document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=112952>Question 112952</A>This question is from textbook <A HREF=http://www.amazon.com/exec/obidos/ASIN/0132321025/algebrahomeworkh>Mathematics for the Trades</A><BR>\n" );
document.write( ": <I><SPAN STYLE=\"word-break: break-all;\"> B-6 Machine Trades On a crankshaft, AB=4.2 in., and the connecting rod AC= 12.5 in. Calculate the size of angle A when the angle at C is 12 degrees. I know the answers are 26 degrees, and 130 degrees, but cannot come up with how they got this. HELP and a Very Big THANKS. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #82393 by </B><A HREF=/tutors/aboutme.mpl?userid=bucky><B>bucky(2189)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=bucky><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=82393\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> I'm not sure that I entirely understand the geometry of this problem, but let me give you <BR>\n" );
document.write( "some thoughts and maybe you can straighten me out or maybe get enough of a clue from the <BR>\n" );
document.write( "following explanation to figure it out.<BR>\n" );
document.write( ".<BR>\n" );
document.write( "I interpret your description as follows. Point B is on the main axis of the crankshaft<BR>\n" );
document.write( "and point A is on the crank. The throw radius of the crankshaft is 4.2 inches. One end<BR>\n" );
document.write( "of the connecting rod connects to the crank at point A. The other end fastens to the piston<BR>\n" );
document.write( "pin at point C and the length of the connecting rod is 12.5 inches.<BR>\n" );
document.write( ".<BR>\n" );
document.write( "Therefore, you have a triangle that is formed by points B, A, and C ... <BR>\n" );
document.write( ".<BR>\n" );
document.write( "The 12 degree angle at point C is formed by the connecting rod and the line that that runs<BR>\n" );
document.write( "from the piston pin (point C) to the main axis of the crankshaft (point B). <BR>\n" );
document.write( ".<BR>\n" );
document.write( "Let's look at this triangle BAC. Side BA is opposite angle C and is 4.2 inches long. Side AC<BR>\n" );
document.write( "is opposite of angle B and is 12.5 inches long. And side BC is opposite of angle A and is of <BR>\n" );
document.write( "unknown length. I am assuming that the crank is approximately in the 11 o'clock position relative to<BR>\n" );
document.write( "its spin about the main axis of the crankshaft ... that is to say that point A is approximately<BR>\n" );
document.write( "at the 11 o'clock position relative to the axis of its revolution at point B.<BR>\n" );
document.write( ".<BR>\n" );
document.write( "We are trying to find angle A ... the angle formed by the crank arm and the connecting rod.<BR>\n" );
document.write( ".<BR>\n" );
document.write( "We can apply the law of sines which says that in triangle ABC there is a relationship:<BR>\n" );
document.write( ".<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=BC%2F%28sinA%29+=+BA%2F%28sinC%29+=+AC%2F%28sinB%29 ALIGN=MIDDLE  ALT=\"BC%2F%28sinA%29+=+BA%2F%28sinC%29+=+AC%2F%28sinB%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( "where the following definitions apply ... <BR>\n" );
document.write( ".<BR>\n" );
document.write( "BC is the length of the side opposite angle A which leads to the relationship <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=BC%2F%28sinA%29 ALIGN=MIDDLE  ALT=\"BC%2F%28sinA%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( "BA is the length of the side opposite angle C which leads to the relationship <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=BA%2F%28sinC%29 ALIGN=MIDDLE  ALT=\"BA%2F%28sinC%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( "AC is the length of the side opposite angle B which leads to the relationship <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=AC%2F%28sinB%29 ALIGN=MIDDLE  ALT=\"AC%2F%28sinB%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( "From the description above and the problem we have BA = 4.2 inches and angle C is 12 degrees.<BR>\n" );
document.write( "Therefore, we can substitute for BA and for C to get the ratio <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4.2%2F%28sin%2812%29%29 ALIGN=MIDDLE  ALT=\"4.2%2F%28sin%2812%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( ".<BR>\n" );
document.write( "Next we know that the connecting rod (side AC) has a length 12.5 inches, but angle B<BR>\n" );
document.write( "which is opposite side AC is unknown. Therefore, the relationship here is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=AC%2F%28sinB%29 ALIGN=MIDDLE  ALT=\"AC%2F%28sinB%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <BR>\n" );
document.write( "by substituting for AC this relationship becomes <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=12.5%2F%28sinB%29 ALIGN=MIDDLE  ALT=\"12.5%2F%28sinB%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >. The law of sines says<BR>\n" );
document.write( "that this relationship has to be equal to the previous ratio we established. That is:<BR>\n" );
document.write( ".<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4.2%2F%28sin%2812%29%29=12.5%2F%28sinB%29 ALIGN=MIDDLE  ALT=\"4.2%2F%28sin%2812%29%29=12.5%2F%28sinB%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( "As usual, you can solve this proportion by cross multiplying ... multiply 4.2 times sinB and <BR>\n" );
document.write( "set that product equal to the product of 12.5 times sin(12). That results in the equation:<BR>\n" );
document.write( ".<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4.2%2A%28sinB%29=+12.5%2A%28sin%2812%29%29 ALIGN=MIDDLE  ALT=\"4.2%2A%28sinB%29=+12.5%2A%28sin%2812%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( "A calculator will tell you that sin(12) = 0.20791169 and when you multiply that by 12.5<BR>\n" );
document.write( "you get 2.598896135. This reduces the equation to:<BR>\n" );
document.write( ".\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4.2%2A%28sinB%29=+2.598896135 ALIGN=MIDDLE  ALT=\"4.2%2A%28sinB%29=+2.598896135\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( "Divide both sides of this equation by 4.2 and you end up with:<BR>\n" );
document.write( ".<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28sinB%29+=+2.598896135%2F4.2+=+0.618784794 ALIGN=MIDDLE  ALT=\"%28sinB%29+=+2.598896135%2F4.2+=+0.618784794\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( "We can now solve for angle B by taking arcsine 0.618784794 to get that angle B is 38.22744791 degrees.<BR>\n" );
document.write( ".<BR>\n" );
document.write( "Now we have that angle C is 12 degrees and angle B is 38.22744791 degrees. The missing<BR>\n" );
document.write( "angle is angle A. But we can find angle A by recognizing that the sum of the three angles<BR>\n" );
document.write( "in the triangle must be 180 degrees. Therefore:<BR>\n" );
document.write( ".<BR>\n" );
document.write( "180 = 12 + 38.22744791 + angle A<BR>\n" );
document.write( ".<BR>\n" );
document.write( "Subtracting 12 + 38.22744791 from both sides results in:<BR>\n" );
document.write( ".<BR>\n" );
document.write( "180 - 12 - 38.22744791 = angle A<BR>\n" );
document.write( ".<BR>\n" );
document.write( "and this reduces to:<BR>\n" );
document.write( ".<BR>\n" );
document.write( "129.7725521 degrees = angle A<BR>\n" );
document.write( ".<BR>\n" );
document.write( "This is pretty close to the 130 degree answer that you gave. <BR>\n" );
document.write( ".<BR>\n" );
document.write( "Next I would suspect that the crank (point A) is at the 5 o'clock position relative to<BR>\n" );
document.write( "point B.<BR>\n" );
document.write( ".<BR>\n" );
document.write( "I haven't figured the geometry for this, but note that 38.22744791 degrees we got previously<BR>\n" );
document.write( "minus the 12 degrees would give an answer of 26.22744791 degrees. I wonder ...?<BR>\n" );
document.write( ".<BR>\n" );
document.write( "Hope this turns you on to something. Sorry about the complex discussion above, but it's <BR>\n" );
document.write( "hard to do a geometry discussion in this forum where the drawing support is limited.<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );