document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=210922>Question 210922</A>This question is from textbook <A HREF=http://www.amazon.com/exec/obidos/ASIN/9780471746553/algebrahomeworkh>Analytic Trigonometry</A><BR>\n" );
document.write( ": <I><SPAN STYLE=\"word-break: break-all;\"> Question #33 asked: From the sunroof of Janet's apartment building, the angle of depression to the base of an office building is 51.4 degrees and the angle of elevation to the top of the office building is 43.2 degrees.  If the office building is 847 ft high, how far apart are the two buildings and how high is the apartment building?\r<BR>\n" );
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document.write( "My answer:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d=847%2F%28tan43.2%2Btan+51.4%29 ALIGN=MIDDLE  ALT=\"d=847%2F%28tan43.2%2Btan+51.4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "d=386 ft - the buildings are about 386 feet apart.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h=tan%2851.4%29386 ALIGN=MIDDLE  ALT=\"h=tan%2851.4%29386\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "h=484 ft - the apartment building is about 484 feet tall.<BR>\n" );
document.write( "***FIGURE FOR #33 LOOKS MORE LIKE THIS:***\r<BR>\n" );
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document.write( "______________/|===========|<BR>\n" );
document.write( "____________/__|===========|<BR>\n" );
document.write( "__________/____|===========|<BR>\n" );
document.write( "________/)43.2__|===========|<BR>\n" );
document.write( "=======|---------|===========|847 ft<BR>\n" );
document.write( "=======|\)51.4__|===========|<BR>\n" );
document.write( "=======|_\_____|===========|<BR>\n" );
document.write( "=======|__\____|===========|<BR>\n" );
document.write( "=======|___\___|===========|<BR>\n" );
document.write( "=======|____\__|===========|<BR>\n" );
document.write( "=======|_____\_|===========|<BR>\n" );
document.write( "***TWO BUILDINGS NEXT TO EACHOTHER WITH A TRIANGLE BETWEEN THEM***<BR>\n" );
document.write( "-------------My question is:------------- \r<BR>\n" );
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document.write( "34. Surveying:<BR>\n" );
document.write( "(A) Using the figure, show that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h=d%2F%28cot+a+%2B+cot+b%29 ALIGN=MIDDLE  ALT=\"h=d%2F%28cot+a+%2B+cot+b%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "_____/|\<BR>\n" );
document.write( "____/_|__\<BR>\n" );
document.write( "___/__|____\<BR>\n" );
document.write( "__/___|h_____\<BR>\n" );
document.write( "_/____|________\<BR>\n" );
document.write( "/_a)__|]_______(b_\<BR>\n" );
document.write( "|--------- d ---------|\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
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document.write( "I started with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h=d%2F%28cot+a+%2B+cot+b%29 ALIGN=MIDDLE  ALT=\"h=d%2F%28cot+a+%2B+cot+b%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and then multiplied <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28cot+a+%2B+cot+b%29 ALIGN=MIDDLE  ALT=\"%28cot+a+%2B+cot+b%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > to both sides and got <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h%28cot+a+%2B+cot+b%29=d ALIGN=MIDDLE  ALT=\"h%28cot+a+%2B+cot+b%29=d\"   DISABLED_style= \"max-width:90%; height:auto;\" >, I somehow got lost after that.<BR>\n" );
document.write( "Then there is the second part:<BR>\n" );
document.write( "(B) Use the results in part (A) to find the distance between the two buildings in Problem 33.<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #159472 by </B><A HREF=/tutors/aboutme.mpl?userid=jdmccarty><B>jdmccarty(1)</B></A><img src=\"/images/stars/iconNewId_16x16.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jdmccarty><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=159472\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Okay, here we go:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h=d%2F%28cota+%2B+cotb%29 ALIGN=MIDDLE  ALT=\"h=d%2F%28cota+%2B+cotb%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "COT is simply the opposite of TAN, correct? this means that: TAN=opp/adj; COT=adj/opp\r<BR>\n" );
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document.write( "So;<BR>\n" );
document.write( "(A) Using the figure, show that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h=d%2F%28cot+a+%2B+cot+b%29 ALIGN=MIDDLE  ALT=\"h=d%2F%28cot+a+%2B+cot+b%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "_____/|\<BR>\n" );
document.write( "____/_|__\<BR>\n" );
document.write( "___/__|____\<BR>\n" );
document.write( "__/___|h_____\<BR>\n" );
document.write( "_/____|________\<BR>\n" );
document.write( "/_a)__|]_______(b_\<BR>\n" );
document.write( "__<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%29 ALIGN=MIDDLE  ALT=\"%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >__|___<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28d-x%29 ALIGN=MIDDLE  ALT=\"%28d-x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "|--------- <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d ALIGN=MIDDLE  ALT=\"d\"   DISABLED_style= \"max-width:90%; height:auto;\" > ---------|<BR>\n" );
document.write( "Notice that we split <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d ALIGN=MIDDLE  ALT=\"d\"   DISABLED_style= \"max-width:90%; height:auto;\" > between the two triangles with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d-x ALIGN=MIDDLE  ALT=\"d-x\"   DISABLED_style= \"max-width:90%; height:auto;\" > That is where we will begin<BR>\n" );
document.write( "We start by recognizing that;<BR>\n" );
document.write( "cota=<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2Fh ALIGN=MIDDLE  ALT=\"x%2Fh\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "cotb=<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28d-x%29%2Fh ALIGN=MIDDLE  ALT=\"%28d-x%29%2Fh\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Plug in our new values to the original equation and:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h=d%2F%28%28x%2Fh%29%2B%28%28d-x%29%2Fh%29%29 ALIGN=MIDDLE  ALT=\"h=d%2F%28%28x%2Fh%29%2B%28%28d-x%29%2Fh%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Simplify, the x cancels and:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h=d%2F%28d%2Fh%29 ALIGN=MIDDLE  ALT=\"h=d%2F%28d%2Fh%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "In order to go any further, we need to realize that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d%2F%28d%2Fh%29 ALIGN=MIDDLE  ALT=\"d%2F%28d%2Fh%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d%2A%28d%2Fh%29%5E%28-1%29 ALIGN=MIDDLE  ALT=\"d%2A%28d%2Fh%29%5E%28-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d ALIGN=MIDDLE  ALT=\"d\"   DISABLED_style= \"max-width:90%; height:auto;\" > is really <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d%2F1 ALIGN=MIDDLE  ALT=\"d%2F1\"   DISABLED_style= \"max-width:90%; height:auto;\" > we end up with:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h=%28d%2F1%29%2A%28d%2Fh%29%5E%28-1%29 ALIGN=MIDDLE  ALT=\"h=%28d%2F1%29%2A%28d%2Fh%29%5E%28-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Negative exponents aren't as scary as they seem, we simply use the inverse, and get: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h=%28d%2F1%29%2A%28h%2Fd%29 ALIGN=MIDDLE  ALT=\"h=%28d%2F1%29%2A%28h%2Fd%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Simplify and find that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d ALIGN=MIDDLE  ALT=\"d\"   DISABLED_style= \"max-width:90%; height:auto;\" > cancels and our answer is: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h=h ALIGN=MIDDLE  ALT=\"h=h\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "____________________________NEXT PART____________________________\r<BR>\n" );
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document.write( "(B) You did this in the last problem, #33 only used TAN.  Remember that COT is the inverse of TAN and to find COT on the calculator, just divide 1 by TAN.<BR>\n" );
document.write( "But you are on the wrong side of the triangles!  Well, we know that a triangle totals 180 degrees and a right triangle uses a 90 degree; leaving us with two other angles to add up to 90 degrees and we have one angle for each triangle already!  What were they?<BR>\n" );
document.write( "43.2 degrees and 51.4 degrees<BR>\n" );
document.write( "Take one angle and subtract it from the remaining 90 degrees left over in our triangle, then repeat with the other:<BR>\n" );
document.write( "cota=<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=90-43.2=46.8 ALIGN=MIDDLE  ALT=\"90-43.2=46.8\"   DISABLED_style= \"max-width:90%; height:auto;\" > and cotb=<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=90-51.4=38.6 ALIGN=MIDDLE  ALT=\"90-51.4=38.6\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Now get out your calculators and start punching digits.<BR>\n" );
document.write( "1/tan(46.8)~.94 and 1/tan(38.6)~1.25 (both to the nearest penny)<BR>\n" );
document.write( "Back to our original problem, plug in the new numbers and we get:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h=847%2F2.19=386 ALIGN=MIDDLE  ALT=\"h=847%2F2.19=386\"   DISABLED_style= \"max-width:90%; height:auto;\" >  The buildings are still about 386 feet apart. </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );