document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=578708>Question 578708</A>: <I><SPAN STYLE=\"word-break: break-all;\"> find the angle(s) that would make each statement true.<BR>\n" );
document.write( "theta equals arc csc (sqrt of 2) </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #370763 by </B><A HREF=/tutors/aboutme.mpl?userid=Theo><B>Theo(13342)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Theo><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=Theo><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=370763\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> let x = theta.<BR>\n" );
document.write( "if x = arccsc(sqrt(2)), this means that:<BR>\n" );
document.write( "csc(x) = sqrt(2).<BR>\n" );
document.write( "since csc(x) = 1/sin(x),this means that:<BR>\n" );
document.write( "1/sin(x) = sqrt(2).<BR>\n" );
document.write( "multiply both sides of this equation by sin(x) and divide both sides of this equation by sqrt(2) to get:<BR>\n" );
document.write( "sin(x) = 1/sqrt(2).<BR>\n" );
document.write( "multiply the expression on the right side of the equation by sqrt(2)/sqrt(2) to get:<BR>\n" );
document.write( "sin(x) = sqrt(2)/2.<BR>\n" );
document.write( "you can either recognize this as 45 degrees or you can use your calculator to see that this is 45 degrees.<BR>\n" );
document.write( "that would be the value of this angle in the first quadrant.<BR>\n" );
document.write( "since the sine of an angle is positive in the first quadrant and the second quadrant, then the angle whose sine is sqrt(2)/2 can be either 45 degrees or 180 - 45 degrees which equals 135 degrees.<BR>\n" );
document.write( "you can use your calculator to verify that the sine of 45 degrees and the sine of 135 degrees are both equal to sqrt(2)/2.<BR>\n" );
document.write( "you can also use your calculator to verify that the cosecant of 45 degrees and the cosecant of 45 degrees are both equal to sqt(2).<BR>\n" );
document.write( "use your calculator to get sin(45) and the answer will be .7071067812<BR>\n" );
document.write( "use your calculator to calculate sqrt(2)/2 and the answer will be .7071067812<BR>\n" );
document.write( "in other words, sqrt(2)/2 and.7071067812 are equivalent.<BR>\n" );
document.write( "use your calculator to get sin(135) and the answer will be .7071067812<BR>\n" );
document.write( "in other words, sin(45) = sin(135).<BR>\n" );
document.write( "to find cosecant(45), you need to get sin(45) and then take the reciprocal of it.<BR>\n" );
document.write( "you get the reciprocal of it by dividing 1 by .7071067812 to get 1.414213562<BR>\n" );
document.write( "use your calculator to get sqrt(2) and the answer will be 1.414213562<BR>\n" );
document.write( "in other words 1.414213562 and sqrt(2) are equivalent.<BR>\n" );
document.write( "your answer is:<BR>\n" );
document.write( "theta equals arccsc (sqrt(2)) if and only if arccsc(theta) = sqrt(2).<BR>\n" );
document.write( "this happens when theta equals 45 degrees and theta equals 135 degrees.<BR>\n" );
document.write( "this assumes that your domain is all angles from 0 to 360.<BR>\n" );
document.write( "if there is no restrictions on the domain, then the answer is answer is:<BR>\n" );
document.write( "45 degrees plus or minus 360 degrees and 135 degrees plus or minus 360 degrees.<BR>\n" );
document.write( "you can graph the equation of y = csc(x).<BR>\n" );
document.write( "since you can't do that directly, then you need to graph the equation of y = 1/sin(x).<BR>\n" );
document.write( "since most graphing software graphs trigonometric functions in radians, you need to convert 45 degrees and 135 degrees to radians.<BR>\n" );
document.write( "the following graph shows you the graph of y = csc(x).<BR>\n" );
document.write( "a horizontal line has been drawn at y = sqrt(2).<BR>\n" );
document.write( "vertical lines have been drawn at the radian equivalent of x = 45, 135, 45 + 360 = 405, 135 + 360 = 495, 45 - 360 = -315, and 135 - 360 = -225 to show you that the the graph of y = csc(x) equals sqrt(2) when x equals those values.<BR>\n" );
document.write( "please note that the graph of y = csc(x) and the graph of y = 1/sin(x) are the same.<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "45 - 360 degrees = pi/4 - 2*pi radians = - (7/4)*pi radians = - 5.5 radians<BR>\n" );
document.write( "45 degrees = pi/4 radians = .8 radians.<BR>\n" );
document.write( "45 + 360 degrees = pi/4 + 2*pi radians = (9/4)*pi radians = 7.1 radians.<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "135 - 360 degrees = (3/4)*pi - 2*pi radians = - (5/4)*pi radians = -3.9 radians.<BR>\n" );
document.write( "135 degrees = (3/4)*pi radians = 2.356194 radians<BR>\n" );
document.write( "135 + 360 degrees = (3/4)*pi + 2*pi radians = + (11/4)*pi radians = 8.6 radians.<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "the graph will show vertical lines at these points and a horizontal line at y = sqrt(2).<BR>\n" );
document.write( "the intersection of these vertical lines and the horizontal line will also intersect the graph of the equation of y = csc(x) which means that the value of csc(x) will be the y values indicated for the x values indicated.<BR>\n" );
document.write( "the table used to plot these vertical lines is shown below:<BR>\n" );
document.write( "<PRE STYLE=\"white-space: pre-wrap; white-space: -moz-pre-wrap; white-space: -pre-wrap; white-space: -o-pre-wrap; word-wrap: break-word;\">\r\n" );
document.write( "        x value     y value    degree equivalent of radians shown under x value\r\n" );
document.write( "           -5.5       1.414     -315 degrees (45 - 360)\r\n" );
document.write( "            0.8       1.414       45 degrees\r\n" );
document.write( "            7.1       1.414      405 degrees (45 + 360)\r\n" );
document.write( "            ---\r\n" );
document.write( "           -3.9       1.414     -225 degrees (135 - 360)\r\n" );
document.write( "            2.4       1.414      135 degrees\r\n" );
document.write( "            8.6       1.414      495 degrees (135 + 360)\r\n" );
document.write( "</PRE><BR>\n" );
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document.write( "since degrees are cyclical, the graph goes on endlessly in both directions which is the reason why the domain of x must be specified for an interval unless you actually want to go on endlessly in both directions.\r<BR>\n" );
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