document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1186939>Question 1186939</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Suppose that O=(0,0), A=(3.4,0), and B=(2.81,1.92). The arc between points B and C is 12.92 units long. \r<BR>\n" );
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document.write( "A. What is the value of (theta)1, the radian measure of Angle AOB?<BR>\n" );
document.write( "B. What is the value of (theta)2, the radian measure of Angle BOC?<BR>\n" );
document.write( "C. What are the x- and y-coordinates of point C? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #849325 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=849325\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Here's the solution, broken down step by step:\r<BR>\n" );
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document.write( "**A. θ₁ (Angle AOB):**\r<BR>\n" );
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document.write( "1.  **Find the lengths of OA and OB:**<BR>\n" );
document.write( "    *   OA = √((3.4 - 0)² + (0 - 0)²) = 3.4<BR>\n" );
document.write( "    *   OB = √((2.81 - 0)² + (1.92 - 0)²) = √(7.8961 + 3.6864) = √11.5825 ≈ 3.403\r<BR>\n" );
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document.write( "2.  **Use the dot product formula:**<BR>\n" );
document.write( "    *   OA • OB = |OA| * |OB| * cos(θ₁)<BR>\n" );
document.write( "    *   (3.4 * 2.81) + (0 * 1.92) = 3.4 * 3.403 * cos(θ₁)<BR>\n" );
document.write( "    *   9.554 = 11.5702 * cos(θ₁)<BR>\n" );
document.write( "    *   cos(θ₁) = 9.554 / 11.5702 ≈ 0.8257\r<BR>\n" );
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document.write( "3.  **Solve for θ₁:**<BR>\n" );
document.write( "    *   θ₁ = arccos(0.8257) ≈ 0.5994 radians\r<BR>\n" );
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document.write( "**B. θ₂ (Angle BOC):**\r<BR>\n" );
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document.write( "1.  **Find the radius of the circle:**  Since OA and OB are radii, and we are working with a circular arc BC, OA is the radius. r = 3.4\r<BR>\n" );
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document.write( "2.  **Use the arc length formula:**<BR>\n" );
document.write( "    *   Arc length (s) = r * θ₂<BR>\n" );
document.write( "    *   12.92 = 3.4 * θ₂<BR>\n" );
document.write( "    *   θ₂ = 12.92 / 3.4 ≈ 3.8 radians\r<BR>\n" );
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document.write( "**C. Coordinates of Point C:**\r<BR>\n" );
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document.write( "1.  **Find the angle from the x-axis to OC:** This is θ₁ + θ₂ ≈ 0.5994 + 3.8 = 4.3994 radians.\r<BR>\n" );
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document.write( "2.  **Use the coordinates formula:**<BR>\n" );
document.write( "    *   x = r * cos(θ₁ + θ₂) = 3.4 * cos(4.3994) ≈ -1.0468<BR>\n" );
document.write( "    *   y = r * sin(θ₁ + θ₂) = 3.4 * sin(4.3994) ≈ -3.2348\r<BR>\n" );
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document.write( "**Therefore:**\r<BR>\n" );
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document.write( "*   θ₁ (Angle AOB) ≈ 0.5994 radians<BR>\n" );
document.write( "*   θ₂ (Angle BOC) ≈ 3.8 radians<BR>\n" );
document.write( "*   Coordinates of point C ≈ (-1.0468, -3.2348)<BR>\n" );
document.write( " </SPAN>\n" );
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