cos^2(x)-sin^2(x)+sin(x)=1

{ "query": { "display": "$$\\cos^{2}\\left(x\\right)-\\sin^{2}\\left(x\\right)+\\sin\\left(x\\right)=1$$", "symbolab_question": "EQUATION#\\cos^{2}(x)-\\sin^{2}(x)+\\sin(x)=1" }, "solution": { "level": "PERFORMED", "subject": "Trigonometry", "topic": "Trig Equations", "subTopic": "Trig Equations", "default": "x=2πn,x=π+2πn,x=\\frac{π}{6}+2πn,x=\\frac{5π}{6}+2πn", "degrees": "x=0^{\\circ }+360^{\\circ }n,x=180^{\\circ }+360^{\\circ }n,x=30^{\\circ }+360^{\\circ }n,x=150^{\\circ }+360^{\\circ }n", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\cos^{2}\\left(x\\right)-\\sin^{2}\\left(x\\right)+\\sin\\left(x\\right)=1{\\quad:\\quad}x=2πn,\\:x=π+2πn,\\:x=\\frac{π}{6}+2πn,\\:x=\\frac{5π}{6}+2πn$$", "input": "\\cos^{2}\\left(x\\right)-\\sin^{2}\\left(x\\right)+\\sin\\left(x\\right)=1", "steps": [ { "type": "step", "primary": "Subtract $$1$$ from both sides", "result": "\\cos^{2}\\left(x\\right)-\\sin^{2}\\left(x\\right)+\\sin\\left(x\\right)-1=0" }, { "type": "interim", "title": "Rewrite using trig identities", "input": "-1+\\cos^{2}\\left(x\\right)+\\sin\\left(x\\right)-\\sin^{2}\\left(x\\right)", "result": "\\sin\\left(x\\right)-2\\sin^{2}\\left(x\\right)=0", "steps": [ { "type": "step", "primary": "Use the Pythagorean identity: $$1=\\cos^{2}\\left(x\\right)+\\sin^{2}\\left(x\\right)$$", "secondary": [ "$$1-\\cos^{2}\\left(x\\right)=\\sin^{2}\\left(x\\right)$$", "$$-1+\\cos^{2}\\left(x\\right)=-\\sin^{2}\\left(x\\right)$$" ], "result": "=\\sin\\left(x\\right)-\\sin^{2}\\left(x\\right)-\\sin^{2}\\left(x\\right)" }, { "type": "step", "primary": "Simplify", "result": "=\\sin\\left(x\\right)-2\\sin^{2}\\left(x\\right)", "meta": { "solvingClass": "Solver" } } ], "meta": { "interimType": "Trig Rewrite Using Trig identities 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7XQoqDkagBdkaGF3BP1M6VCH+prMD3oJfvXUn3MkBUiK/SZuHgqchNgrvosICC0xGOdhXIglekP2bk8wRyD3Q8umxY48eSLsbyMu/5yTExUbC4HIK63JpB+rGbO1V7H0Qs6oSK3z3AmGSSoStcBJUPzwpEU56/Q6pej0GHL58RHsPvT5rKSPA+SQZpUDyew3u23Xi3IvT385kAYFK2CQOxBkJGB3SwIAzffMMCti9BhrWM+R/5vzPLegxuhGAmoA+" } }, { "type": "interim", "title": "Solve by substitution", "input": "\\sin\\left(x\\right)-2\\sin^{2}\\left(x\\right)=0", "result": "\\sin\\left(x\\right)=0,\\:\\sin\\left(x\\right)=\\frac{1}{2}", "steps": [ { "type": "step", "primary": "Let: $$\\sin\\left(x\\right)=u$$", "result": "u-2u^{2}=0" }, { "type": "interim", "title": "$$u-2u^{2}=0{\\quad:\\quad}u=0,\\:u=\\frac{1}{2}$$", "input": "u-2u^{2}=0", "steps": [ { "type": "step", "primary": "Write in the standard form $$ax^{2}+bx+c=0$$", "result": "-2u^{2}+u=0" }, { "type": "interim", "title": "Solve with the quadratic formula", "input": "-2u^{2}+u=0", "result": "{u}_{1,\\:2}=\\frac{-1\\pm\\:\\sqrt{1^{2}-4\\left(-2\\right)\\cdot\\:0}}{2\\left(-2\\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 $${\\quad}a=-2,\\:b=1,\\:c=0$$", "result": "{u}_{1,\\:2}=\\frac{-1\\pm\\:\\sqrt{1^{2}-4\\left(-2\\right)\\cdot\\:0}}{2\\left(-2\\right)}" } ], "meta": { "interimType": "Solving The Quadratic Equation With Quadratic Formula Definition 0Eq", "gptData": "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" } }, { "type": "interim", "title": "$$\\sqrt{1^{2}-4\\left(-2\\right)\\cdot\\:0}=1$$", "input": "\\sqrt{1^{2}-4\\left(-2\\right)\\cdot\\:0}", "result": "{u}_{1,\\:2}=\\frac{-1\\pm\\:1}{2\\left(-2\\right)}", "steps": [ { "type": "step", "primary": "Apply rule $$1^{a}=1$$", "secondary": [ "$$1^{2}=1$$" ], "result": "=\\sqrt{1-4\\left(-2\\right)\\cdot\\:0}" }, { "type": "step", "primary": "Apply rule $$-\\left(-a\\right)=a$$", "result": "=\\sqrt{1+4\\cdot\\:2\\cdot\\:0}" }, { "type": "step", "primary": "Apply rule $$0\\cdot\\:a=0$$", "result": "=\\sqrt{1+0}" }, { "type": "step", "primary": "Add the numbers: $$1+0=1$$", "result": "=\\sqrt{1}" }, { "type": "step", "primary": "Apply rule $$\\sqrt{1}=1$$", "result": "=1" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7Uac2nmABaurcWf8xH8OQyyBqAK+u1uUA6eWfx20RApzehkKrn0era9rz8TlL+x/vj+lG4r/2yK6aJdxvUCCu4btCR5dIjxQ5ASg+ZPFVSsdlyvzZvmgr6BbflRzL8H8tZyb36vpKwrPyssOZsvwtrg==" } }, { "type": "step", "primary": "Separate the solutions", "result": "{u}_{1}=\\frac{-1+1}{2\\left(-2\\right)},\\:{u}_{2}=\\frac{-1-1}{2\\left(-2\\right)}" }, { "type": "interim", "title": "$$u=\\frac{-1+1}{2\\left(-2\\right)}:{\\quad}0$$", "input": "\\frac{-1+1}{2\\left(-2\\right)}", "steps": [ { "type": "step", "primary": "Remove parentheses: $$\\left(-a\\right)=-a$$", "result": "=\\frac{-1+1}{-2\\cdot\\:2}" }, { "type": "step", "primary": "Add/Subtract the numbers: $$-1+1=0$$", "result": "=\\frac{0}{-2\\cdot\\:2}" }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:2=4$$", "result": "=\\frac{0}{-4}" }, { "type": "step", "primary": "Apply the fraction rule: $$\\frac{a}{-b}=-\\frac{a}{b}$$", "result": "=-\\frac{0}{4}" }, { "type": "step", "primary": "Apply rule $$\\frac{0}{a}=0,\\:a\\ne\\:0$$", "result": "=-0" }, { "type": "step", "result": "=0" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7ZILxh0n9XGu+UX30rV52KVBraIXtDlgD3G/CwhQUjohwkKGJWEPFPk38sdJMsyPIc1L1JfkzeAMH8Sv8wAfVX6QS8+Ejzws6A1XwOMup5uLxfayEPhINvNr8uCW/1LTC" } }, { "type": "interim", "title": "$$u=\\frac{-1-1}{2\\left(-2\\right)}:{\\quad}\\frac{1}{2}$$", "input": "\\frac{-1-1}{2\\left(-2\\right)}", "steps": [ { "type": "step", "primary": "Remove parentheses: $$\\left(-a\\right)=-a$$", "result": "=\\frac{-1-1}{-2\\cdot\\:2}" }, { "type": "step", "primary": "Subtract the numbers: $$-1-1=-2$$", "result": "=\\frac{-2}{-2\\cdot\\:2}" }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:2=4$$", "result": "=\\frac{-2}{-4}" }, { "type": "step", "primary": "Apply the fraction rule: $$\\frac{-a}{-b}=\\frac{a}{b}$$", "result": "=\\frac{2}{4}" }, { "type": "step", "primary": "Cancel the common factor: $$2$$", "result": "=\\frac{1}{2}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7CoFrCplKlG9JQtC7YmAoI1BraIXtDlgD3G/CwhQUjohwkKGJWEPFPk38sdJMsyPI4zPT6LZm4vvilNzqSUf5kUZjT4S6OaNQYcHY9HJJkC4fxUj6O9O80LGkpcT/GM7iwhKbDnZDRLLlBw2jEV2ywg==" } }, { "type": "step", "primary": "The solutions to the quadratic equation are:", "result": "u=0,\\:u=\\frac{1}{2}" } ], "meta": { "solvingClass": "Equations", "interimType": "Equations" } }, { "type": "step", "primary": "Substitute back $$u=\\sin\\left(x\\right)$$", "result": "\\sin\\left(x\\right)=0,\\:\\sin\\left(x\\right)=\\frac{1}{2}" } ], "meta": { "interimType": "Substitution Method 0Eq" } }, { "type": "interim", "title": "$$\\sin\\left(x\\right)=0{\\quad:\\quad}x=2πn,\\:x=π+2πn$$", "input": "\\sin\\left(x\\right)=0", "steps": [ { "type": "interim", "title": "General solutions for $$\\sin\\left(x\\right)=0$$", "result": "x=0+2πn,\\:x=π+2πn", "steps": [ { "type": "step", "primary": "$$\\sin\\left(x\\right)$$ periodicity table with $$2πn$$ cycle:<br/>$$\\begin{array}{|c|c|c|c|}\\hline x&\\sin(x)&x&\\sin(x)\\\\\\hline 0&0&π&0\\\\\\hline \\frac{π}{6}&\\frac{1}{2}&\\frac{7π}{6}&-\\frac{1}{2}\\\\\\hline \\frac{π}{4}&\\frac{\\sqrt{2}}{2}&\\frac{5π}{4}&-\\frac{\\sqrt{2}}{2}\\\\\\hline \\frac{π}{3}&\\frac{\\sqrt{3}}{2}&\\frac{4π}{3}&-\\frac{\\sqrt{3}}{2}\\\\\\hline \\frac{π}{2}&1&\\frac{3π}{2}&-1\\\\\\hline \\frac{2π}{3}&\\frac{\\sqrt{3}}{2}&\\frac{5π}{3}&-\\frac{\\sqrt{3}}{2}\\\\\\hline \\frac{3π}{4}&\\frac{\\sqrt{2}}{2}&\\frac{7π}{4}&-\\frac{\\sqrt{2}}{2}\\\\\\hline \\frac{5π}{6}&\\frac{1}{2}&\\frac{11π}{6}&-\\frac{1}{2}\\\\\\hline \\end{array}$$" }, { "type": "step", "result": "x=0+2πn,\\:x=π+2πn" } ], "meta": { "interimType": "Trig General Solutions sin 1Eq" } }, { "type": "interim", "title": "Solve $$x=0+2πn:{\\quad}x=2πn$$", "input": "x=0+2πn", "steps": [ { "type": "step", "primary": "$$0+2πn=2πn$$", "result": "x=2πn" } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve Title 1Eq" } }, { "type": "step", "result": "x=2πn,\\:x=π+2πn" } ], "meta": { "interimType": "N/A" } }, { "type": "interim", "title": "$$\\sin\\left(x\\right)=\\frac{1}{2}{\\quad:\\quad}x=\\frac{π}{6}+2πn,\\:x=\\frac{5π}{6}+2πn$$", "input": "\\sin\\left(x\\right)=\\frac{1}{2}", "steps": [ { "type": "interim", "title": "General solutions for $$\\sin\\left(x\\right)=\\frac{1}{2}$$", "result": "x=\\frac{π}{6}+2πn,\\:x=\\frac{5π}{6}+2πn", "steps": [ { "type": "step", "primary": "$$\\sin\\left(x\\right)$$ periodicity table with $$2πn$$ cycle:<br/>$$\\begin{array}{|c|c|c|c|}\\hline x&\\sin(x)&x&\\sin(x)\\\\\\hline 0&0&π&0\\\\\\hline \\frac{π}{6}&\\frac{1}{2}&\\frac{7π}{6}&-\\frac{1}{2}\\\\\\hline \\frac{π}{4}&\\frac{\\sqrt{2}}{2}&\\frac{5π}{4}&-\\frac{\\sqrt{2}}{2}\\\\\\hline \\frac{π}{3}&\\frac{\\sqrt{3}}{2}&\\frac{4π}{3}&-\\frac{\\sqrt{3}}{2}\\\\\\hline \\frac{π}{2}&1&\\frac{3π}{2}&-1\\\\\\hline \\frac{2π}{3}&\\frac{\\sqrt{3}}{2}&\\frac{5π}{3}&-\\frac{\\sqrt{3}}{2}\\\\\\hline \\frac{3π}{4}&\\frac{\\sqrt{2}}{2}&\\frac{7π}{4}&-\\frac{\\sqrt{2}}{2}\\\\\\hline \\frac{5π}{6}&\\frac{1}{2}&\\frac{11π}{6}&-\\frac{1}{2}\\\\\\hline \\end{array}$$" }, { "type": "step", "result": "x=\\frac{π}{6}+2πn,\\:x=\\frac{5π}{6}+2πn" } ], "meta": { "interimType": "Trig General Solutions sin 1Eq" } } ], "meta": { "interimType": "N/A" } }, { "type": "step", "primary": "Combine all the solutions", "result": "x=2πn,\\:x=π+2πn,\\:x=\\frac{π}{6}+2πn,\\:x=\\frac{5π}{6}+2πn" } ], "meta": { "solvingClass": "Trig Equations", "practiceLink": "/practice/trigonometry-practice#area=main&subtopic=Trig%20Equations", "practiceTopic": "Trig Equations" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "\\cos^{2}(x)-\\sin^{2}(x)+\\sin(x)-1" }, "showViewLarger": true } }, "meta": { "showVerify": true } }