document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1174508>Question 1174508</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Astronomers often measure large distances using astronomical units (AU) where 1 AU is the average distance from Earth to the Sun. In the image, d represents the distance from a star to the Sun. Using a technique called “stellar parallax,” astronomers determined \large \theta is 0.00001389 degrees.  How far away is the star from the Sun in astronomical units? Show your reasoning. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #850639 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1959)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=CPhill><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=CPhill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=850639\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Here's how to calculate the distance to the star using stellar parallax:\r<BR>\n" );
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document.write( "**Understanding Stellar Parallax**\r<BR>\n" );
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document.write( "* Stellar parallax is the apparent shift in the position of a star when viewed from different points in Earth's orbit around the Sun.<BR>\n" );
document.write( "* This shift is very small and is measured as an angle (θ).<BR>\n" );
document.write( "* The distance to the star is inversely proportional to the parallax angle.\r<BR>\n" );
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document.write( "**Calculations**\r<BR>\n" );
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document.write( "1.  **Relating Parallax to Distance:**<BR>\n" );
document.write( "    * When the parallax angle (θ) is very small, we can use the approximation tan(θ) ≈ θ (in radians).<BR>\n" );
document.write( "    * The relationship between distance (d), the baseline (1 AU), and the parallax angle (θ) is: tan(θ) = (1 AU) / d.<BR>\n" );
document.write( "    * Therefore, d = 1 / tan(θ) AU.<BR>\n" );
document.write( "2.  **Convert Degrees to Radians:**<BR>\n" );
document.write( "    * θ = 0.00001389 degrees<BR>\n" );
document.write( "    * To convert to radians, multiply by π / 180:<BR>\n" );
document.write( "        * θ (radians) = 0.00001389 * (π / 180) ≈ 2.422 × 10⁻⁷ radians.<BR>\n" );
document.write( "3.  **Calculate Distance:**<BR>\n" );
document.write( "    * d = 1 / tan(2.422 × 10⁻⁷)<BR>\n" );
document.write( "    * Since the angle is so small, tan(θ) is very close to θ.<BR>\n" );
document.write( "    * d = 1 / (2.422 × 10⁻⁷) AU<BR>\n" );
document.write( "    * d ≈ 4,128,819 AU.\r<BR>\n" );
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document.write( "**Therefore, the star is approximately 4,128,819 astronomical units away from the Sun.**<BR>\n" );
document.write( " </SPAN>\n" );
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