cosh(z)=1,cosh(z)=-2

{ "query": { "display": "$$\\cosh\\left(z\\right)=1,\\:\\cosh\\left(z\\right)=-2$$", "symbolab_question": "EQUATION#\\cosh(z)=1,\\cosh(z)=-2" }, "solution": { "level": "PERFORMED", "subject": "Trigonometry", "topic": "Trig Equations", "subTopic": "Trig Equations", "default": "z=0", "degrees": "z=0^{\\circ }", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\cosh\\left(z\\right)=1,\\:\\cosh\\left(z\\right)=-2{\\quad:\\quad}z=0$$", "input": "\\cosh\\left(z\\right)=1,\\:\\cosh\\left(z\\right)=-2", "steps": [ { "type": "interim", "title": "Rewrite using trig identities", "input": "\\cosh\\left(z\\right)=1", "result": "\\frac{e^{z}+e^{-z}}{2}=1", "steps": [ { "type": "step", "primary": "Use the Hyperbolic identity: $$\\cosh\\left(x\\right)=\\frac{e^{x}+e^{-x}}{2}$$", "result": "\\frac{e^{z}+e^{-z}}{2}=1" } ], "meta": { "interimType": "Trig Rewrite Using Trig identities 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s71EZrVCDfZjh0S8ARhjtEX9ayDryVlU65JXUnw0AXhfCAYUxCRRMw2cScUTF07kq8MvMnPiuRieO/X9beCVqRKbWzVFA6VxLY8LrAw/d4eKv7Cukb8E7dn1xMQmgqOSwf04mRy+XfAsV1buHdoreJne9sGZu5A1MXROmEpnxG69oGqP+/Ugc9TvNWR3IfFZuE9mth+xoYA4yaoGna981gpoZ+y+YEwVMxeIPlP7C0eko=" } }, { "type": "interim", "title": "$$\\frac{e^{z}+e^{-z}}{2}=1{\\quad:\\quad}z=0$$", "input": "\\frac{e^{z}+e^{-z}}{2}=1", "steps": [ { "type": "step", "primary": "Multiply both sides by $$2$$", "result": "\\frac{e^{z}+e^{-z}}{2}\\cdot\\:2=1\\cdot\\:2" }, { "type": "step", "primary": "Simplify", "result": "e^{z}+e^{-z}=2" }, { "type": "interim", "title": "Apply exponent rules", "input": "e^{z}+e^{-z}=2", "result": "e^{z}+\\left(e^{z}\\right)^{-1}=2", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^{bc}=\\left(a^{b}\\right)^{c}$$", "secondary": [ "$$e^{-z}=\\left(e^{z}\\right)^{-1}$$" ], "result": "e^{z}+\\left(e^{z}\\right)^{-1}=2", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } } ], "meta": { "interimType": "Apply Exp Rules Title 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7lbhRT+9hrUVb2O84M6qfZDkGDiJNY5bVYW55vAe8dwDMjpQnG3kDEFW4j3KHbbIaK73FMT/8JGTXWZSUiKGEX6sijAPLIMZ8QtMnwl2eie2FhnfdDO00sQQagQNVquKW6SW5aOnxG/gNDgOjato5sN0plL1+PmcY7wXX3UhgrDtfToiN8LJK2ayApoXEPeA+vzIPeEtDfcHv/z8uls8Teg==" } }, { "type": "step", "primary": "Rewrite the equation with $$e^{z}=u$$", "result": "u+\\left(u\\right)^{-1}=2" }, { "type": "interim", "title": "Solve $$u+u^{-1}=2:{\\quad}u=1$$", "input": "u+u^{-1}=2", "steps": [ { "type": "step", "primary": "Refine", "result": "u+\\frac{1}{u}=2" }, { "type": "interim", "title": "Multiply both sides by $$u$$", "input": "u+\\frac{1}{u}=2", "result": "u^{2}+1=2u", "steps": [ { "type": "step", "primary": "Multiply both sides by $$u$$", "result": "uu+\\frac{1}{u}u=2u" }, { "type": "interim", "title": "Simplify", "input": "uu+\\frac{1}{u}u=2u", "result": "u^{2}+1=2u", "steps": [ { "type": "interim", "title": "Simplify $$uu:{\\quad}u^{2}$$", "input": "uu", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$uu=\\:u^{1+1}$$" ], "result": "=u^{1+1}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$1+1=2$$", "result": "=u^{2}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7/E93FIYHpq26Gj2mwfeoqMzBWJotReR4P4m6RE6FZ2Oes25OoAq8kAwRqD36EFJeHjb2+5NLFZrsH9fcPWg/TcJNRMYaC/08wZ47yP7pQyE=" } }, { "type": "interim", "title": "Simplify $$\\frac{1}{u}u:{\\quad}1$$", "input": "\\frac{1}{u}u", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{1\\cdot\\:u}{u}" }, { "type": "step", "primary": "Cancel the common factor: $$u$$", "result": "=1" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7EeLivs/RMqUCyuozV/Q3AC061ljBSPJeENOw2efoSWs8auWUd4WNoosLPjGkhvjRRSpN33oxZMojoqvYhvSJACLkpiVA7S8UT8ieezZKu6oGJz3eKnhJxw90n+P5vsKL" } }, { "type": "step", "result": "u^{2}+1=2u" } ], "meta": { "interimType": "Generic Simplify 0Eq" } } ], "meta": { "interimType": "Multiply Both Sides Specific 1Eq", "gptData": "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" } }, { "type": "interim", "title": "Solve $$u^{2}+1=2u:{\\quad}u=1$$", "input": "u^{2}+1=2u", "steps": [ { "type": "interim", "title": "Move $$2u\\:$$to the left side", "input": "u^{2}+1=2u", "result": "u^{2}+1-2u=0", "steps": [ { "type": "step", "primary": "Subtract $$2u$$ from both sides", "result": "u^{2}+1-2u=2u-2u" }, { "type": "step", "primary": "Simplify", "result": "u^{2}+1-2u=0" } ], "meta": { "interimType": "Move to the Left Title 1Eq", "gptData": "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" } }, { "type": "step", "primary": "Write in the standard form $$ax^{2}+bx+c=0$$", "result": "u^{2}-2u+1=0" }, { "type": "interim", "title": "Solve with the quadratic formula", "input": "u^{2}-2u+1=0", "result": "{u}_{1,\\:2}=\\frac{-\\left(-2\\right)\\pm\\:\\sqrt{\\left(-2\\right)^{2}-4\\cdot\\:1\\cdot\\:1}}{2\\cdot\\:1}", "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 $${\\quad}a=1,\\:b=-2,\\:c=1$$", "result": "{u}_{1,\\:2}=\\frac{-\\left(-2\\right)\\pm\\:\\sqrt{\\left(-2\\right)^{2}-4\\cdot\\:1\\cdot\\:1}}{2\\cdot\\:1}" } ], "meta": { "interimType": "Solving The Quadratic Equation With Quadratic Formula Definition 0Eq", "gptData": "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" } }, { "type": "interim", "title": "$$\\left(-2\\right)^{2}-4\\cdot\\:1\\cdot\\:1=0$$", "input": "\\left(-2\\right)^{2}-4\\cdot\\:1\\cdot\\:1", "result": "{u}_{1,\\:2}=\\frac{-\\left(-2\\right)\\pm\\:\\sqrt{0}}{2\\cdot\\:1}", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$\\left(-a\\right)^{n}=a^{n},\\:$$if $$n$$ is even", "secondary": [ "$$\\left(-2\\right)^{2}=2^{2}$$" ], "result": "=2^{2}-4\\cdot\\:1\\cdot\\:1" }, { "type": "step", "primary": "Multiply the numbers: $$4\\cdot\\:1\\cdot\\:1=4$$", "result": "=2^{2}-4" }, { "type": "step", "primary": "$$2^{2}=4$$", "result": "=4-4" }, { "type": "step", "primary": "Subtract the numbers: $$4-4=0$$", "result": "=0" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7DPXAC421UjJhhGOo7DnYa25PLVy0IEOU1w5RhPMODOwgJ/ZZA32ZInFBpDtxBfiKf7LqB9CcyvYCWDsGseX09ngWpZrrqhM2PY7ADOWfoCdBY+wojvBUDQ0rdjc9Zpxevn7iF8w34il58z5Kk+cQTQ==" } }, { "type": "step", "result": "u=\\frac{-\\left(-2\\right)}{2\\cdot\\:1}" }, { "type": "interim", "title": "$$\\frac{-\\left(-2\\right)}{2\\cdot\\:1}=1$$", "input": "\\frac{-\\left(-2\\right)}{2\\cdot\\:1}", "result": "u=1", "steps": [ { "type": "step", "primary": "Apply rule $$-\\left(-a\\right)=a$$", "result": "=\\frac{2}{2\\cdot\\:1}" }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:1=2$$", "result": "=\\frac{2}{2}" }, { "type": "step", "primary": "Apply rule $$\\frac{a}{a}=1$$", "result": "=1" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7v228JUa8d3c6dzWBzNQkPLFZak7+g3aGbe9C49/LWHDNGoPE9TME3q+OPmgkv2RQHce6XHdcf2wFqSC/IQIp7Y+MAqL37zfLm0yQ/WPDUpmwkktQA3aNGhf00gW9Ij7KIrIyDjh72nkbsZF8nZ6f2A==" } }, { "type": "step", "primary": "The solution to the quadratic equation is:", "result": "u=1" } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve Title 1Eq" } }, { "type": "step", "result": "u=1" }, { "type": "step", "primary": "Verify Solutions" }, { "type": "interim", "title": "Find undefined (singularity) points:$${\\quad}u=0$$", "steps": [ { "type": "step", "primary": "Take the denominator(s) of $$u+u^{-1}$$ and compare to zero" }, { "type": "step", "result": "u=0" }, { "type": "step", "primary": "The following points are undefined", "result": "u=0" } ], "meta": { "interimType": "Undefined Points 0Eq" } }, { "type": "step", "primary": "Combine undefined points with solutions:" }, { "type": "step", "result": "u=1" } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve Title 1Eq" } }, { "type": "step", "result": "u=1" }, { "type": "step", "primary": "Substitute back $$u=e^{z},\\:$$solve for $$z$$" }, { "type": "interim", "title": "Solve $$e^{z}=1:{\\quad}z=0$$", "input": "e^{z}=1", "steps": [ { "type": "interim", "title": "Apply exponent rules", "input": "e^{z}=1", "result": "z=0", "steps": [ { "type": "step", "primary": "If $$f\\left(x\\right)=g\\left(x\\right)$$, then $$\\ln\\left(f\\left(x\\right)\\right)=\\ln\\left(g\\left(x\\right)\\right)$$", "result": "\\ln\\left(e^{z}\\right)=\\ln\\left(1\\right)" }, { "type": "step", "primary": "Apply log rule: $$\\ln\\left(e^a\\right)=a$$", "secondary": [ "$$\\ln\\left(e^{z}\\right)=z$$" ], "result": "z=\\ln\\left(1\\right)", "meta": { "practiceLink": "/practice/logarithms-practice", "practiceTopic": "Expand FOIL" } }, { "type": "interim", "title": "Simplify $$\\ln\\left(1\\right):{\\quad}0$$", "input": "\\ln\\left(1\\right)", "steps": [ { "type": "step", "primary": "Apply log rule: $$\\log_a\\left(1\\right)=0$$", "result": "=0", "meta": { "practiceLink": "/practice/logarithms-practice", "practiceTopic": "Expand FOIL" } } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7cJO4Mw9PX2rs+FfXhju0IwOfOVs9mPIqDLV5QIWwt3m4DS9snDRdGFIEJoiNCqQWTeQKHeh69S6dnv9vSoUoFAJyfalcNsA5/wfkdc07YubiAEmXhYw7WsDRrfT9tRiW" } }, { "type": "step", "result": "z=0" } ], "meta": { "interimType": "Apply Exp Rules Title 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7jXv/RNGy03eAa6mfboPuPG9f69mFY4KpYZIl0v+ArtOhHeJ1xBi321LjY4vsXC9em3FAiPzxVy0umodhDNEdMootQFPTqsfMc6aQSblBSMTWwPs1+Gw97t4MeuaNjSYT0iUaFOhwum4uivlyOmSIiBxUFhZkDT22BzQg28RPQec=" } } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve Title 1Eq" } }, { "type": "step", "result": "z=0" } ], "meta": { "solvingClass": "Equations", "interimType": "Equations" } }, { "type": "step", "result": "z=0" } ], "meta": { "solvingClass": "Trig Equations", "practiceLink": "/practice/trigonometry-practice#area=main&subtopic=Trig%20Equations", "practiceTopic": "Trig Equations" } }, "plot_output": { "meta": { "plotInfo": { "variable": "z", "plotRequest": "\\cosh(z)-1,\\cosh(z)=-2" }, "showViewLarger": true } }, "meta": { "showVerify": true } }