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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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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