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

{ "query": { "display": "$$\\sin\\left(x\\right)+1=2\\sqrt{1-\\sin^{2}\\left(x\\right)}$$", "symbolab_question": "EQUATION#\\sin(x)+1=2\\sqrt{1-\\sin^{2}(x)}" }, "solution": { "level": "PERFORMED", "subject": "Trigonometry", "topic": "Trig Equations", "subTopic": "Trig Equations", "default": "x=0.64350…+2πn,x=π-0.64350…+2πn,x=\\frac{3π}{2}+2πn", "degrees": "x=36.86989…^{\\circ }+360^{\\circ }n,x=143.13010…^{\\circ }+360^{\\circ }n,x=270^{\\circ }+360^{\\circ }n", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\sin\\left(x\\right)+1=2\\sqrt{1-\\sin^{2}\\left(x\\right)}{\\quad:\\quad}x=0.64350…+2πn,\\:x=π-0.64350…+2πn,\\:x=\\frac{3π}{2}+2πn$$", "input": "\\sin\\left(x\\right)+1=2\\sqrt{1-\\sin^{2}\\left(x\\right)}", "steps": [ { "type": "interim", "title": "Solve by substitution", "input": "\\sin\\left(x\\right)+1=2\\sqrt{1-\\sin^{2}\\left(x\\right)}", "result": "\\sin\\left(x\\right)=\\frac{3}{5},\\:\\sin\\left(x\\right)=-1", "steps": [ { "type": "step", "primary": "Let: $$\\sin\\left(x\\right)=u$$", "result": "u+1=2\\sqrt{1-u^{2}}" }, { "type": "interim", "title": "$$u+1=2\\sqrt{1-u^{2}}{\\quad:\\quad}u=\\frac{3}{5},\\:u=-1$$", "input": "u+1=2\\sqrt{1-u^{2}}", "steps": [ { "type": "interim", "title": "Square both sides:$${\\quad}u^{2}+2u+1=4-4u^{2}$$", "input": "u+1=2\\sqrt{1-u^{2}}", "result": "u^{2}+2u+1=4-4u^{2}", "steps": [ { "type": "step", "result": "\\left(u+1\\right)^{2}=\\left(2\\sqrt{1-u^{2}}\\right)^{2}" }, { "type": "interim", "title": "Expand $$\\left(u+1\\right)^{2}:{\\quad}u^{2}+2u+1$$", "input": "\\left(u+1\\right)^{2}", "steps": [ { "type": "step", "primary": "Apply Perfect Square Formula: $$\\left(a+b\\right)^{2}=a^{2}+2ab+b^{2}$$", "secondary": [ "$$a=u,\\:\\:b=1$$" ], "meta": { "practiceLink": "/practice/expansion-practice#area=main&subtopic=Perfect%20Square", "practiceTopic": "Expand Perfect Square" } }, { "type": "step", "result": "=u^{2}+2u\\cdot\\:1+1^{2}" }, { "type": "interim", "title": "Simplify $$u^{2}+2u\\cdot\\:1+1^{2}:{\\quad}u^{2}+2u+1$$", "input": "u^{2}+2u\\cdot\\:1+1^{2}", "result": "=u^{2}+2u+1", "steps": [ { "type": "step", "primary": "Apply rule $$1^{a}=1$$", "secondary": [ "$$1^{2}=1$$" ], "result": "=u^{2}+2\\cdot\\:1\\cdot\\:u+1" }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:1=2$$", "result": "=u^{2}+2u+1" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Expand Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7WohdluKCYMv4V7XZj6EqH/lEY8POVHTWz55nO3/9OPFAmCwc12jAO81YHmMfL6YV9thEz2uS+PindP4p4ZFTuU3kCh3oevUunZ7/b0qFKBT8eG5MTQUZ/YATQRzrac3aialcV/dI5TH4fXyp+ncwuA==" } }, { "type": "interim", "title": "Expand $$\\left(2\\sqrt{1-u^{2}}\\right)^{2}:{\\quad}4-4u^{2}$$", "input": "\\left(2\\sqrt{1-u^{2}}\\right)^{2}", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$\\left(a\\cdot\\:b\\right)^{n}=a^{n}b^{n}$$", "result": "=2^{2}\\left(\\sqrt{1-u^{2}}\\right)^{2}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "interim", "title": "$$\\left(\\sqrt{1-u^{2}}\\right)^{2}:{\\quad}1-u^{2}$$", "steps": [ { "type": "step", "primary": "Apply radical rule: $$\\sqrt{a}=a^{\\frac{1}{2}}$$", "result": "=\\left(\\left(1-u^{2}\\right)^{\\frac{1}{2}}\\right)^{2}", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } }, { "type": "step", "primary": "Apply exponent rule: $$\\left(a^{b}\\right)^{c}=a^{bc}$$", "result": "=\\left(1-u^{2}\\right)^{\\frac{1}{2}\\cdot\\:2}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "interim", "title": "$$\\frac{1}{2}\\cdot\\:2=1$$", "input": "\\frac{1}{2}\\cdot\\:2", "result": "=1-u^{2}", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{1\\cdot\\:2}{2}" }, { "type": "step", "primary": "Cancel the common factor: $$2$$", "result": "=1" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l3vTdf410Ywhq1vZ0kzF8e30Fwl9QKPJxyO/TFRCb5Grju+5Z51e/ZZSD3gRHwjBE9/03SOiEv+BIHutWLr6nUfz18ijmoplMAomfJM9x8W1GdKgiNs+PolKvTuWzYk/" } } ], "meta": { "interimType": "N/A" } }, { "type": "step", "result": "=2^{2}\\left(1-u^{2}\\right)" }, { "type": "step", "primary": "$$2^{2}=4$$", "result": "=4\\left(1-u^{2}\\right)" }, { "type": "interim", "title": "Expand $$4\\left(1-u^{2}\\right):{\\quad}4-4u^{2}$$", "input": "4\\left(1-u^{2}\\right)", "steps": [ { "type": "step", "primary": "Apply the distributive law: $$a\\left(b-c\\right)=ab-ac$$", "secondary": [ "$$a=4,\\:b=1,\\:c=u^{2}$$" ], "result": "=4\\cdot\\:1-4u^{2}", "meta": { "practiceLink": "/practice/expansion-practice", "practiceTopic": "Expand Rules" } }, { "type": "step", "primary": "Multiply the numbers: $$4\\cdot\\:1=4$$", "result": "=4-4u^{2}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Expand Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7sI9wTliw5jQrXHLErH9g5Kx7HZ0MmmPPWIJzAK2VkHJ1DFKHA6Ddb5RE0Ay7mw9rpVjCuiAuXb1FPodh3Y0nfXql8XXPq6bNQlMm+36iNhkThSZbpYbjm8YuezXG3fXKW0rIZ5zdqE9boP35ZESaDw==" } }, { "type": "step", "result": "=4-4u^{2}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Expand Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7ZzqUGJQUlh00WhiOyofxX8tLJeqYnyftcJLfYZ/WwFILAZlDhoAdFQF6AF4pPagvRKAKbro8EavzLN8GuIDU/PWVUjtAdd6EqSvsyX6r+5AezFilETfuNygjs0XPkV2ha6dvsGDFR3H1x4Hnqml/vsNNA2zBOCYfUf0rhXmKVE4=" } }, { "type": "step", "result": "u^{2}+2u+1=4-4u^{2}" } ], "meta": { "interimType": "Radicals Square Both Sides Title 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDmP8XZX7cH+SrMN7zcXz+emNN9F1s3wXrd19zJ0jG6bZz3WXcdK/Diu3GUdCHfBGghkQY9Af7SiXjCPVqjDrZVVPTIH4EzjLo2mzzL6wGKcPC30sSftAIFS6Qkpy19IkppD+IQ+trcYYxyfOPi4jvW3hL3hU5yO5/xrtTXJ807Bw==" } }, { "type": "interim", "title": "Solve $$u^{2}+2u+1=4-4u^{2}:{\\quad}u=\\frac{3}{5},\\:u=-1$$", "input": "u^{2}+2u+1=4-4u^{2}", "steps": [ { "type": "interim", "title": "Move $$4u^{2}\\:$$to the left side", "input": "u^{2}+2u+1=4-4u^{2}", "result": "5u^{2}+2u+1=4", "steps": [ { "type": "step", "primary": "Add $$4u^{2}$$ to both sides", "result": "u^{2}+2u+1+4u^{2}=4-4u^{2}+4u^{2}" }, { "type": "step", "primary": "Simplify", "result": "5u^{2}+2u+1=4" } ], "meta": { "interimType": "Move to the Left Title 1Eq", "gptData": "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" } }, { "type": "interim", "title": "Move $$4\\:$$to the left side", "input": "5u^{2}+2u+1=4", "result": "5u^{2}+2u-3=0", "steps": [ { "type": "step", "primary": "Subtract $$4$$ from both sides", "result": "5u^{2}+2u+1-4=4-4" }, { "type": "step", "primary": "Simplify", "result": "5u^{2}+2u-3=0" } ], "meta": { "interimType": "Move to the Left Title 1Eq", "gptData": "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" } }, { "type": "interim", "title": "Solve with the quadratic formula", "input": "5u^{2}+2u-3=0", "result": "{u}_{1,\\:2}=\\frac{-2\\pm\\:\\sqrt{2^{2}-4\\cdot\\:5\\left(-3\\right)}}{2\\cdot\\:5}", "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 $${\\quad}a=5,\\:b=2,\\:c=-3$$", "result": "{u}_{1,\\:2}=\\frac{-2\\pm\\:\\sqrt{2^{2}-4\\cdot\\:5\\left(-3\\right)}}{2\\cdot\\:5}" } ], "meta": { "interimType": "Solving The Quadratic Equation With Quadratic Formula Definition 0Eq", "gptData": "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" } }, { "type": "interim", "title": "$$\\sqrt{2^{2}-4\\cdot\\:5\\left(-3\\right)}=8$$", "input": "\\sqrt{2^{2}-4\\cdot\\:5\\left(-3\\right)}", "result": "{u}_{1,\\:2}=\\frac{-2\\pm\\:8}{2\\cdot\\:5}", "steps": [ { "type": "step", "primary": "Apply rule $$-\\left(-a\\right)=a$$", "result": "=\\sqrt{2^{2}+4\\cdot\\:5\\cdot\\:3}" }, { "type": "step", "primary": "Multiply the numbers: $$4\\cdot\\:5\\cdot\\:3=60$$", "result": "=\\sqrt{2^{2}+60}" }, { "type": "step", "primary": "$$2^{2}=4$$", "result": "=\\sqrt{4+60}" }, { "type": "step", "primary": "Add the numbers: $$4+60=64$$", "result": "=\\sqrt{64}" }, { "type": "step", "primary": "Factor the number: $$64=8^{2}$$", "result": "=\\sqrt{8^{2}}" }, { "type": "step", "primary": "Apply radical rule: $$\\sqrt[n]{a^n}=a$$", "secondary": [ "$$\\sqrt{8^{2}}=8$$" ], "result": "=8", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s74YGLq7ZDKNPpn08HXk5Zm482b3HPw2ePrwllylr1hy7ehkKrn0era9rz8TlL+x/vkLiX3CLfIG78zwY25m2BNrtCR5dIjxQ5ASg+ZPFVSsddEzskrWlCIieu/GPZsDgnogb1xzrhieGGrczvhsLuEQ==" } }, { "type": "step", "primary": "Separate the solutions", "result": "{u}_{1}=\\frac{-2+8}{2\\cdot\\:5},\\:{u}_{2}=\\frac{-2-8}{2\\cdot\\:5}" }, { "type": "interim", "title": "$$u=\\frac{-2+8}{2\\cdot\\:5}:{\\quad}\\frac{3}{5}$$", "input": "\\frac{-2+8}{2\\cdot\\:5}", "steps": [ { "type": "step", "primary": "Add/Subtract the numbers: $$-2+8=6$$", "result": "=\\frac{6}{2\\cdot\\:5}" }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:5=10$$", "result": "=\\frac{6}{10}" }, { "type": "step", "primary": "Cancel the common factor: $$2$$", "result": "=\\frac{3}{5}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7ugzqT4wZ5lJdh8CwAuVAS1TwL+rfdKeYnjNSoNdZgTkDnzlbPZjyKgy1eUCFsLd5ou0fH4IfJpokJqvIcA4Iqh3iMvWYwwTTugkbawoy4uNgAYUJ/HHPxVoInxHoigvz2O5TxxFzs4G12i4LSZg9Nw==" } }, { "type": "interim", "title": "$$u=\\frac{-2-8}{2\\cdot\\:5}:{\\quad}-1$$", "input": "\\frac{-2-8}{2\\cdot\\:5}", "steps": [ { "type": "step", "primary": "Subtract the numbers: $$-2-8=-10$$", "result": "=\\frac{-10}{2\\cdot\\:5}" }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:5=10$$", "result": "=\\frac{-10}{10}" }, { "type": "step", "primary": "Apply the fraction rule: $$\\frac{-a}{b}=-\\frac{a}{b}$$", "result": "=-\\frac{10}{10}" }, { "type": "step", "primary": "Apply rule $$\\frac{a}{a}=1$$", "result": "=-1" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7AkKjGTji69+5/bpsxVKdVFTwL+rfdKeYnjNSoNdZgTkDnzlbPZjyKgy1eUCFsLd5pmhpfHqzxU6UVoC7gLWdYz/AeYEIX4I11Ba3Yz03hSAKSy3GIq50EhbVm6eu0dlYialcV/dI5TH4fXyp+ncwuA==" } }, { "type": "step", "primary": "The solutions to the quadratic equation are:", "result": "u=\\frac{3}{5},\\:u=-1" } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve Title 1Eq" } }, { "type": "step", "result": "u=\\frac{3}{5},\\:u=-1" }, { "type": "interim", "title": "Verify Solutions:$${\\quad}u=\\frac{3}{5}\\:$$True$$,\\:\\:u=-1\\:$$True", "steps": [ { "type": "step", "primary": "Check the solutions by plugging them into $$u+1=2\\sqrt{1-u^{2}}$$<br/>Remove the ones that don't agree with the equation." }, { "type": "interim", "title": "Plug in $$u=\\frac{3}{5}:{\\quad}$$True", "input": "\\left(\\frac{3}{5}\\right)+1=2\\sqrt{1-\\left(\\frac{3}{5}\\right)^{2}}", "steps": [ { "type": "interim", "title": "$$\\left(\\frac{3}{5}\\right)+1=\\frac{8}{5}$$", "input": "\\left(\\frac{3}{5}\\right)+1", "steps": [ { "type": "step", "primary": "Remove parentheses: $$\\left(a\\right)=a$$", "result": "=\\frac{3}{5}+1" }, { "type": "step", "primary": "Convert element to fraction: $$1=\\frac{1\\cdot\\:5}{5}$$", "result": "=\\frac{1\\cdot\\:5}{5}+\\frac{3}{5}" }, { "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{1\\cdot\\:5+3}{5}" }, { "type": "interim", "title": "$$1\\cdot\\:5+3=8$$", "input": "1\\cdot\\:5+3", "steps": [ { "type": "step", "primary": "Multiply the numbers: $$1\\cdot\\:5=5$$", "result": "=5+3" }, { "type": "step", "primary": "Add the numbers: $$5+3=8$$", "result": "=8" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7CyW57ak5AulCtFLSBgOhq96GQqufR6tr2vPxOUv7H++QuJfcIt8gbvzPBjbmbYE2P/n/sT8Hudl/0KJRqY9qeTgE3ql4+CsDvWWVKmqbeX4=" } }, { "type": "step", "result": "=\\frac{8}{5}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7F8XivpmVR/CWM7w89tv6vROGsTQeYliHAv5CNHRd5kf9ovYKijQYhJDCbxu/nAOJ5sqzH2pg8adhcc133yUY6cy82qTZUEi5vCQD0nP8naYyxa/q98ANWLQf0rhyAjwGS1fjLUpy/xoroxIIKjK6Dg==" } }, { "type": "interim", "title": "$$2\\sqrt{1-\\left(\\frac{3}{5}\\right)^{2}}=\\frac{8}{5}$$", "input": "2\\sqrt{1-\\left(\\frac{3}{5}\\right)^{2}}", "steps": [ { "type": "interim", "title": "$$\\sqrt{1-\\left(\\frac{3}{5}\\right)^{2}}=\\frac{4}{5}$$", "input": "\\sqrt{1-\\left(\\frac{3}{5}\\right)^{2}}", "steps": [ { "type": "interim", "title": "$$\\left(\\frac{3}{5}\\right)^{2}=\\frac{9}{25}$$", "input": "\\left(\\frac{3}{5}\\right)^{2}", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$\\left(\\frac{a}{b}\\right)^{c}=\\frac{a^{c}}{b^{c}}$$", "result": "=\\frac{3^{2}}{5^{2}}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "$$3^{2}=9$$", "result": "=\\frac{9}{5^{2}}" }, { "type": "step", "primary": "$$5^{2}=25$$", "result": "=\\frac{9}{25}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7F8XivpmVR/CWM7w89tv6vY5IpdliG1E4K4EtDGLN9yvMwViaLUXkeD+JukROhWdjajSNL8gwtl8uwYDXGS049uE9C24zoT9xMnOqMMSSaanwfNriGAVwTyfSfYx1u9JsIf5iaBRHoQWGSEdnGQKtXyS3daIZHtloJpe/PvtsyNI=" } }, { "type": "step", "result": "=\\sqrt{1-\\frac{9}{25}}" }, { "type": "interim", "title": "Join $$1-\\frac{9}{25}:{\\quad}\\frac{16}{25}$$", "input": "1-\\frac{9}{25}", "result": "=\\sqrt{\\frac{16}{25}}", "steps": [ { "type": "step", "primary": "Convert element to fraction: $$1=\\frac{1\\cdot\\:25}{25}$$", "result": "=\\frac{1\\cdot\\:25}{25}-\\frac{9}{25}" }, { "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{1\\cdot\\:25-9}{25}" }, { "type": "interim", "title": "$$1\\cdot\\:25-9=16$$", "input": "1\\cdot\\:25-9", "steps": [ { "type": "step", "primary": "Multiply the numbers: $$1\\cdot\\:25=25$$", "result": "=25-9" }, { "type": "step", "primary": "Subtract the numbers: $$25-9=16$$", "result": "=16" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7mVCUjWOZKkD0T38qZm2Ap1XTSum/z5kLpMzXS1UJIex8LCe1KOCcL5OIveXdCO/joHMgHqp2TL2vVYhONoO4NBrxV4yQqxqVAbzG6oBWFUQ=" } }, { "type": "step", "result": "=\\frac{16}{25}" } ], "meta": { "interimType": "Algebraic Manipulation Join Concise Title 1Eq" } }, { "type": "step", "primary": "Apply radical rule: $$\\sqrt[n]{\\frac{a}{b}}=\\frac{\\sqrt[n]{a}}{\\sqrt[n]{b}},\\:\\quad$$ assuming $$a\\ge0,\\:b\\ge0$$", "result": "=\\frac{\\sqrt{16}}{\\sqrt{25}}", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } }, { "type": "interim", "title": "$$\\sqrt{25}=5$$", "input": "\\sqrt{25}", "result": "=\\frac{\\sqrt{16}}{5}", "steps": [ { "type": "step", "primary": "Factor the number: $$25=5^{2}$$", "result": "=\\sqrt{5^{2}}" }, { "type": "step", "primary": "Apply radical rule: $$\\sqrt[n]{a^n}=a$$", "secondary": [ "$$\\sqrt{5^{2}}=5$$" ], "result": "=5", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } } ], "meta": { "interimType": "N/A" } }, { "type": "interim", "title": "$$\\sqrt{16}=4$$", "input": "\\sqrt{16}", "result": "=\\frac{4}{5}", "steps": [ { "type": "step", "primary": "Factor the number: $$16=4^{2}$$", "result": "=\\sqrt{4^{2}}" }, { "type": "step", "primary": "Apply radical rule: $$\\sqrt[n]{a^n}=a$$", "secondary": [ "$$\\sqrt{4^{2}}=4$$" ], "result": "=4", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } } ], "meta": { "interimType": "N/A" } } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7c0c3OZU478PEGHP8tj6SwaQ3rWNRD1Tqoo8TEB3riUJV00rpv8+ZC6TM10tVCSHs0xDS+Y5aj0hl+F6LvDaAlkSnHSTXJQZPzYrRXZ0wHXPjDQ4W9yWDssBvjiC+XrWYMLA8Rm/X489SJMYQ71Ttv59fR5bGcjZ13oliKrC6bHGwiNrEngO+NNvZ9sqNu+2V" } }, { "type": "step", "result": "=2\\cdot\\:\\frac{4}{5}" }, { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{4\\cdot\\:2}{5}" }, { "type": "step", "primary": "Multiply the numbers: $$4\\cdot\\:2=8$$", "result": "=\\frac{8}{5}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7sIKgEMIFucUA2YNlB/GnJpD1RwhihQK06S4cM4zfiAktOtZYwUjyXhDTsNnn6ElrSGpx9e3Lolh3WlYiBDdmOA/QaI57/PQy3BGWbwKPl9g2UBVbTQK2kC3stEeLcU0SAXlrto1v+pabEc0AZRkANGi1JRe8WKFPQgCLsXgwPX8cVfaNUbNnvQHnbUTm4aYo" } }, { "type": "step", "primary": "$$\\frac{8}{5}=\\frac{8}{5}$$" }, { "type": "step", "result": "\\mathrm{True}" } ], "meta": { "interimType": "Generic Plug 1Eq" } }, { "type": "interim", "title": "Plug in $$u=-1:{\\quad}$$True", "input": "\\left(-1\\right)+1=2\\sqrt{1-\\left(-1\\right)^{2}}", "steps": [ { "type": "interim", "title": "$$\\left(-1\\right)+1=0$$", "input": "\\left(-1\\right)+1", "steps": [ { "type": "step", "primary": "Remove parentheses: $$\\left(-a\\right)=-a$$", "result": "=-1+1" }, { "type": "step", "primary": "Add/Subtract the numbers: $$-1+1=0$$", "result": "=0" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7ydBWjEXZuGKiDESYcPgxE3WD310L1+P2yDQQfMEhENF3v1SuRJVPRIIuzzQuCKGXULkF4s6F1ex11sb6l8WTCCS3daIZHtloJpe/PvtsyNI=" } }, { "type": "interim", "title": "$$2\\sqrt{1-\\left(-1\\right)^{2}}=0$$", "input": "2\\sqrt{1-\\left(-1\\right)^{2}}", "steps": [ { "type": "interim", "title": "$$\\sqrt{1-\\left(-1\\right)^{2}}=0$$", "input": "\\sqrt{1-\\left(-1\\right)^{2}}", "steps": [ { "type": "interim", "title": "$$\\left(-1\\right)^{2}=1$$", "input": "\\left(-1\\right)^{2}", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$\\left(-a\\right)^{n}=a^{n},\\:$$if $$n$$ is even", "secondary": [ "$$\\left(-1\\right)^{2}=1^{2}$$" ], "result": "=1^{2}" }, { "type": "step", "primary": "Apply rule $$1^{a}=1$$", "result": "=1" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s78E1FVQW6YvXK7raPRxih+c0ag8T1MwTer44+aCS/ZFAdx7pcd1x/bAWpIL8hAintf05A2GsVmPba4FjoW22b4iKyMg44e9p5G7GRfJ2en9g=" } }, { "type": "step", "result": "=\\sqrt{1-1}" }, { "type": "step", "primary": "Subtract the numbers: $$1-1=0$$", "result": "=\\sqrt{0}" }, { "type": "step", "primary": "Apply rule $$\\sqrt{0}=0$$", "result": "=0" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7BFR+9VZsOp4IEf1jC/TTJCkWnF/8WxWzZ96351/38mDMwViaLUXkeD+JukROhWdjguVaeA0vkFtAR3gzqI9IZ/gAMeVwd9YtsoQyaxdPKx389O67RWU0odhnGINNvoei" } }, { "type": "step", "result": "=2\\cdot\\:0" }, { "type": "step", "primary": "Apply rule $$0\\cdot\\:a=0$$", "result": "=0" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7hYN2L9zcRIr/vYiIuQyU7F1iJRJdZU8Snq+IOvMxYQFwkKGJWEPFPk38sdJMsyPIc1L1JfkzeAMH8Sv8wAfVX1uDefCyAL74XZNHvW5bWtVVfrHM/WjtlmmD/uTQMUgf" } }, { "type": "step", "primary": "$$0=0$$" }, { "type": "step", "result": "\\mathrm{True}" } ], "meta": { "interimType": "Generic Plug 1Eq" } } ], "meta": { "interimType": "Check Solutions Plug Preface 1Eq" } }, { "type": "step", "primary": "The solutions are", "result": "u=\\frac{3}{5},\\:u=-1" } ], "meta": { "solvingClass": "Equations", "interimType": "Equations" } }, { "type": "step", "primary": "Substitute back $$u=\\sin\\left(x\\right)$$", "result": "\\sin\\left(x\\right)=\\frac{3}{5},\\:\\sin\\left(x\\right)=-1" } ], "meta": { "interimType": "Substitution Method 0Eq" } }, { "type": "interim", "title": "$$\\sin\\left(x\\right)=\\frac{3}{5}{\\quad:\\quad}x=\\arcsin\\left(\\frac{3}{5}\\right)+2πn,\\:x=π-\\arcsin\\left(\\frac{3}{5}\\right)+2πn$$", "input": "\\sin\\left(x\\right)=\\frac{3}{5}", "steps": [ { "type": "interim", "title": "Apply trig inverse properties", "input": "\\sin\\left(x\\right)=\\frac{3}{5}", "result": "x=\\arcsin\\left(\\frac{3}{5}\\right)+2πn,\\:x=π-\\arcsin\\left(\\frac{3}{5}\\right)+2πn", "steps": [ { "type": "step", "primary": "General solutions for $$\\sin\\left(x\\right)=\\frac{3}{5}$$", "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{3}{5}\\right)+2πn,\\:x=π-\\arcsin\\left(\\frac{3}{5}\\right)+2πn" } ], "meta": { "interimType": "Trig Apply Inverse Props 0Eq" } } ], "meta": { "interimType": "N/A" } }, { "type": "interim", "title": "$$\\sin\\left(x\\right)=-1{\\quad:\\quad}x=\\frac{3π}{2}+2πn$$", "input": "\\sin\\left(x\\right)=-1", "steps": [ { "type": "interim", "title": "General solutions for $$\\sin\\left(x\\right)=-1$$", "result": "x=\\frac{3π}{2}+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{3π}{2}+2πn" } ], "meta": { "interimType": "Trig General Solutions sin 1Eq" } } ], "meta": { "interimType": "N/A" } }, { "type": "step", "primary": "Combine all the solutions", "result": "x=\\arcsin\\left(\\frac{3}{5}\\right)+2πn,\\:x=π-\\arcsin\\left(\\frac{3}{5}\\right)+2πn,\\:x=\\frac{3π}{2}+2πn" }, { "type": "step", "primary": "Show solutions in decimal form", "result": "x=0.64350…+2πn,\\:x=π-0.64350…+2πn,\\:x=\\frac{3π}{2}+2πn" } ], "meta": { "solvingClass": "Trig Equations", "practiceLink": "/practice/trigonometry-practice#area=main&subtopic=Trig%20Equations", "practiceTopic": "Trig Equations" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "\\sin(x)+1-2\\sqrt{1-\\sin^{2}(x)}" }, "showViewLarger": true } }, "meta": { "showVerify": true } }