document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1174893>Question 1174893</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Solve the following trigonometric equations without using a calculator. Find all solutions where 0 ≤ x < 2 π . If no solution exists write \"DNE\". (Hint: isolate the trig function then draw triangles.)\r<BR>\n" );
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document.write( "tan^2(x)-1=0\r<BR>\n" );
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document.write( "*(I got π/4,3π/4,and 5π/4, I am not sure if I have all the solutions)*<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #800387 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3817)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><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=800387\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "tan^2(x) - 1 = 0<BR>\n" );
document.write( "tan^2(x) = 1<BR>\n" );
document.write( "tan(x) = sqrt(1) or tan(x) = -sqrt(1)<BR>\n" );
document.write( "tan(x) = 1 or tan(x) = -1\r<BR>\n" );
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document.write( "If tan(x) = 1, then x = pi/4 or x = 5pi/4<BR>\n" );
document.write( "Use the unit circle.<BR>\n" );
document.write( "Note the jump from pi/4 to 5pi/4 is exactly pi units<BR>\n" );
document.write( "pi/4+pi = 5pi/4<BR>\n" );
document.write( "If we know one solution for tangent, then we add or subtract pi units to get to another solution. This is because the period of the tangent function is exactly pi units. It repeats itself every pi units.\r<BR>\n" );
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document.write( "If tan(x) = -1, then x = 3pi/4 or x = 7pi/4<BR>\n" );
document.write( "3pi/4 + pi = 7pi/4\r<BR>\n" );
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document.write( "There are four solutions in the interval <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0+%3C=+x+%3C+2pi ALIGN=MIDDLE  ALT=\"0+%3C=+x+%3C+2pi\"   DISABLED_style= \"max-width:90%; height:auto;\" > that satisfy tan^2(x) - 1 = 0 and those four solutions are:<BR>\n" );
document.write( "<font color=red>pi/4, 3pi/4, 5pi/4, 7pi/4</font><BR>\n" );
document.write( "The nice pattern here is that the coefficients for the pi terms in the numerator is the set {1,3,5,7}\r<BR>\n" );
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document.write( "Something like 9pi/4 doesn't work because it's too large<BR>\n" );
document.write( "9pi/4 > 2pi<BR>\n" );
document.write( "9pi > 4*2pi<BR>\n" );
document.write( "9pi > 8pi<BR>\n" );
document.write( "9 > 8 <BR>\n" );
document.write( "In other words, x = 9pi/4 is outside the interval <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0+%3C=+x+%3C+2pi ALIGN=MIDDLE  ALT=\"0+%3C=+x+%3C+2pi\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Or you could note that 9pi/4 = 7.07 approximately and 2pi = 6.28 approximately, to help see that 9pi/4 is too big. <BR>\n" );
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