http://wiki.sommarmatte.se/wikis/2009/bridgecourse1-ImperialCollege/index.php?action=history&feed=atom&title=Solution_4.3%3A4d 2026-04-08T05:48:30Z Revision history for this page on the wiki MediaWiki 1.11.1 http://wiki.sommarmatte.se/wikis/2009/bridgecourse1-ImperialCollege/index.php?title=Solution_4.3:4d&diff=5849&oldid=prev 2008-10-09T14:16:19Z

<p></p> <table style="background-color: white; color:black;"> <col class='diff-marker' /> <col class='diff-content' /> <col class='diff-marker' /> <col class='diff-content' /> <tr> <td colspan='2' style="background-color: white; color:black;">←Older revision</td> <td colspan='2' style="background-color: white; color:black;">Revision as of 14:16, 9 October 2008</td> </tr> <tr><td colspan="2" class="diff-lineno">Line 1:</td> <td colspan="2" class="diff-lineno">Line 1:</td></tr> <tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>With the formula for double angles and the Pythagorean identity</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>With the formula for double angles and the Pythagorean identity &lt;math&gt;\cos^2<ins style="color: red; font-weight: bold; text-decoration: none;">\!</ins>v + \sin^2<ins style="color: red; font-weight: bold; text-decoration: none;">\!</ins>v = 1&lt;/math&gt;, we can express &lt;math&gt;\cos <ins style="color: red; font-weight: bold; text-decoration: none;">2v</ins>&lt;/math&gt; in terms of &lt;math&gt;\cos v&lt;/math&gt;,</div></td></tr> <tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>&lt;math&gt;\cos ^<del style="color: red; font-weight: bold; text-decoration: none;">{</del>2<del style="color: red; font-weight: bold; text-decoration: none;">}</del>v+\sin ^<del style="color: red; font-weight: bold; text-decoration: none;">{</del>2<del style="color: red; font-weight: bold; text-decoration: none;">}</del>v=1&lt;/math&gt;, we can express </div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div></div></td></tr> <tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>&lt;math&gt;\<del style="color: red; font-weight: bold; text-decoration: none;">text{</del>cos <del style="color: red; font-weight: bold; text-decoration: none;">2}v\text{ }</del>&lt;/math&gt;</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div></div></td></tr> <tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>in terms of </div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div></div></td></tr> <tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>&lt;math&gt;\<del style="color: red; font-weight: bold; text-decoration: none;">text{</del>cos <del style="color: red; font-weight: bold; text-decoration: none;">}</del>v&lt;/math&gt;,</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div></div></td></tr> <tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr> <tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>&#160;</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">{{Displayed math||</ins>&lt;math&gt;\begin{align}</div></td></tr> <tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>&lt;math&gt;\begin{align}</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>\cos <ins style="color: red; font-weight: bold; text-decoration: none;">2v &amp;</ins>= \cos^2<ins style="color: red; font-weight: bold; text-decoration: none;">\!</ins>v - \sin^2<ins style="color: red; font-weight: bold; text-decoration: none;">\!</ins>v<ins style="color: red; font-weight: bold; text-decoration: none;">\\[5pt]</ins></div></td></tr> <tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del style="color: red; font-weight: bold; text-decoration: none;">&amp; </del>\<del style="color: red; font-weight: bold; text-decoration: none;">text{</del>cos <del style="color: red; font-weight: bold; text-decoration: none;">2}v</del>=\cos ^<del style="color: red; font-weight: bold; text-decoration: none;">{</del>2<del style="color: red; font-weight: bold; text-decoration: none;">}</del>v-\sin ^<del style="color: red; font-weight: bold; text-decoration: none;">{</del>2<del style="color: red; font-weight: bold; text-decoration: none;">}</del>v=\cos ^<del style="color: red; font-weight: bold; text-decoration: none;">{</del>2<del style="color: red; font-weight: bold; text-decoration: none;">}</del>v-<del style="color: red; font-weight: bold; text-decoration: none;">\left</del>( 1-\cos ^<del style="color: red; font-weight: bold; text-decoration: none;">{</del>2<del style="color: red; font-weight: bold; text-decoration: none;">}</del>v <del style="color: red; font-weight: bold; text-decoration: none;">\right</del>) \\ </div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">&amp;</ins>= \cos^2<ins style="color: red; font-weight: bold; text-decoration: none;">\!</ins>v - (1-\cos^2<ins style="color: red; font-weight: bold; text-decoration: none;">\!</ins>v)\\<ins style="color: red; font-weight: bold; text-decoration: none;">[5pt] </ins></div></td></tr> <tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>&amp; =2\cos ^<del style="color: red; font-weight: bold; text-decoration: none;">{</del>2<del style="color: red; font-weight: bold; text-decoration: none;">}</del>v-1=2b^<del style="color: red; font-weight: bold; text-decoration: none;">{</del>2<del style="color: red; font-weight: bold; text-decoration: none;">}</del>-1 \\ </div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>&amp;= 2\cos^2<ins style="color: red; font-weight: bold; text-decoration: none;">\!</ins>v-1<ins style="color: red; font-weight: bold; text-decoration: none;">\\[5pt]</ins></div></td></tr> <tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>\end{align}&lt;/math&gt;</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">&amp;</ins>= 2b^2-1\<ins style="color: red; font-weight: bold; text-decoration: none;">,</ins>\<ins style="color: red; font-weight: bold; text-decoration: none;">textrm{.} </ins></div></td></tr> <tr><td colspan="2">&nbsp;</td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>\end{align}&lt;/math&gt;<ins style="color: red; font-weight: bold; text-decoration: none;">}}</ins></div></td></tr> </table>

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<p></p> <table style="background-color: white; color:black;"> <col class='diff-marker' /> <col class='diff-content' /> <col class='diff-marker' /> <col class='diff-content' /> <tr> <td colspan='2' style="background-color: white; color:black;">←Older revision</td> <td colspan='2' style="background-color: white; color:black;">Revision as of 11:48, 29 September 2008</td> </tr> <tr><td colspan="2" class="diff-lineno">Line 1:</td> <td colspan="2" class="diff-lineno">Line 1:</td></tr> <tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>{{<del style="color: red; font-weight: bold; text-decoration: none;">NAVCONTENT_START</del>}}</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">With the formula for double angles and the Pythagorean identity</ins></div></td></tr> <tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>&lt;<del style="color: red; font-weight: bold; text-decoration: none;">center</del>&gt; <del style="color: red; font-weight: bold; text-decoration: none;">[[Image:4_3_4d.gif]] </del>&lt;/<del style="color: red; font-weight: bold; text-decoration: none;">center</del>&gt;</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">&lt;math&gt;\cos ^</ins>{<ins style="color: red; font-weight: bold; text-decoration: none;">2}v+\sin ^</ins>{<ins style="color: red; font-weight: bold; text-decoration: none;">2</ins>}<ins style="color: red; font-weight: bold; text-decoration: none;">v=1&lt;/math&gt;, we can express </ins></div></td></tr> <tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>{{<del style="color: red; font-weight: bold; text-decoration: none;">NAVCONTENT_STOP</del>}}</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">&lt;math&gt;\text{cos 2</ins>}<ins style="color: red; font-weight: bold; text-decoration: none;">v\text{ }&lt;/math&gt;</ins></div></td></tr> <tr><td colspan="2">&nbsp;</td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">in terms of </ins></div></td></tr> <tr><td colspan="2">&nbsp;</td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>&lt;<ins style="color: red; font-weight: bold; text-decoration: none;">math</ins>&gt;<ins style="color: red; font-weight: bold; text-decoration: none;">\text{cos }v</ins>&lt;/<ins style="color: red; font-weight: bold; text-decoration: none;">math</ins>&gt;<ins style="color: red; font-weight: bold; text-decoration: none;">,</ins></div></td></tr> <tr><td colspan="2">&nbsp;</td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>&#160;</div></td></tr> <tr><td colspan="2">&nbsp;</td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>&#160;</div></td></tr> <tr><td colspan="2">&nbsp;</td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">&lt;math&gt;\begin</ins>{<ins style="color: red; font-weight: bold; text-decoration: none;">align}</ins></div></td></tr> <tr><td colspan="2">&nbsp;</td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">&amp; \text</ins>{<ins style="color: red; font-weight: bold; text-decoration: none;">cos 2</ins>}<ins style="color: red; font-weight: bold; text-decoration: none;">v=\cos ^{2</ins>}<ins style="color: red; font-weight: bold; text-decoration: none;">v-\sin ^{2}v=\cos ^{2}v-\left( 1-\cos ^{2}v \right) \\ </ins></div></td></tr> <tr><td colspan="2">&nbsp;</td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">&amp; =2\cos ^{2}v-1=2b^{2}-1 \\ </ins></div></td></tr> <tr><td colspan="2">&nbsp;</td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">\end{align}&lt;/math&gt;</ins></div></td></tr> </table>

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<p><a href="/wikis/2009/bridgecourse1-ImperialCollege/index.php?title=L%C3%B6sning_4.3:4d&amp;action=edit" class="new" title="Lösning 4.3:4d">Lösning 4.3:4d</a> moved to <a href="/wikis/2009/bridgecourse1-ImperialCollege/index.php/Solution_4.3:4d" title="Solution 4.3:4d">Solution 4.3:4d</a>: Robot: moved page</p> <table style="background-color: white; color:black;"> <col class='diff-marker' /> <col class='diff-content' /> <col class='diff-marker' /> <col class='diff-content' /> <tr> <td colspan='2' style="background-color: white; color:black;">←Older revision</td> <td colspan='2' style="background-color: white; color:black;">Revision as of 09:55, 9 September 2008</td> </tr> </table>

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