<html><head><title>Math Addendum 1</title><style type="text/css"> p { margin: 0 0 0 0; font-family: Arial, sans-serif; font-size:10pt; font-weight: normal; font-style: normal; vertical-align: baseline; color: black; text-decoration: none; } div { border-width:0px; border-style: solid; border-color: red; } body { background-color: #fff; } p img { vertical-align: middle; } p b, li b { font-weight : bold; } p i, li i { font-style : italic; } p u, li u { text-decoration : underline; } p so, li so { text-decoration : line-through; } p sub, p sub { font-size : 70%; vertical-align: sub; } p sup, p sup { font-size : 70%; vertical-align: super; } ul { padding-left : 2em; /* for mozilla list marker box */ margin-left : 0; /* for IE list marker box */ } ol { padding-left : 2em; /* for mozilla list marker box */ margin-right : -0.5em; margin-left : 0; /* for IE list marker box */ } p.Style1, li.Style1 /*Normal*/ { margin-left: 0pt; margin-right: 0pt; text-align: left; font-family: 'Arial', sans-serif;; font-size: 10pt;font-weight: normal; font-style: normal; vertical-align: baseline; } p.Style2, li.Style2 /*Heading 1*/ { margin-left: 0pt; margin-right: 0pt; text-align: left; font-family: 'Arial', sans-serif;; font-size: 13.26315789473684pt;font-weight: bold; font-style: normal; vertical-align: baseline; } p.Style3, li.Style3 /*Heading 2*/ { margin-left: 0pt; margin-right: 0pt; text-align: left; font-family: 'Arial', sans-serif;; font-size: 12pt;font-weight: bold; font-style: italic; vertical-align: baseline; } p.Style4, li.Style4 /*Heading 3*/ { margin-left: 0pt; margin-right: 0pt; text-align: left; font-family: 'Arial', sans-serif;; font-size: 12pt;font-weight: normal; font-style: normal; vertical-align: baseline; } p.Style5, li.Style5 /*Paragraph*/ { margin-left: 0pt; margin-right: 0pt; text-indent: 21pt; text-align: left; font-family: 'Arial', sans-serif;; font-size: 10pt;font-weight: normal; font-style: normal; vertical-align: baseline; } p.Style6, li.Style6 /*List*/ { margin-left: 14.25pt; margin-right: 0pt; text-indent: -14.25pt; text-align: left; font-family: 'Arial', sans-serif;; font-size: 10pt;font-weight: normal; font-style: normal; vertical-align: baseline; } p.Style7, li.Style7 /*Indent*/ { margin-left: 108pt; margin-right: 0pt; text-align: left; font-family: 'Arial', sans-serif;; font-size: 10pt;font-weight: normal; font-style: normal; vertical-align: baseline; } p.Style8, li.Style8 /*Title*/ { margin-left: 0pt; margin-right: 0pt; text-align: center; font-family: 'Times New Roman', sans-serif;; font-size: 24pt;font-weight: bold; font-style: normal; vertical-align: baseline; } p.Style9, li.Style9 /*Subtitle*/ { margin-left: 0pt; margin-right: 0pt; text-align: center; font-family: 'Times New Roman', sans-serif;; font-size: 18pt;font-weight: normal; font-style: normal; vertical-align: baseline; } </style></head><body><div id="mc-region-2" style="position: absolute; top: 24pt; left: 18pt; width: 273.75pt; "><a name="" /><img border="0" src=".\addendum 1 angles_images/IMG0003_105261812.PNG" id="generatedImage2" path=".\addendum 1 angles_images/IMG0003_105261812.PNG" style="width: 273.75pt;height: 41.25pt;"></img></div><div id="mc-region-5" style="position: absolute; top: 86.25pt; left: 18pt; width: 111.5pt; "><a name="" /><p class="Style1"><b>MathCAD Addendum 1</b></p><p class="Style1"><i>Version: 12 July 2006</i></p></div><div id="mc-region-39" style="position: absolute; top: 128.25pt; left: 12pt; width: 223.25pt; "><a name="" /><p class="Style1">1. <b><i>Angular measurement units and conversions</i></b></p></div><div id="mc-region-91" style="position: absolute; top: 152.25pt; left: 12pt; width: 408.5pt; "><a name="" /><p class="Style1">A <b>degree</b> is probably most familiar angular unit: There are 90 degrees in a right angle and 360 degrees in a circle. A more mathematically natural unit is the <b>radian:</b> There are 2<span style="font-size: 10pt; font-family: 'Symbol', sans-serif;"><span style="font-weight: normal; font-style: normal;">p</span></span><span style="font-size: 10pt; font-family: 'Arial', sans-serif;"><span style="font-weight: normal; font-style: normal;"> (~6.28) radians in a circle. Hence</span></span></p></div><div id="mc-region-8" style="position: absolute; top: 195pt; left: 90pt; width: 78pt; "><a name="" /><img border="0" src=".\addendum 1 angles_images/IMG0004_105261828.PNG" id="generatedImage8" path=".\addendum 1 angles_images/IMG0004_105261828.PNG" style="width: 78pt;height: 12.75pt;"></img></div><div id="mc-region-12" style="position: absolute; top: 194.25pt; left: 186pt; width: 15.5pt; "><a name="" /><p class="Style1">so</p></div><div id="mc-region-11" style="position: absolute; top: 195pt; left: 216pt; width: 73.5pt; "><a name="" /><img border="0" src=".\addendum 1 angles_images/IMG0005_105261843.PNG" id="generatedImage11" path=".\addendum 1 angles_images/IMG0005_105261843.PNG" style="width: 73.5pt;height: 12.75pt;"></img></div><div id="mc-region-92" style="position: absolute; top: 218.25pt; left: 12pt; width: 378.5pt; "><a name="" /><p class="Style1">Most computer math languages (including MathCAD) expect that the arguments of trig functions are in radians. Hence, the familiar sin(30deg) = 0.5 doesn't give the expected answer: </p></div><div id="mc-region-15" style="position: absolute; top: 261pt; left: 90pt; width: 72pt; "><a name="" /><img border="0" src=".\addendum 1 angles_images/IMG0006_105261859.PNG" id="generatedImage15" path=".\addendum 1 angles_images/IMG0006_105261859.PNG" style="width: 72pt;height: 12.75pt;"></img></div><div id="mc-region-93" style="position: absolute; top: 284.25pt; left: 12pt; width: 196.25pt; "><a name="" /><p class="Style1">unless we specify the angular unit explicitly:</p></div><div id="mc-region-19" style="position: absolute; top: 285pt; left: 258pt; width: 75.75pt; "><a name="" /><img border="0" src=".\addendum 1 angles_images/IMG0007_105261859.PNG" id="generatedImage19" path=".\addendum 1 angles_images/IMG0007_105261859.PNG" style="width: 75.75pt;height: 12.75pt;"></img></div><div id="mc-region-94" style="position: absolute; top: 308.25pt; left: 12pt; width: 396.5pt; "><a name="" /><p class="Style1">In astronomy, most angles are very small (&lt;&lt;1 deg), so we commonly express angles in either <b>arcminutes</b> (1/60 deg) or <b>arcseconds</b> (1/3600 deg). MathCAD has both units buil-in: </p></div><div id="mc-region-95" style="position: absolute; top: 363pt; left: 6pt; width: 82.5pt; "><a name="" /><img border="0" src=".\addendum 1 angles_images/IMG0008_105261875.PNG" id="generatedImage95" path=".\addendum 1 angles_images/IMG0008_105261875.PNG" style="width: 82.5pt;height: 12.75pt;"></img></div><div id="mc-region-88" style="position: absolute; top: 356.25pt; left: 126pt; width: 50.25pt; "><a name="" /><img border="0" src=".\addendum 1 angles_images/IMG0009_105261890.PNG" id="generatedImage88" path=".\addendum 1 angles_images/IMG0009_105261890.PNG" style="width: 50.25pt;height: 27.75pt;"></img></div><div id="mc-region-89" style="position: absolute; top: 357pt; left: 186pt; width: 114pt; "><a name="" /><img border="0" src=".\addendum 1 angles_images/IMG0010_105261906.PNG" id="generatedImage89" path=".\addendum 1 angles_images/IMG0010_105261906.PNG" style="width: 114pt;height: 18.75pt;"></img></div><div id="mc-region-90" style="position: absolute; top: 356.25pt; left: 318pt; width: 91.5pt; "><a name="" /><img border="0" src=".\addendum 1 angles_images/IMG0011_105261921.PNG" id="generatedImage90" path=".\addendum 1 angles_images/IMG0011_105261921.PNG" style="width: 91.5pt;height: 27.75pt;"></img></div><div id="mc-region-60" style="position: absolute; top: 398.25pt; left: 6pt; width: 110.75pt; "><a name="" /><p class="Style1">2. <b><i>Small angle formula</i></b></p></div><div id="mc-region-57" style="position: absolute; top: 434.25pt; left: 6pt; width: 402.5pt; "><a name="" /><p class="Style1">For small angles (<span style="font-size: 10pt; font-family: 'Symbol', sans-serif;"><span style="font-weight: normal; font-style: normal;">q</span></span><span style="font-size: 10pt; font-family: 'Arial', sans-serif;"><span style="font-weight: normal; font-style: normal;">&lt;&lt;1 radian), the sine or tangent of an angle is approximately equal to the angle itself i</span></span><span style="font-size: 10pt; font-family: 'Arial', sans-serif;"><i><span style="font-weight: normal;">f expressed in radians. </span></i></span><span style="font-size: 10pt; font-family: 'Arial', sans-serif;"><span style="font-weight: normal; font-style: normal;">For example:</span></span></p></div><div id="mc-region-61" style="position: absolute; top: 471pt; left: 42pt; width: 84.75pt; "><a name="" /><img border="0" src=".\addendum 1 angles_images/IMG0012_105261937.PNG" id="generatedImage61" path=".\addendum 1 angles_images/IMG0012_105261937.PNG" style="width: 84.75pt;height: 12.75pt;"></img></div><div id="mc-region-62" style="position: absolute; top: 471pt; left: 144pt; width: 85.5pt; "><a name="" /><img border="0" src=".\addendum 1 angles_images/IMG0013_105261953.PNG" id="generatedImage62" path=".\addendum 1 angles_images/IMG0013_105261953.PNG" style="width: 85.5pt;height: 12.75pt;"></img></div><div id="mc-region-63" style="position: absolute; top: 471pt; left: 270pt; width: 65.25pt; "><a name="" /><img border="0" src=".\addendum 1 angles_images/IMG0014_105261968.PNG" id="generatedImage63" path=".\addendum 1 angles_images/IMG0014_105261968.PNG" style="width: 65.25pt;height: 12.75pt;"></img></div><div id="mc-region-58" style="position: absolute; top: 506.25pt; left: 6pt; width: 361.25pt; "><a name="" /><p class="Style1">This means that we can approximate either the sine or tangent by the angle itself. </p></div><div id="mc-region-59" style="position: absolute; top: 536.25pt; left: 6pt; width: 396.5pt; "><a name="" /><p class="Style1">The 'small angle formula' is a relation between the size of an object, its distance, and its angualr size. If the linear size <i>D</i> of an object is given in AU, and the angular size <span style="font-size: 10pt; font-family: 'Symbol', sans-serif;"><span style="font-weight: normal; font-style: normal;">a</span></span><span style="font-size: 10pt; font-family: 'Arial', sans-serif;"><span style="font-weight: normal; font-style: normal;"> </span></span>is in arcseconds, and the distance <i>d</i> is in parsecs, the relationship is simply written: </p></div><div id="mc-region-64" style="position: absolute; top: 584.25pt; left: 174pt; width: 33.75pt; "><a name="" /><img border="0" src=".\addendum 1 angles_images/IMG0015_105261984.PNG" id="generatedImage64" path=".\addendum 1 angles_images/IMG0015_105261984.PNG" style="width: 33.75pt;height: 13.5pt;"></img></div><div id="mc-region-65" style="position: absolute; top: 630pt; left: 66pt; width: 303pt; "><a name="" /><img border="0" src=".\addendum 1 angles_images/IMG0017_105262000.PNG" id="generatedImage65" path=".\addendum 1 angles_images/IMG0017_105262000.PNG" style="width: 303pt;height: 213pt;"></img></div><div id="mc-region-66" style="position: absolute; top: 854.25pt; left: 12pt; width: 390.5pt; "><a name="" /><p class="Style1"><i>Example: The planet Neptune is observed to have an angular size </i><span style="font-size: 10pt; font-family: 'Symbol', sans-serif;"><i><span style="font-weight: normal;">a</span></i></span><span style="font-size: 10pt; font-family: 'Arial', sans-serif;"><i><span style="font-weight: normal;"> = 2.3 arcsec, and a distance of 29.5 AU. What is its diameter?</span></i></span></p></div><div id="mc-region-80" style="position: absolute; top: 896.25pt; left: 18pt; width: 164pt; "><a name="" /><p class="Style1">First let's define an astronimical unit:</p></div><div id="mc-region-79" style="position: absolute; top: 891pt; left: 204pt; width: 69.75pt; "><a name="" /><img border="0" src=".\addendum 1 angles_images/IMG0018_105262015.PNG" id="generatedImage79" path=".\addendum 1 angles_images/IMG0018_105262015.PNG" style="width: 69.75pt;height: 18.75pt;"></img></div><div id="mc-region-74" style="position: absolute; top: 932.25pt; left: 36pt; width: 60pt; "><a name="" /><img border="0" src=".\addendum 1 angles_images/IMG0019_105262031.PNG" id="generatedImage74" path=".\addendum 1 angles_images/IMG0019_105262031.PNG" style="width: 60pt;height: 13.5pt;"></img></div><div id="mc-region-75" style="position: absolute; top: 933pt; left: 120pt; width: 51.75pt; "><a name="" /><img border="0" src=".\addendum 1 angles_images/IMG0020_105262031.PNG" id="generatedImage75" path=".\addendum 1 angles_images/IMG0020_105262031.PNG" style="width: 51.75pt;height: 12.75pt;"></img></div><div id="mc-region-81" style="position: absolute; top: 932.25pt; left: 186pt; width: 35.25pt; "><a name="" /><img border="0" src=".\addendum 1 angles_images/IMG0021_105262046.PNG" id="generatedImage81" path=".\addendum 1 angles_images/IMG0021_105262046.PNG" style="width: 35.25pt;height: 13.5pt;"></img></div><div id="mc-region-82" style="position: absolute; top: 927pt; left: 246pt; width: 76.5pt; background-color: inherit; "><a name="" /><img border="0" src=".\addendum 1 angles_images/IMG0022_105262062.PNG" id="generatedImage82" path=".\addendum 1 angles_images/IMG0022_105262062.PNG" style="width: 76.5pt;height: 18.75pt;"></img></div><div></div></body></html>