document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=627375>Question 627375</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find the exact value of tan[cos^-1(1/4)-csc^-1(-3/2)] </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #394916 by </B><A HREF=/tutors/aboutme.mpl?userid=jsmallt9><B>jsmallt9(3758)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jsmallt9><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=394916\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> First of all, we should recognize that the expression \"exact value\" means \"put away your calculator and solve this by hand\". (If \"exact value\" was not used then we could just use our calculators and get a quick decimal approximation.)<br><BR>\n" );
document.write( "An exponent of -1 on the name of a function indicates an inverse function. Inverse trig functions take a ratio as input and return the angle within a specific range which would have that ratio. For example:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%5E%28-1%29%281%2F4%29 ALIGN=MIDDLE  ALT=\"cos%5E%28-1%29%281%2F4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > would return the one and only angle between zero and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=pi ALIGN=MIDDLE  ALT=\"pi\"   DISABLED_style= \"max-width:90%; height:auto;\" > which has a cos ratio of 1/4. And <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=csc%5E%28-1%29%28-3%2F2%29 ALIGN=MIDDLE  ALT=\"csc%5E%28-1%29%28-3%2F2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > would return the one and only angle between <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-pi%2F2 ALIGN=MIDDLE  ALT=\"-pi%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=pi%2F2 ALIGN=MIDDLE  ALT=\"pi%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > whose csc ratio is -3/2. (Note how the two inverse functions have different ranges. Some have ranges between 0 and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=pi ALIGN=MIDDLE  ALT=\"pi\"   DISABLED_style= \"max-width:90%; height:auto;\" > and the other have a range between <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-pi%2F2 ALIGN=MIDDLE  ALT=\"-pi%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=pi%2F2 ALIGN=MIDDLE  ALT=\"pi%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >. When you work with these inverse Trig functions you have to learn which range goes with which inverse function.)<br><BR>\n" );
document.write( "In summary, the inverse Trig functions return angles as output. So your expression:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=tan%28cos%5E%28-1%29%281%2F4%29-csc%5E%28-1%29%28-3%2F2%29%29 ALIGN=MIDDLE  ALT=\"tan%28cos%5E%28-1%29%281%2F4%29-csc%5E%28-1%29%28-3%2F2%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "represents the tan of the difference/subtraction of two angles. We have a formula for this:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=tan%28A-B%29+=+%28tan%28A%29-tan%28B%29%29%2F%281%2Btan%28A%29%2Atan%28B%29%29 ALIGN=MIDDLE  ALT=\"tan%28A-B%29+=+%28tan%28A%29-tan%28B%29%29%2F%281%2Btan%28A%29%2Atan%28B%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "This formula tells how we can find that tan of a difference of angles using the tan's of the individual angles. So to find the exact value of your expression we will need the tan's of each the two given angles.<br><BR>\n" );
document.write( "Let's start with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=tan%28cos%5E%28-1%29%281%2F4%29%29 ALIGN=MIDDLE  ALT=\"tan%28cos%5E%28-1%29%281%2F4%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.  The range of inverse cos is 0 to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=pi ALIGN=MIDDLE  ALT=\"pi\"   DISABLED_style= \"max-width:90%; height:auto;\" > and within this range only angles between 0 and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=pi%2F2 ALIGN=MIDDLE  ALT=\"pi%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > are positive. So we know that the angle in question is between 0 and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=pi%2F2 ALIGN=MIDDLE  ALT=\"pi%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > (i.e. it is a 1st quadrant angle). The tan of a first quadrant angle will be positive.  But which positive number? For this we can use a reference (right) triangle. Pick one of the acute angles to be the angle we are working with. We know from <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%5E%28-1%29%281%2F4%29 ALIGN=MIDDLE  ALT=\"cos%5E%28-1%29%281%2F4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > that the cos ratio is 1/4. Since cos is adjacent/hypotenuse we will use 1 for the adjacent side and 4 for the hypotenuse. To find the tan ratio we need opposite/adjacent. We have the adjacent side. We need the opposite side. Use the Pythagorean Theorem to find the opposite side. You should get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%2815%29 ALIGN=MIDDLE  ALT=\"sqrt%2815%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >. So<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=tan%28cos%5E%28-1%29%281%2F4%29%29+=+sqrt%2815%29%2F1+=+sqrt%2815%29 ALIGN=MIDDLE  ALT=\"tan%28cos%5E%28-1%29%281%2F4%29%29+=+sqrt%2815%29%2F1+=+sqrt%2815%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "We can use a similar process for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=tan%28csc%5E%28-1%29%28-3%2F2%29%29 ALIGN=MIDDLE  ALT=\"tan%28csc%5E%28-1%29%28-3%2F2%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >. The range for inverse csc is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-pi%2F2 ALIGN=MIDDLE  ALT=\"-pi%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=pi%2F2 ALIGN=MIDDLE  ALT=\"pi%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Our angle has a negative csc, -3/2. Within the range, only angles between <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=pi%2F2 ALIGN=MIDDLE  ALT=\"pi%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > and 0 (i.e. a 4th quadrant angle) will have a negative csc. Fourth quadrant angles will have negative tan's. But which negative number? Again we can use a reference triangle. (Note: We already know we are supposed to get a negative result. So we can ignore signs in our triangle. csc is hypotenuse/opposite. So our hypotenuse is 3 and our opposite side is 2. tan is still opposite/adjacent so we will use the Pythagorean Theorem again to find the missing side. We should get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%285%29 ALIGN=MIDDLE  ALT=\"sqrt%285%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > for the adjacent side. So<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=tan%28csc%5E%28-1%29%28-3%2F2%29%29+=+-2%2Fsqrt%285%29 ALIGN=MIDDLE  ALT=\"tan%28csc%5E%28-1%29%28-3%2F2%29%29+=+-2%2Fsqrt%285%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "(Note: if this was the final answer we should rationalize the denominator. But this is not our final answer and I prefer to postpone issues like rationalizing until the very end. (Sometimes these issues work themselves out on their own.))<br><BR>\n" );
document.write( "Now that we have values for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=tan%28cos%5E%28-1%29%281%2F4%29%29 ALIGN=MIDDLE  ALT=\"tan%28cos%5E%28-1%29%281%2F4%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=tan%28csc%5E%28-1%29%28-3%2F2%29%29 ALIGN=MIDDLE  ALT=\"tan%28csc%5E%28-1%29%28-3%2F2%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > we are ready to use our tan(A-B) formula on:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=tan%28cos%5E%28-1%29%281%2F4%29-+csc%5E%28-1%29%28-3%2F2%29%29 ALIGN=MIDDLE  ALT=\"tan%28cos%5E%28-1%29%281%2F4%29-+csc%5E%28-1%29%28-3%2F2%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "From the formula we get:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28tan%28cos%5E%28-1%29%281%2F4%29%29+-+tan%28csc%5E%28-1%29%28-3%2F2%29%29%29%2F%281%2Btan%28cos%5E%28-1%29%281%2F4%29%29%2Atan%28csc%5E%28-1%29%28-3%2F2%29%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Substituting in our values we get:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28sqrt%2815%29+-+%28-2%2Fsqrt%285%29%29%29%2F%281%2Bsqrt%2815%29%2A%28-2%2Fsqrt%285%29%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Simplifying we get:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28sqrt%2815%29+%2B+2%2Fsqrt%285%29%29%2F%281-%282%2Asqrt%2815%29%29%2Fsqrt%285%29%29 ALIGN=MIDDLE  ALT=\"%28sqrt%2815%29+%2B+2%2Fsqrt%285%29%29%2F%281-%282%2Asqrt%2815%29%29%2Fsqrt%285%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28sqrt%2815%29+%2B+2%2Fsqrt%285%29%29%2F%281-2%2Asqrt%283%29%29 ALIGN=MIDDLE  ALT=\"%28sqrt%2815%29+%2B+2%2Fsqrt%285%29%29%2F%281-2%2Asqrt%283%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "We can eliminate the fraction within a fraction by multiplying the numerator and denominator by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%285%29 ALIGN=MIDDLE  ALT=\"sqrt%285%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28%28sqrt%2815%29+%2B+2%2Fsqrt%285%29%29%2F%281-2%2Asqrt%283%29%29%29%28sqrt%285%29%2Fsqrt%285%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28sqrt%2875%29+%2B+2%29%2F%28sqrt%285%29-2%2Asqrt%2815%29%29 ALIGN=MIDDLE  ALT=\"%28sqrt%2875%29+%2B+2%29%2F%28sqrt%285%29-2%2Asqrt%2815%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Now we can deal with rationalizing. To rationalize a two-term denominator we multiply the top and bottom by the denominators conjugate: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28sqrt%285%29%2B2%2Asqrt%2815%29%29 ALIGN=MIDDLE  ALT=\"%28sqrt%285%29%2B2%2Asqrt%2815%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28%28sqrt%2875%29+%2B+2%29%2F%28sqrt%285%29-2%2Asqrt%2815%29%29%29%28%28sqrt%285%29%2B2%2Asqrt%2815%29%29%2F%28sqrt%285%29%2B2%2Asqrt%2815%29%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "On top we must use FOIL to multiply. On the bottom we can use the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28a%2Bb%29%28a-b%29+=+a%5E2-b%5E2 ALIGN=MIDDLE  ALT=\"%28a%2Bb%29%28a-b%29+=+a%5E2-b%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > pattern to multiply more quickly than FOIL:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28sqrt%2875%29%2Asqrt%285%29%2Bsqrt%2875%29%2A2sqrt%2815%29+%2B+2%2Asqrt%285%29+%2B+2%2A2sqrt%2815%29%29%2F%28%28sqrt%285%29%29%5E2+-+%282sqrt%2815%29%29%5E2%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Simplifying:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28sqrt%28375%29%2B2sqrt%281175%29+%2B+2sqrt%285%29+%2B+4sqrt%2815%29%29%2F%285+-+%284%2A15%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28sqrt%28375%29%2B2sqrt%281175%29+%2B+2sqrt%285%29+%2B+4sqrt%2815%29%29%2F%285+-+60%29 ALIGN=MIDDLE  ALT=\"%28sqrt%28375%29%2B2sqrt%281175%29+%2B+2sqrt%285%29+%2B+4sqrt%2815%29%29%2F%285+-+60%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28sqrt%28375%29%2B2sqrt%281175%29+%2B+2sqrt%285%29+%2B+4sqrt%2815%29%29%2F%28-55%29 ALIGN=MIDDLE  ALT=\"%28sqrt%28375%29%2B2sqrt%281175%29+%2B+2sqrt%285%29+%2B+4sqrt%2815%29%29%2F%28-55%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Last of all we simplify, if possible, the square roots that remain in the numerator. The first two have perfect square factors:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28sqrt%2825%2A15%29%2B2sqrt%2825%2A9%2A5%29+%2B+2sqrt%285%29+%2B+4sqrt%2815%29%29%2F%28-55%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28sqrt%2825%29%2Asqrt%2815%29%2B2sqrt%2825%29%2Asqrt%289%29%2Asqrt%285%29+%2B+2sqrt%285%29+%2B+4sqrt%2815%29%29%2F%28-55%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%285sqrt%2815%29%2B2%2A5%2A3%2Asqrt%285%29+%2B+2sqrt%285%29+%2B+4sqrt%2815%29%29%2F%28-55%29 ALIGN=MIDDLE  ALT=\"%285sqrt%2815%29%2B2%2A5%2A3%2Asqrt%285%29+%2B+2sqrt%285%29+%2B+4sqrt%2815%29%29%2F%28-55%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%285sqrt%2815%29%2B30sqrt%285%29+%2B+2sqrt%285%29+%2B+4sqrt%2815%29%29%2F%28-55%29 ALIGN=MIDDLE  ALT=\"%285sqrt%2815%29%2B30sqrt%285%29+%2B+2sqrt%285%29+%2B+4sqrt%2815%29%29%2F%28-55%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "And finally, adding like terms:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%289sqrt%2815%29%2B32sqrt%285%29%29%2F%28-55%29 ALIGN=MIDDLE  ALT=\"%289sqrt%2815%29%2B32sqrt%285%29%29%2F%28-55%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "or<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-%289sqrt%2815%29%2B32sqrt%285%29%29%2F55 ALIGN=MIDDLE  ALT=\"-%289sqrt%2815%29%2B32sqrt%285%29%29%2F55\"   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );