document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1099758>Question 1099758</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find the exact value of the expression or state that it does not exist:<BR>\n" );
document.write( "csc^-1 (-2)\r<BR>\n" );
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document.write( "a.The value of csc^-1 x is an angle in what interval? \r<BR>\n" );
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document.write( "b. Let csc^-1(-2) = &#952; such that sin^-1 = ___ and sin&#952;= ____\r<BR>\n" );
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document.write( "c.  Is the value of sin&#952; positive, negative, or equal to 0? \r<BR>\n" );
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document.write( "d. does the angle &#952;: lie in the interval 0<&#952;<&#960;/2 , is equal to &#960;/2, or lies in the interval -&#960;/2<&#952;<0\r<BR>\n" );
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document.write( "e. Where does the terminal side of angle &#952; lie?  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #714259 by </B><A HREF=/tutors/aboutme.mpl?userid=KMST><B>KMST(5328)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=KMST><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=KMST><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=714259\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=csc%5E%28-1%29%28x%29 ALIGN=MIDDLE  ALT=\"y=csc%5E%28-1%29%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > (alternatively represented as <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=arccsc%28x%29 ALIGN=MIDDLE  ALT=\"y=arccsc%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "is the inverse function of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=csc%28x%29=1%2Fsin%28x%29 ALIGN=MIDDLE  ALT=\"y=csc%28x%29=1%2Fsin%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "The graph of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=csc%28x%29=1%2Fsin%28x%29 ALIGN=MIDDLE  ALT=\"y=csc%28x%29=1%2Fsin%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > looks like this:<BR>\n" );
document.write( " <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28600%2C200%2C-4%2C6%2C-5%2C5%2C%0D%0Ablue%28line%281.571%2C-5%2C1.571%2C5%29%29%2Clocate%281.625%2C-2%2Cblue%28x=pi%2F2%29%29%2C%0D%0Ablue%28line%28-1.571%2C-5%2C-1.571%2C5%29%29%2Clocate%28-1.5%2C3%2Cblue%28x=-pi%2F2%29%29%2C%0D%0Agraph%28600%2C200%2C-4%2C6%2C-5%2C5%2C1%2Fsin%28x%29%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > .\r<BR>\n" );
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document.write( "The function <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=csc%28x%29=1%2Fsin%28x%29 ALIGN=MIDDLE  ALT=\"y=csc%28x%29=1%2Fsin%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > is a periodic function,<BR>\n" );
document.write( "like all trigonometric functions,<BR>\n" );
document.write( "so the inverse can only be defined by restricting the domains for the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" > values, so as not to repeat <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y ALIGN=MIDDLE  ALT=\"y\"   DISABLED_style= \"max-width:90%; height:auto;\" > values.<BR>\n" );
document.write( "That would be taking only the part of the graph between the blue lines,<BR>\n" );
document.write( "to get a function defined as<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=csc%28x%29=1%2Fsin%28x%29 ALIGN=MIDDLE  ALT=\"y=csc%28x%29=1%2Fsin%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=for ALIGN=MIDDLE  ALT=\"for\"   DISABLED_style= \"max-width:90%; height:auto;\" > <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-pi%2F2%3C=x%3C=pi%2F2 ALIGN=MIDDLE  ALT=\"-pi%2F2%3C=x%3C=pi%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=only ALIGN=MIDDLE  ALT=\"only\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "(quadrants I and IV).<BR>\n" );
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document.write( "Then we, can interchange the variables and define<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=csc%5E%28-1%29%28x%29 ALIGN=MIDDLE  ALT=\"y=csc%5E%28-1%29%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > is the angle <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y ALIGN=MIDDLE  ALT=\"y\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "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;\" > <BR>\n" );
document.write( "such that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=csc%28y%29=x ALIGN=MIDDLE  ALT=\"csc%28y%29=x\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "That is the same approach used to define the inverse of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=sin%28x%29 ALIGN=MIDDLE  ALT=\"y=sin%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .\r<BR>\n" );
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document.write( " <BR>\n" );
document.write( "So,<BR>\n" );
document.write( "a. The value of the function <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=csc%5E%28-1%29%28x%29 ALIGN=MIDDLE  ALT=\"y=csc%5E%28-1%29%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > is an angle <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=theta ALIGN=MIDDLE  ALT=\"theta\"   DISABLED_style= \"max-width:90%; height:auto;\" > in the interval <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22%5B%22 ALIGN=MIDDLE  ALT=\"%22%5B%22\"   DISABLED_style= \"max-width:90%; height:auto;\" ><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;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22%2C%22 ALIGN=MIDDLE  ALT=\"%22%2C%22\"   DISABLED_style= \"max-width:90%; height:auto;\" ><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;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22%5D%22 ALIGN=MIDDLE  ALT=\"%22%5D%22\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28-pi%2F2%3C=x%3C=pi%2F2%29 ALIGN=MIDDLE  ALT=\"highlight%28-pi%2F2%3C=x%3C=pi%2F2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( " <BR>\n" );
document.write( "b. What is sin^-1 supposed to mean? <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2Fsine ALIGN=MIDDLE  ALT=\"1%2Fsine\"   DISABLED_style= \"max-width:90%; height:auto;\" > ?<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=csc%5E%28-1%29%282%29 ALIGN=MIDDLE  ALT=\"csc%5E%28-1%29%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > is the angle <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=theta ALIGN=MIDDLE  ALT=\"theta\"   DISABLED_style= \"max-width:90%; height:auto;\" > , with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-pi%2F2%3C=x%3C=pi%2F2 ALIGN=MIDDLE  ALT=\"-pi%2F2%3C=x%3C=pi%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > that has<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=csc%28theta%29=1%2Fsin%28theta%29=%28sin%28theta%29%29%5E%28-1%29=highlight%28-2%29 ALIGN=MIDDLE  ALT=\"csc%28theta%29=1%2Fsin%28theta%29=%28sin%28theta%29%29%5E%28-1%29=highlight%28-2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%28theta%29=highlight%28-1%2F2%29 ALIGN=MIDDLE  ALT=\"sin%28theta%29=highlight%28-1%2F2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( " <BR>\n" );
document.write( "NOTE:<BR>\n" );
document.write( "We know that in quadrant I <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%28pi%2F6%29=1%2F2 ALIGN=MIDDLE  ALT=\"sin%28pi%2F6%29=1%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > (or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%2830%5Eo%29=1%2F2 ALIGN=MIDDLE  ALT=\"sin%2830%5Eo%29=1%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "In quadrant IV, using the symmetrical quadrant I angle as reference,<BR>\n" );
document.write( "we find that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%28-pi%2F6%29=-1%2F2 ALIGN=MIDDLE  ALT=\"sin%28-pi%2F6%29=-1%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "so <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=csc%5E%28-1%29%282%29=-pi%2F6 ALIGN=MIDDLE  ALT=\"csc%5E%28-1%29%282%29=-pi%2F6\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "However, this question wants you to crawl to the answer very, very slowly.<BR>\n" );
document.write( " <BR>\n" );
document.write( "c.  Is the value of sin&#952; positive, negative, or equal to 0?<BR>\n" );
document.write( "In part b. we found that we are looking for an angle whose sine is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-1%2F2 ALIGN=MIDDLE  ALT=\"-1%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > , so we know that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-1%2F2 ALIGN=MIDDLE  ALT=\"-1%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28negative%29 ALIGN=MIDDLE  ALT=\"highlight%28negative%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( " <BR>\n" );
document.write( "d. An angle <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=theta ALIGN=MIDDLE  ALT=\"theta\"   DISABLED_style= \"max-width:90%; height:auto;\" > that has a negative sine,<BR>\n" );
document.write( "and is in the interval we agreed to in part b. <BR>\n" );
document.write( "has to be in the interval <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0%3Ctheta%3C=pi%2F2 ALIGN=MIDDLE  ALT=\"0%3Ctheta%3C=pi%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "e. The terminal side of angle <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=theta ALIGN=MIDDLE  ALT=\"theta\"   DISABLED_style= \"max-width:90%; height:auto;\" > lies in quadrant IV.<BR>\n" );
document.write( "Here it is:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28300%2C300%2C-1.2%2C1.2%2C-1.2%2C1.2%2Cgrid%280%29%2C%0D%0Acircle%280%2C0%2C1%29%2Carrow%280%2C0%2C1.169%2C-0.675%29%2C%0D%0Ared%28arc%280%2C0%2C1.62%2C1.62%2C0%2C30%29%29%2Clocate%280.65%2C-0.15%2Cred%28theta%29%29%2C%0D%0Ared%28triangle%280.693%2C-0.4%2C0.743%2C-0.4%2C0.693%2C-0.35%29%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > It is a negative angle because it is a counterclockwise turn from the positive x-axis.\r<BR>\n" );
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document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );