Editing Appendix/Ramblings/TrigFunctions

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=Analytic Expressions for Selected Trigonometric Functions=
In what follows we generally will provide expressions that result from evaluating <math>\sin\theta</math>, with the understanding that <math>\cos\theta</math> and <math>\tan\theta</math> then also can be evaluated straightforwardly via the familiar relations,

<table border="0" align="center" cellpadding="5">
<tr>
  <td align="right"><math>\cos\theta</math></td>
  <td align="right"><math>=</math></td>
  <td align="right"><math>\pm (1 - \sin^2\theta)^{1 / 2} \, ,</math></td>
<td align="center">&nbsp; &nbsp; &nbsp; and, &nbsp; &nbsp; &nbsp;
  <td align="right"><math>\tan\theta</math></td>
  <td align="right"><math>=</math></td>
  <td align="right"><math>\frac{\sin\theta}{\pm (1 - \sin^2\theta )^{1 / 2}} \, .</math></td>
</tr>
</table>

==Widely Used Evaluations==

<table border="1" align="center" cellpadding="8">
<tr>
  <td align="center" colspan="2"><math>0 \le</math> Angle <math>(\theta) \le \frac{\pi}{2}</math></td>
  <td align="center" rowspan="2"><math>\sin\theta</math></td>
</tr>
<tr>
  <td align="center">Radians</td>
  <td align="center">Degrees</td>
</tr>

<tr>
  <td align="center"><math>0</math></td>
  <td align="center"><math>0^\circ</math></td>
  <td align="center"><math>0</math></td>
</tr>

<tr>
  <td align="center"><math>\frac{\pi}{6}</math></td>
  <td align="center"><math>30^\circ</math></td>
  <td align="center"><math>\frac{1}{2}</math></td>
</tr>

<tr>
  <td align="center"><math>\frac{\pi}{4}</math></td>
  <td align="center"><math>45^\circ</math></td>
  <td align="center"><math>\frac{\sqrt{2}}{2}</math></td>
</tr>

<tr>
  <td align="center"><math>\frac{\pi}{3}</math></td>
  <td align="center"><math>60^\circ</math></td>
  <td align="center"><math>\frac{\sqrt{3}}{2}</math></td>
</tr>

<tr>
  <td align="center"><math>\frac{\pi}{2}</math></td>
  <td align="center"><math>90^\circ</math></td>
  <td align="center"><math>1</math></td>
</tr>
</table>

==Integer-Degree Angles==

A PDF-formatted document generated by James T. Parent<sup>&dagger;</sup> lists [https://www.intmath.com/blog/wp-content/images/2011/06/exact-values-sin-degrees.pdf exact values for the sine of all integer-degree angles] between zero and ninety degrees, inclusive.  An explanation of how the expressions in this document were derived, can be found on the [https://www.intmath.com/blog/mathematics/how-do-you-find-exact-values-for-the-sine-of-all-angles-6212 ''SquareCirclez'' IntMath blog].


<table border="1" align="center" cellpadding="8" width="75%">
<tr>
  <td align="center" colspan="3">Examples Extracted from the [https://www.intmath.com/blog/wp-content/images/2011/06/exact-values-sin-degrees.pdf James T. Parent Document]<sup>&dagger;</sup></td>
</tr>

<tr>
  <td align="center" colspan="2"><math>0 \le</math> Angle <math>(\theta) \le \frac{\pi}{2}</math></td>
  <td align="center" rowspan="2"><math>\sin\theta</math></td>
</tr>
<tr>
  <td align="center">Radians</td>
  <td align="center">Degrees</td>
</tr>

<tr>
  <td align="center"><math>\frac{\pi}{60}</math></td>
  <td align="center"><math>3^\circ</math></td>
  <td align="center">
<math>
\frac{\sqrt{6}}{48} \biggl(\sqrt{5} - 1\biggr)\biggl(3 + \sqrt{3}\biggr)
- \frac{\sqrt{3}}{24}\biggl(3 - \sqrt{3}\biggr)\biggl(5 + \sqrt{5}\biggr)^{1 / 2}
</math>
  </td>
</tr>

<tr>
  <td align="center"><math>\frac{\pi}{12}</math></td>
  <td align="center"><math>15^\circ</math></td>
  <td align="center">
<math>
\frac{\sqrt{2}}{4}\biggl( \sqrt{3} - 1\biggr)
=
\frac{1}{2}\biggl(2 - \sqrt{3}\biggr)^{1 / 2}
</math>
  </td>
</tr>

<tr>
  <td align="center"><math>\frac{\pi}{5}</math></td>
  <td align="center"><math>36^\circ</math></td>
  <td align="center">
<math>
\frac{\sqrt{2}}{8} \biggl( \sqrt{5} - 1\biggr) \biggl( 5 + \sqrt{5} \biggr)^{ 1 /2} 
=
\frac{1}{4}\biggl(10 - 2\sqrt{5}\biggr)^{1 / 2}
</math>
  </td>
</tr>

<tr>
  <td align="center"><math>\frac{5\pi}{12}</math></td>
  <td align="center"><math>75^\circ</math></td>
  <td align="center"><math>\frac{\sqrt{3} + 1}{2\sqrt{2}}</math></td>
</tr>

<tr>
<td align="left" colspan="3">
<sup>&dagger;</sup>James T. Parent has previously taught mathematics at Schenectady County Community College, Schenectady, New York, and at Great Bay Community College, Portsmouth, New Hampshire.
</td>
</tr>
</table>


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