P9oLHcpsNTAM+xJ5qyMHBL5bpvifv5vGBw1SAwf9w2RwnnYtP4vmA==" } }, { "type": "step", "primary": "Separate the solutions", "result": "{u}_{1}=\\frac{-17+15}{2\\left(-4\\right)},\\:{u}_{2}=\\frac{-17-15}{2\\left(-4\\right)}" }, { "type": "interim", "title": "$$u=\\frac{-17+15}{2\\left(-4\\right)}:{\\quad}\\frac{1}{4}$$", "input": "\\frac{-17+15}{2\\left(-4\\right)}", "steps": [ { "type": "step", "primary": "Remove parentheses: $$\\left(-a\\right)=-a$$", "result": "=\\frac{-17+15}{-2\\cdot\\:4}" }, { "type": "step", "primary": "Add/Subtract the numbers: $$-17+15=-2$$", "result": "=\\frac{-2}{-2\\cdot\\:4}" }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:4=8$$", "result": "=\\frac{-2}{-8}" }, { "type": "step", "primary": "Apply the fraction rule: $$\\frac{-a}{-b}=\\frac{a}{b}$$", "result": "=\\frac{2}{8}" }, { "type": "step", "primary": "Cancel the common factor: $$2$$", "result": "=\\frac{1}{4}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7fv0iKAk2zYR0bZ6y1WtQ6hiXCLvzVp08cg9hAay0X1cJQJZuTAY5js+oqjdT8ksllomW24Kpw3yEAlwZxxIYkD/L0MoYg+CUn6oyL3EO7Yooalh5PhgqGlcNXVERojN6qKGKHT+FAp7dHdJOch/6gw==" } }, { "type": "interim", "title": "$$u=\\frac{-17-15}{2\\left(-4\\right)}:{\\quad}4$$", "input": "\\frac{-17-15}{2\\left(-4\\right)}", "steps": [ { "type": "step", "primary": "Remove parentheses: $$\\left(-a\\right)=-a$$", "result": "=\\frac{-17-15}{-2\\cdot\\:4}" }, { "type": "step", "primary": "Subtract the numbers: $$-17-15=-32$$", "result": "=\\frac{-32}{-2\\cdot\\:4}" }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:4=8$$", "result": "=\\frac{-32}{-8}" }, { "type": "step", "primary": "Apply the fraction rule: $$\\frac{-a}{-b}=\\frac{a}{b}$$", "result": "=\\frac{32}{8}" }, { "type": "step", "primary": "Divide the numbers: $$\\frac{32}{8}=4$$", "result": "=4" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s781Yn6gjbKPnxwYjTy+ZohhiXCLvzVp08cg9hAay0X1cJQJZuTAY5js+oqjdT8kslhVvTh05gpKATMeKyHd4bYY6EN1BhUKLfjcWGu4MByuUZwI5lKexvQmxJi0jSzCfP" } }, { "type": "step", "primary": "The solutions to the quadratic equation are:", "result": "u=\\frac{1}{4},\\:u=4" } ], "meta": { "solvingClass": "Equations", "interimType": "Equations" } }, { "type": "step", "primary": "Substitute back $$u=\\sin\\left(x\\right)$$", "result": "\\sin\\left(x\\right)=\\frac{1}{4},\\:\\sin\\left(x\\right)=4" } ], "meta": { "interimType": "Substitution Method 0Eq" } }, { "type": "interim", "title": "$$\\sin\\left(x\\right)=\\frac{1}{4}{\\quad:\\quad}x=\\arcsin\\left(\\frac{1}{4}\\right)+2πn,\\:x=π-\\arcsin\\left(\\frac{1}{4}\\right)+2πn$$", "input": "\\sin\\left(x\\right)=\\frac{1}{4}", "steps": [ { "type": "interim", "title": "Apply trig inverse properties", "input": "\\sin\\left(x\\right)=\\frac{1}{4}", "result": "x=\\arcsin\\left(\\frac{1}{4}\\right)+2πn,\\:x=π-\\arcsin\\left(\\frac{1}{4}\\right)+2πn", "steps": [ { "type": "step", "primary": "General solutions for $$\\sin\\left(x\\right)=\\frac{1}{4}$$", "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}{4}\\right)+2πn,\\:x=π-\\arcsin\\left(\\frac{1}{4}\\right)+2πn" } ], "meta": { "interimType": "Trig Apply Inverse Props 0Eq" } } ], "meta": { "interimType": "N/A" } }, { "type": "interim", "title": "$$\\sin\\left(x\\right)=4{\\quad:\\quad}$$No Solution", "input": "\\sin\\left(x\\right)=4", "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}{4}\\right)+2πn,\\:x=π-\\arcsin\\left(\\frac{1}{4}\\right)+2πn" }, { "type": "step", "primary": "Show solutions in decimal form", "result": "x=0.25268…+2πn,\\:x=π-0.25268…+2πn" } ], "meta": { "solvingClass": "Trig Equations", "practiceLink": "/practice/trigonometry-practice#area=main&subtopic=Trig%20Equations", "practiceTopic": "Trig Equations" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "4\\cos^{2}(x)+17\\sin(x)-8" }, "showViewLarger": true } }, "meta": { "showVerify": true } }