tan(θ)=sin(2θ)

{ "query": { "display": "$$\\tan\\left(θ\\right)=\\sin\\left(2θ\\right)$$", "symbolab_question": "EQUATION#\\tan(θ)=\\sin(2θ)" }, "solution": { "level": "PERFORMED", "subject": "Trigonometry", "topic": "Trig Equations", "subTopic": "Trig Equations", "default": "θ=2πn,θ=π+2πn,θ=\\frac{3π}{4}+2πn,θ=\\frac{5π}{4}+2πn,θ=\\frac{π}{4}+2πn,θ=\\frac{7π}{4}+2πn", "degrees": "θ=0^{\\circ }+360^{\\circ }n,θ=180^{\\circ }+360^{\\circ }n,θ=135^{\\circ }+360^{\\circ }n,θ=225^{\\circ }+360^{\\circ }n,θ=45^{\\circ }+360^{\\circ }n,θ=315^{\\circ }+360^{\\circ }n", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\tan\\left(θ\\right)=\\sin\\left(2θ\\right){\\quad:\\quad}θ=2πn,\\:θ=π+2πn,\\:θ=\\frac{3π}{4}+2πn,\\:θ=\\frac{5π}{4}+2πn,\\:θ=\\frac{π}{4}+2πn,\\:θ=\\frac{7π}{4}+2πn$$", "input": "\\tan\\left(θ\\right)=\\sin\\left(2θ\\right)", "steps": [ { "type": "step", "primary": "Subtract $$\\sin\\left(2θ\\right)$$ from both sides", "result": "\\tan\\left(θ\\right)-\\sin\\left(2θ\\right)=0" }, { "type": "interim", "title": "Express with sin, cos", "input": "-\\sin\\left(2θ\\right)+\\tan\\left(θ\\right)", "result": "\\frac{\\sin\\left(θ\\right)-\\cos\\left(θ\\right)\\sin\\left(2θ\\right)}{\\cos\\left(θ\\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": "=-\\sin\\left(2θ\\right)+\\frac{\\sin\\left(θ\\right)}{\\cos\\left(θ\\right)}" }, { "type": "interim", "title": "Simplify $$-\\sin\\left(2θ\\right)+\\frac{\\sin\\left(θ\\right)}{\\cos\\left(θ\\right)}:{\\quad}\\frac{-\\sin\\left(2θ\\right)\\cos\\left(θ\\right)+\\sin\\left(θ\\right)}{\\cos\\left(θ\\right)}$$", "input": "-\\sin\\left(2θ\\right)+\\frac{\\sin\\left(θ\\right)}{\\cos\\left(θ\\right)}", "result": "=\\frac{-\\sin\\left(2θ\\right)\\cos\\left(θ\\right)+\\sin\\left(θ\\right)}{\\cos\\left(θ\\right)}", "steps": [ { "type": "step", "primary": "Convert element to fraction: $$\\sin\\left(2θ\\right)=\\frac{\\sin\\left(2θ\\right)\\cos\\left(θ\\right)}{\\cos\\left(θ\\right)}$$", "result": "=-\\frac{\\sin\\left(2θ\\right)\\cos\\left(θ\\right)}{\\cos\\left(θ\\right)}+\\frac{\\sin\\left(θ\\right)}{\\cos\\left(θ\\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{-\\sin\\left(2θ\\right)\\cos\\left(θ\\right)+\\sin\\left(θ\\right)}{\\cos\\left(θ\\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": "\\sin\\left(θ\\right)-\\cos\\left(θ\\right)\\sin\\left(2θ\\right)=0" }, { "type": "interim", "title": "Rewrite using trig identities", "input": "\\sin\\left(θ\\right)-\\cos\\left(θ\\right)\\sin\\left(2θ\\right)", "result": "\\sin\\left(θ\\right)-2\\cos^{2}\\left(θ\\right)\\sin\\left(θ\\right)=0", "steps": [ { "type": "step", "primary": "Use the Double Angle identity: $$\\sin\\left(2x\\right)=2\\sin\\left(x\\right)\\cos\\left(x\\right)$$", "result": "=\\sin\\left(θ\\right)-\\cos\\left(θ\\right)\\cdot\\:2\\sin\\left(θ\\right)\\cos\\left(θ\\right)" }, { "type": "interim", "title": "$$\\cos\\left(θ\\right)\\cdot\\:2\\sin\\left(θ\\right)\\cos\\left(θ\\right)=2\\cos^{2}\\left(θ\\right)\\sin\\left(θ\\right)$$", "input": "\\cos\\left(θ\\right)\\cdot\\:2\\sin\\left(θ\\right)\\cos\\left(θ\\right)", "result": "=\\sin\\left(θ\\right)-2\\cos^{2}\\left(θ\\right)\\sin\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\cos\\left(θ\\right)\\cos\\left(θ\\right)=\\:\\cos^{1+1}\\left(θ\\right)$$" ], "result": "=2\\sin\\left(θ\\right)\\cos^{1+1}\\left(θ\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$1+1=2$$", "result": "=2\\sin\\left(θ\\right)\\cos^{2}\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7+R+zZ8/JbWROQCA5DEnJcPh/xJq9dkg3QS6kAbDTRkxLC0A8HE3alKIjt7EgOk1ncJChiVhDxT5N/LHSTLMjyKmpvGO0fG4klduKVcXoKAyK8BFA8e2j0rMZLzt9f3uyOKwPc8gDVaC03mkkwusZngpXuAYxnWioySZrfrRc/B7Eb5EXvHcIaC0echt0q/S/DyOb0N9am8QDxAikRZt+97PvNHPzVKpEPTYcRQThaqE=" } } ], "meta": { "interimType": "Trig Rewrite Using Trig identities 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7Awm8OHaUUE5X73KqyTFjFI7LGsuKKgX02M/jQ/bzISOrOewQ85AorY6M6laY/bZE8wQyZ2Br8agbY8qkLUt6WPm0tu7qq7Iur/j8bUReEEOq5I4KKx/r65u7kTSvPypsqTMPfD81kqA3Z3fIw8s8sv9erWKAzbv1idSFCvcuuLRutCcg9w/J9jG04p9ZNzWXTeQKHeh69S6dnv9vSoUoFMI82VRiPna1zhjwHzQjfQHmXrQXKsikqlLypKmWBjE8JLd1ohke2Wgml78++2zI0g==" } }, { "type": "interim", "title": "Factor $$\\sin\\left(θ\\right)-2\\cos^{2}\\left(θ\\right)\\sin\\left(θ\\right):{\\quad}-\\sin\\left(θ\\right)\\left(\\sqrt{2}\\cos\\left(θ\\right)+1\\right)\\left(\\sqrt{2}\\cos\\left(θ\\right)-1\\right)$$", "input": "\\sin\\left(θ\\right)-2\\cos^{2}\\left(θ\\right)\\sin\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Factor out common term $$-\\sin\\left(θ\\right)$$", "result": "=-\\sin\\left(θ\\right)\\left(-1+2\\cos^{2}\\left(θ\\right)\\right)", "meta": { "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } }, { "type": "interim", "title": "Factor $$2\\cos^{2}\\left(θ\\right)-1:{\\quad}\\left(\\sqrt{2}\\cos\\left(θ\\right)+1\\right)\\left(\\sqrt{2}\\cos\\left(θ\\right)-1\\right)$$", "input": "2\\cos^{2}\\left(θ\\right)-1", "steps": [ { "type": "interim", "title": "Rewrite $$2\\cos^{2}\\left(θ\\right)-1$$ as $$\\left(\\sqrt{2}\\cos\\left(θ\\right)\\right)^{2}-1^{2}$$", "input": "2\\cos^{2}\\left(θ\\right)-1", "result": "=\\left(\\sqrt{2}\\cos\\left(θ\\right)\\right)^{2}-1^{2}", "steps": [ { "type": "step", "primary": "Apply radical rule: $$a=\\left(\\sqrt{a}\\right)^{2}$$", "secondary": [ "$$2=\\left(\\sqrt{2}\\right)^{2}$$" ], "result": "=\\left(\\sqrt{2}\\right)^{2}\\cos^{2}\\left(θ\\right)-1", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } }, { "type": "step", "primary": "Rewrite $$1$$ as $$1^{2}$$", "result": "=\\left(\\sqrt{2}\\right)^{2}\\cos^{2}\\left(θ\\right)-1^{2}" }, { "type": "step", "primary": "Apply exponent rule: $$a^{m}b^{m}=\\left(ab\\right)^{m}$$", "secondary": [ "$$\\left(\\sqrt{2}\\right)^{2}\\cos^{2}\\left(θ\\right)=\\left(\\sqrt{2}\\cos\\left(θ\\right)\\right)^{2}$$" ], "result": "=\\left(\\sqrt{2}\\cos\\left(θ\\right)\\right)^{2}-1^{2}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } } ], "meta": { "interimType": "Generic Rewrite As Specific 2Eq" } }, { "type": "step", "primary": "Apply Difference of Two Squares Formula: $$x^{2}-y^{2}=\\left(x+y\\right)\\left(x-y\\right)$$", "secondary": [ "$$\\left(\\sqrt{2}\\cos\\left(θ\\right)\\right)^{2}-1^{2}=\\left(\\sqrt{2}\\cos\\left(θ\\right)+1\\right)\\left(\\sqrt{2}\\cos\\left(θ\\right)-1\\right)$$" ], "result": "=\\left(\\sqrt{2}\\cos\\left(θ\\right)+1\\right)\\left(\\sqrt{2}\\cos\\left(θ\\right)-1\\right)", "meta": { "practiceLink": "/practice/factoring-practice#area=main&subtopic=Difference%20of%20Two%20Squares", "practiceTopic": "Factor Difference of Squares" } } ], "meta": { "interimType": "Algebraic Manipulation Factor Title 1Eq" } }, { "type": "step", "result": "=-\\sin\\left(θ\\right)\\left(\\sqrt{2}\\cos\\left(θ\\right)+1\\right)\\left(\\sqrt{2}\\cos\\left(θ\\right)-1\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Factor Title 1Eq" } }, { "type": "step", "result": "-\\sin\\left(θ\\right)\\left(\\sqrt{2}\\cos\\left(θ\\right)+1\\right)\\left(\\sqrt{2}\\cos\\left(θ\\right)-1\\right)=0" }, { "type": "step", "primary": "Solving each part separately", "result": "\\sin\\left(θ\\right)=0\\lor\\:\\sqrt{2}\\cos\\left(θ\\right)+1=0\\lor\\:\\sqrt{2}\\cos\\left(θ\\right)-1=0" }, { "type": "interim", "title": "$$\\sin\\left(θ\\right)=0{\\quad:\\quad}θ=2πn,\\:θ=π+2πn$$", "input": "\\sin\\left(θ\\right)=0", "steps": [ { "type": "interim", "title": "General solutions for $$\\sin\\left(θ\\right)=0$$", "result": "θ=0+2πn,\\:θ=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": "θ=0+2πn,\\:θ=π+2πn" } ], "meta": { "interimType": "Trig General Solutions sin 1Eq" } }, { "type": "interim", "title": "Solve $$θ=0+2πn:{\\quad}θ=2πn$$", "input": "θ=0+2πn", "steps": [ { "type": "step", "primary": "$$0+2πn=2πn$$", "result": "θ=2πn" } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve Title 1Eq" } }, { "type": "step", "result": "θ=2πn,\\:θ=π+2πn" } ], "meta": { "solvingClass": "Trig Equations", "interimType": "Trig Equations" } }, { "type": "interim", "title": "$$\\sqrt{2}\\cos\\left(θ\\right)+1=0{\\quad:\\quad}θ=\\frac{3π}{4}+2πn,\\:θ=\\frac{5π}{4}+2πn$$", "input": "\\sqrt{2}\\cos\\left(θ\\right)+1=0", "steps": [ { "type": "interim", "title": "Move $$1\\:$$to the right side", "input": "\\sqrt{2}\\cos\\left(θ\\right)+1=0", "result": "\\sqrt{2}\\cos\\left(θ\\right)=-1", "steps": [ { "type": "step", "primary": "Subtract $$1$$ from both sides", "result": "\\sqrt{2}\\cos\\left(θ\\right)+1-1=0-1" }, { "type": "step", "primary": "Simplify", "result": "\\sqrt{2}\\cos\\left(θ\\right)=-1" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": "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" } }, { "type": "interim", "title": "Divide both sides by $$\\sqrt{2}$$", "input": "\\sqrt{2}\\cos\\left(θ\\right)=-1", "result": "\\cos\\left(θ\\right)=-\\frac{\\sqrt{2}}{2}", "steps": [ { "type": "step", "primary": "Divide both sides by $$\\sqrt{2}$$", "result": "\\frac{\\sqrt{2}\\cos\\left(θ\\right)}{\\sqrt{2}}=\\frac{-1}{\\sqrt{2}}" }, { "type": "interim", "title": "Simplify", "input": "\\frac{\\sqrt{2}\\cos\\left(θ\\right)}{\\sqrt{2}}=\\frac{-1}{\\sqrt{2}}", "result": "\\cos\\left(θ\\right)=-\\frac{\\sqrt{2}}{2}", "steps": [ { "type": "interim", "title": "Simplify $$\\frac{\\sqrt{2}\\cos\\left(θ\\right)}{\\sqrt{2}}:{\\quad}\\cos\\left(θ\\right)$$", "input": "\\frac{\\sqrt{2}\\cos\\left(θ\\right)}{\\sqrt{2}}", "steps": [ { "type": "step", "primary": "Cancel the common factor: $$\\sqrt{2}$$", "result": "=\\cos\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s74DwjiBDVBIb8+yh6fkkMMdtK58IrruNZZpc41yUjaDO9eXg0wY6q+jIeOHWzjTOZCUCWbkwGOY7PqKo3U/JLJbfVxaVZWweWngVAZ8hO2bpFKk3fejFkyiOiq9iG9IkAIuSmJUDtLxRPyJ57Nkq7qi6yyeeeKwmd0ICbjP7Oejw1IGGNbmAWFGmhlV4gYgiXCavHWqV0CcO3cTEIFbZUKQ==" } }, { "type": "interim", "title": "Simplify $$\\frac{-1}{\\sqrt{2}}:{\\quad}-\\frac{\\sqrt{2}}{2}$$", "input": "\\frac{-1}{\\sqrt{2}}", "steps": [ { "type": "step", "primary": "Apply the fraction rule: $$\\frac{-a}{b}=-\\frac{a}{b}$$", "result": "=-\\frac{1}{\\sqrt{2}}" }, { "type": "interim", "title": "Rationalize $$-\\frac{1}{\\sqrt{2}}:{\\quad}-\\frac{\\sqrt{2}}{2}$$", "input": "-\\frac{1}{\\sqrt{2}}", "result": "=-\\frac{\\sqrt{2}}{2}", "steps": [ { "type": "step", "primary": "Multiply by the conjugate $$\\frac{\\sqrt{2}}{\\sqrt{2}}$$", "result": "=-\\frac{1\\cdot\\:\\sqrt{2}}{\\sqrt{2}\\sqrt{2}}", "meta": { "title": { "extension": "To rationalize the denominator, multiply numerator and denominator by the conjugate of the radical $$\\sqrt{2}$$" } } }, { "type": "step", "primary": "$$1\\cdot\\:\\sqrt{2}=\\sqrt{2}$$" }, { "type": "interim", "title": "$$\\sqrt{2}\\sqrt{2}=2$$", "input": "\\sqrt{2}\\sqrt{2}", "result": "=-\\frac{\\sqrt{2}}{2}", "steps": [ { "type": "step", "primary": "Apply radical rule: $$\\sqrt{a}\\sqrt{a}=a$$", "secondary": [ "$$\\sqrt{2}\\sqrt{2}=2$$" ], "result": "=2", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7OXRSLY4uXiMbvMfICP5T88aFSKFknkW7qg0U6E21MorMwViaLUXkeD+JukROhWdjBJz+17+I6kHbIEeqwhy4mdC5tTYOsrqRy+4iBsRgZTQLmgWGjI523GgzSlA2k6Cv" } } ], "meta": { "interimType": "Rationalize Title 1Eq" } } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7tkkS6M9CCT9LrMl7pDU/tr15eDTBjqr6Mh44dbONM5kJQJZuTAY5js+oqjdT8kslDa1hUvtzgBcwD7zey8pjX4xmKsrTmr07seOuQC7eif7WwPs1+Gw97t4MeuaNjSYTwPBBw3byB5RPtpSy6YLt2mmZmrN+TIHczZDNlT/7/mGJqVxX90jlMfh9fKn6dzC4" } }, { "type": "step", "result": "\\cos\\left(θ\\right)=-\\frac{\\sqrt{2}}{2}" } ], "meta": { "interimType": "Generic Simplify 0Eq" } } ], "meta": { "interimType": "Divide Both Sides Specific 1Eq", "gptData": "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" } }, { "type": "interim", "title": "General solutions for $$\\cos\\left(θ\\right)=-\\frac{\\sqrt{2}}{2}$$", "result": "θ=\\frac{3π}{4}+2πn,\\:θ=\\frac{5π}{4}+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": "θ=\\frac{3π}{4}+2πn,\\:θ=\\frac{5π}{4}+2πn" } ], "meta": { "interimType": "Trig General Solutions cos 1Eq" } } ], "meta": { "solvingClass": "Trig Equations", "interimType": "Trig Equations" } }, { "type": "interim", "title": "$$\\sqrt{2}\\cos\\left(θ\\right)-1=0{\\quad:\\quad}θ=\\frac{π}{4}+2πn,\\:θ=\\frac{7π}{4}+2πn$$", "input": "\\sqrt{2}\\cos\\left(θ\\right)-1=0", "steps": [ { "type": "interim", "title": "Move $$1\\:$$to the right side", "input": "\\sqrt{2}\\cos\\left(θ\\right)-1=0", "result": "\\sqrt{2}\\cos\\left(θ\\right)=1", "steps": [ { "type": "step", "primary": "Add $$1$$ to both sides", "result": "\\sqrt{2}\\cos\\left(θ\\right)-1+1=0+1" }, { "type": "step", "primary": "Simplify", "result": "\\sqrt{2}\\cos\\left(θ\\right)=1" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": "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" } }, { "type": "interim", "title": "Divide both sides by $$\\sqrt{2}$$", "input": "\\sqrt{2}\\cos\\left(θ\\right)=1", "result": "\\cos\\left(θ\\right)=\\frac{\\sqrt{2}}{2}", "steps": [ { "type": "step", "primary": "Divide both sides by $$\\sqrt{2}$$", "result": "\\frac{\\sqrt{2}\\cos\\left(θ\\right)}{\\sqrt{2}}=\\frac{1}{\\sqrt{2}}" }, { "type": "interim", "title": "Simplify", "input": "\\frac{\\sqrt{2}\\cos\\left(θ\\right)}{\\sqrt{2}}=\\frac{1}{\\sqrt{2}}", "result": "\\cos\\left(θ\\right)=\\frac{\\sqrt{2}}{2}", "steps": [ { "type": "interim", "title": "Simplify $$\\frac{\\sqrt{2}\\cos\\left(θ\\right)}{\\sqrt{2}}:{\\quad}\\cos\\left(θ\\right)$$", "input": "\\frac{\\sqrt{2}\\cos\\left(θ\\right)}{\\sqrt{2}}", "steps": [ { "type": "step", "primary": "Cancel the common factor: $$\\sqrt{2}$$", "result": "=\\cos\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s74DwjiBDVBIb8+yh6fkkMMdtK58IrruNZZpc41yUjaDO9eXg0wY6q+jIeOHWzjTOZCUCWbkwGOY7PqKo3U/JLJbfVxaVZWweWngVAZ8hO2bpFKk3fejFkyiOiq9iG9IkAIuSmJUDtLxRPyJ57Nkq7qi6yyeeeKwmd0ICbjP7Oejw1IGGNbmAWFGmhlV4gYgiXCavHWqV0CcO3cTEIFbZUKQ==" } }, { "type": "interim", "title": "Simplify $$\\frac{1}{\\sqrt{2}}:{\\quad}\\frac{\\sqrt{2}}{2}$$", "input": "\\frac{1}{\\sqrt{2}}", "steps": [ { "type": "step", "primary": "Multiply by the conjugate $$\\frac{\\sqrt{2}}{\\sqrt{2}}$$", "result": "=\\frac{1\\cdot\\:\\sqrt{2}}{\\sqrt{2}\\sqrt{2}}", "meta": { "title": { "extension": "To rationalize the denominator, multiply numerator and denominator by the conjugate of the radical $$\\sqrt{2}$$" } } }, { "type": "step", "primary": "$$1\\cdot\\:\\sqrt{2}=\\sqrt{2}$$" }, { "type": "interim", "title": "$$\\sqrt{2}\\sqrt{2}=2$$", "input": "\\sqrt{2}\\sqrt{2}", "result": "=\\frac{\\sqrt{2}}{2}", "steps": [ { "type": "step", "primary": "Apply radical rule: $$\\sqrt{a}\\sqrt{a}=a$$", "secondary": [ "$$\\sqrt{2}\\sqrt{2}=2$$" ], "result": "=2", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7OXRSLY4uXiMbvMfICP5T88aFSKFknkW7qg0U6E21MorMwViaLUXkeD+JukROhWdjBJz+17+I6kHbIEeqwhy4mdC5tTYOsrqRy+4iBsRgZTQLmgWGjI523GgzSlA2k6Cv" } } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s78B+yE0cCl8hT7YErNdnZGSQocUa/Khw1SLN1OZoDI7yrju+5Z51e/ZZSD3gRHwjBnvjDY21b5XBQ44AG3rKeoJPz7fKEZJdwb5+eH5xAg+lN5Aod6Hr1Lp2e/29KhSgUgQHNT2nz+ImvmnBFu276+QAWYM1eO6oSaF4qo2LhF0Akt3WiGR7ZaCaXvz77bMjS" } }, { "type": "step", "result": "\\cos\\left(θ\\right)=\\frac{\\sqrt{2}}{2}" } ], "meta": { "interimType": "Generic Simplify 0Eq" } } ], "meta": { "interimType": "Divide Both Sides Specific 1Eq", "gptData": "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" } }, { "type": "interim", "title": "General solutions for $$\\cos\\left(θ\\right)=\\frac{\\sqrt{2}}{2}$$", "result": "θ=\\frac{π}{4}+2πn,\\:θ=\\frac{7π}{4}+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": "θ=\\frac{π}{4}+2πn,\\:θ=\\frac{7π}{4}+2πn" } ], "meta": { "interimType": "Trig General Solutions cos 1Eq" } } ], "meta": { "solvingClass": "Trig Equations", "interimType": "Trig Equations" } }, { "type": "step", "primary": "Combine all the solutions", "result": "θ=2πn,\\:θ=π+2πn,\\:θ=\\frac{3π}{4}+2πn,\\:θ=\\frac{5π}{4}+2πn,\\:θ=\\frac{π}{4}+2πn,\\:θ=\\frac{7π}{4}+2πn" } ], "meta": { "solvingClass": "Trig Equations", "practiceLink": "/practice/trigonometry-practice#area=main&subtopic=Trig%20Equations", "practiceTopic": "Trig Equations" } }, "plot_output": { "meta": { "plotInfo": { "variable": "θ", "plotRequest": "\\tan(θ)-\\sin(2θ)" }, "showViewLarger": true } }, "meta": { "showVerify": true } }