Problem 52467. Easy Sequences 2: Trigonometric function with integral input and output

Created by Ramon Villamangca

Solve Later

{"parts":[{"partUri":"/matlab/document.xml","contentType":"application/vnd.mathworks.matlab.code.document+xml","content":"<?xml version=\"1.0\" encoding=\"UTF-8\"?><w:document xmlns:w=\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\"><w:body><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>The function 'F', defined as: </w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t></w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>                </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>F(x) = -\\cos(2\\pi x/3)/9+(2x\\sqrt{3}/9+1/\\sqrt{3})\\cdot\\sin(2\\pi x/3)+1/9</w:t></w:r></w:customXml><w:r><w:t>,</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t></w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>will always return an integer if the input is a natural number (in radians). Furthermore, since the cosine and sine of 0, are 1 and 0 respectively, therefore F(0) = 0.  </w:t></w:r><w:r><w:rPr><w:b/></w:rPr><w:t>For a given natural number 'n'  your task is to find the value of F(n).</w:t></w:r></w:p></w:body></w:document>","relationship":null}],"relationships":[{"relationshipType":"http://schemas.mathworks.com/matlab/code/2013/relationships/document","target":"/matlab/document.xml","relationshipId":"rId1"}]}

<div style = "text-align: start; line-height: 20.440000534057617px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: normal; text-decoration: none; white-space: normal; "><div style="block-size: 163px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; vertical-align: baseline; "><div style="block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; text-align: left; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; "><span style="">The function 'F', defined as: </span></span></div><div style="block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; text-align: left; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; "><span style=""></span></span></div><div style="block-size: 22px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; 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" width="403.5" height="21" style="width: 403.5px; height: 21px;"></span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; "><span style="">,</span></span></div><div style="block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; text-align: left; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; "><span style=""></span></span></div><div style="block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; text-align: left; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; "><span style="">will always return an integer if the input is a natural number (in radians). Furthermore, since the cosine and sine of 0, are 1 and 0 respectively, therefore F(0) = 0.  </span></span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; "><span style="font-weight: bold; ">For a given natural number 'n'  your task is to find the value of F(n).</span></span></div></div></div>

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Last Solution submitted on Feb 21, 2023

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Mark Posen on 10 Dec 2021

Can someone please explain test case 4? I don't see what is going on, as intmax-4 is clearly not 1234567891011.

I could only pass this case by bodging x inside the function, which is clearly unsatisfactory.

Dyuman Joshi on 10 Dec 2021

intmax - 4 is not equal to 1234567891011. But
F(intmax-4) is equal to F(1234567891011). The given function can have same output for different inputs.

Ramon Villamangca on 14 Dec 2021

Hi Mark,
Trigonometric functions are naturally periodic. So we can expect same function return from different inputs.

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Problem 52467. Easy Sequences 2: Trigonometric function with integral input and output

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Problem 52467. Easy Sequences 2: Trigonometric function with integral input and output

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