document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=883651>Question 883651</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find the slope of the line bisecting the angle from line1, with slope 2, to line2, with no slope.\r<BR>\n" );
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document.write( "Hope you'll help me. Thanks! Will wait asap. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #533682 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=533682\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> You are only given slopes, so it must mean that y-intercepts do not matter.<BR>\n" );
document.write( "The lines could intersect each other anywhere, and they could intersect the x- and y-axes anywhere,<BR>\n" );
document.write( "but for convenience I will make the lines intersect at the origin.<BR>\n" );
document.write( "(Whoever does not like my drawing can move the axes up, down, and left or right to his/her taste. As long as nothing is rotated, the slopes remain the same).<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28300%2C300%2C-2%2C3%2C-1%2C4%2C%0D%0Agrid%280%29%2Ctriangle%280%2C0%2C1%2C0%2C1%2C2%29%2C%0D%0Arectangle%281%2C0%2C0.9%2C0.1%29%2Clocate%281.02%2C1.2%2C2%29%2C%0D%0Aline%28-1%2C-2%2C2%2C4%29%2Cline%280%2C-1%2C0%2C4%29%2C%0D%0Alocate%281.9%2C2.5%2Cline%29%2Clocate%282.5%2C2.5%2C1%29%2C%0D%0Aarrow%281.8%2C2.36%2C1.2%2C2.36%29%2C%0D%0Alocate%28-2%2C2.8%2Cline%29%2Clocate%28-1.3%2C2.8%2C2%29%2C%0D%0Aarrow%28-1%2C2.66%2C-0.1%2C2.66%29%2C%0D%0Agreen%28line%28-0.3%2C-1.27%2C1%2C4.236%29%29%2C%0D%0Alocate%281.9%2C3.5%2Cgreen%28bisector%29%29%2C%0D%0Agreen%28arrow%281.85%2C3.36%2C0.8%2C3.36%29%29%2C%0D%0Ared%28arc%280%2C0%2C1.5%2C1.5%2C-63.4%2C0%29%29%2C%0D%0Alocate%280.5%2C0.4%2Cred%28A%29%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > The slope of line 1 is the tangent of the angle <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=red%28A%29%29 ALIGN=MIDDLE  ALT=\"red%28A%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > it makes with the x-axis (measured counterclockwise from the positive x-axis).<BR>\n" );
document.write( "The slope of the y-axis is undefined, and line 2 is the y-axis.<BR>\n" );
document.write( "The bisector of the acute angle formed by lines 1 and 2 is drawn in green.<BR>\n" );
document.write( "The angle it forms with the positive x-axis is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28A%2B90%5Eo%29%2F2 ALIGN=MIDDLE  ALT=\"%28A%2B90%5Eo%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "(If you don't like degrees, it is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28A%2Bpi%2F2%29%2F2 ALIGN=MIDDLE  ALT=\"%28A%2Bpi%2F2%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > in radians).<BR>\n" );
document.write( "The slope of the bisector is the tangent of that angle.<BR>\n" );
document.write( " <BR>\n" );
document.write( "If you only need an approximate value, you could use inverse trigonometric functions.<BR>\n" );
document.write( "For an exact value, I would use trigonometric identities.<BR>\n" );
document.write( " <BR>\n" );
document.write( "USING INVERSE FUNCTIONS:<BR>\n" );
document.write( "We use the inverse of tangent from calculator or computer to find the angle that has a tangent of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2 ALIGN=MIDDLE  ALT=\"2\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "Hopefully you know how to do that with your calculator and/or computer.<BR>\n" );
document.write( "The inverse function of tangent is represented as <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22tan%22%5E%28-1%29 ALIGN=MIDDLE  ALT=\"%22tan%22%5E%28-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > , or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=arctan ALIGN=MIDDLE  ALT=\"arctan\"   DISABLED_style= \"max-width:90%; height:auto;\" > , or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ATAN ALIGN=MIDDLE  ALT=\"ATAN\"   DISABLED_style= \"max-width:90%; height:auto;\" > , and in some calculators you have to press a key for inverse functions and then the tangent key.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=tan%28A%29=2 ALIGN=MIDDLE  ALT=\"tan%28A%29=2\"   DISABLED_style= \"max-width:90%; height:auto;\" >---><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=63.4%5Eo ALIGN=MIDDLE  ALT=\"A=63.4%5Eo\"   DISABLED_style= \"max-width:90%; height:auto;\" >(rounded),<BR>\n" );
document.write( "or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=1.107 ALIGN=MIDDLE  ALT=\"A=1.107\"   DISABLED_style= \"max-width:90%; height:auto;\" >(rounded) in radians, if you prefer radians.<BR>\n" );
document.write( "Using <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=63.4%5Eo ALIGN=MIDDLE  ALT=\"A=63.4%5Eo\"   DISABLED_style= \"max-width:90%; height:auto;\" >:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28A%2B90%5Eo%29%2F2=76.7%5Eo ALIGN=MIDDLE  ALT=\"%28A%2B90%5Eo%29%2F2=76.7%5Eo\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=tan%2876.7%5Eo%29=4.23 ALIGN=MIDDLE  ALT=\"tan%2876.7%5Eo%29=4.23\"   DISABLED_style= \"max-width:90%; height:auto;\" >(rounded) is the approximate value for the slope of the bisector.<BR>\n" );
document.write( "Using <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=1.107 ALIGN=MIDDLE  ALT=\"A=1.107\"   DISABLED_style= \"max-width:90%; height:auto;\" >(in radians):<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281.107%2Bpi%2F2%29%2F2=1.339 ALIGN=MIDDLE  ALT=\"%281.107%2Bpi%2F2%29%2F2=1.339\"   DISABLED_style= \"max-width:90%; height:auto;\" >(rounding the result but not <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<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=tan%281.339%29=4.24 ALIGN=MIDDLE  ALT=\"tan%281.339%29=4.24\"   DISABLED_style= \"max-width:90%; height:auto;\" >(rounded) is the approximate value for the slope of the bisector.<BR>\n" );
document.write( "You could round to more digits to get a better approximation.<BR>\n" );
document.write( " <BR>\n" );
document.write( "USING TRIGONOMETRIC IDENTITIES FOR EXACT VALUE:<BR>\n" );
document.write( "From the values of the trigonometric functions for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A ALIGN=MIDDLE  ALT=\"A\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<BR>\n" );
document.write( "and the well known values for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=90%5Eo=pi.2 ALIGN=MIDDLE  ALT=\"90%5Eo=pi.2\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "we would like to find the trigonometric functions for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28A%2B90%5Eo%29%2F2=%28A%2Bpi%2F2%29%2F2 ALIGN=MIDDLE  ALT=\"%28A%2B90%5Eo%29%2F2=%28A%2Bpi%2F2%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "For easier typing, I am going to use <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=90%5Eo+rather+%7B%7B%7Bpi%2F2 ALIGN=MIDDLE  ALT=\"90%5Eo+rather+%7B%7B%7Bpi%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > from now on.<BR>\n" );
document.write( "There are probably many ways to get to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=tan%28%28A%2B90%5Eo%29%2F2%29 ALIGN=MIDDLE  ALT=\"tan%28%28A%2B90%5Eo%29%2F2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "but I am going to show the one that came to my mind first,<BR>\n" );
document.write( "and that requires starting from <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%28A%29 ALIGN=MIDDLE  ALT=\"sin%28A%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%28A%29 ALIGN=MIDDLE  ALT=\"cos%28A%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "The right triangle with angle <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A ALIGN=MIDDLE  ALT=\"A\"   DISABLED_style= \"max-width:90%; height:auto;\" > that I included in my sketch<BR>\n" );
document.write( "has legs measuring <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1 ALIGN=MIDDLE  ALT=\"1\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2 ALIGN=MIDDLE  ALT=\"2\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "so the Pythagorean theorem says that the length of the hypotenuse is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%281%5E2%2B2%5E2%29=sqrt%281%2B4%29=sqrt%285%29 ALIGN=MIDDLE  ALT=\"sqrt%281%5E2%2B2%5E2%29=sqrt%281%2B4%29=sqrt%285%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "Trigonometric ratios tell us that<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%28A%29=2%2Fsqrt%285%29 ALIGN=MIDDLE  ALT=\"sin%28A%29=2%2Fsqrt%285%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%28A%29=1%2Fsqrt%285%29 ALIGN=MIDDLE  ALT=\"cos%28A%29=1%2Fsqrt%285%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Trigonometric identities for sum of angles tell us that<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%28A%2B90%5Eo%29=cos%28A%29cos%2890%5Eo%29-sin%28A%29sin%2890%5Eo%29=%281%2Fsqrt%285%29%29%2A0-%282%2Fsqrt%285%29%29%2A1=-2%2Fsqrt%285%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "We are half of he way there<BR>\n" );
document.write( "Trigonometric identities for half angles are<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=abs%28sin%28B%2F2%29%29=sqrt%28%281-cos%28B%29%29%2F2%29 ALIGN=MIDDLE  ALT=\"abs%28sin%28B%2F2%29%29=sqrt%28%281-cos%28B%29%29%2F2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=abs%28cos%28B%2F2%29%29=sqrt%28%281%2Bcos%28B%29%29%2F2%29 ALIGN=MIDDLE  ALT=\"abs%28cos%28B%2F2%29%29=sqrt%28%281%2Bcos%28B%29%29%2F2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "and we have to figure out the sign for ourselves.<BR>\n" );
document.write( "Since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28A%2B90%5Eo%29%2F2 ALIGN=MIDDLE  ALT=\"%28A%2B90%5Eo%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > is in the first quadrant, we know that all its trigonometric functions have positive values, so<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%28%28A%2B90%5Eo%29%2F2%29=sqrt%28%281-cos%28A%2B90%5Eo%29%29%2F2%29=sqrt%28%281-%281%2Fsqrt%285%29%29%29%2F2%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=cos%28%28A%2B90%5Eo%29%2F2%29=sqrt%28%281%2Bcos%28A%2B90%5Eo%29%29%2F2%29=sqrt%28%281%2B%281%2Fsqrt%285%29%29%29%2F2%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Tangent is sine divided by cosine, so<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=tan%28%28A%2B90%5Eo%29%2F2%29=sqrt%28%281-%282%2Fsqrt%285%29%29%29%2F2%29%2Fsqrt%28%281%2B%282%2Fsqrt%285%29%29%29%2F2%29=sqrt%28%281-%282%2Fsqrt%285%29%29%29%2F%281%2B%282%2Fsqrt%285%29%29%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >=<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28%28sqrt%285%29%2B2%29%2F%28sqrt%285%29-2%29%29=sqrt%28%28sqrt%285%29%2B2%29%28sqrt%285%29%2B2%29%2F%28%28sqrt%285%29-2%29%28sqrt%285%29%2B2%29%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >=<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28%28sqrt%285%29%2B2%29%5E2%2F%28%28sqrt%285%29%2B2%29%28sqrt%285%29-2%29%29%29 ALIGN=MIDDLE  ALT=\"sqrt%28%28sqrt%285%29%2B2%29%5E2%2F%28%28sqrt%285%29%2B2%29%28sqrt%285%29-2%29%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >=<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28%28sqrt%285%29%2B2%29%5E2%2F%285-2%5E2%29%29 ALIGN=MIDDLE  ALT=\"sqrt%28%28sqrt%285%29%2B2%29%5E2%2F%285-2%5E2%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >=<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28%28sqrt%285%29%2B2%29%5E2%2F%285-4%29%29 ALIGN=MIDDLE  ALT=\"sqrt%28%28sqrt%285%29%2B2%29%5E2%2F%285-4%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >=<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28%28sqrt%285%29%2B2%29%5E2%29 ALIGN=MIDDLE  ALT=\"sqrt%28%28sqrt%285%29%2B2%29%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >=<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28sqrt%285%29%2B2%29 ALIGN=MIDDLE  ALT=\"highlight%28sqrt%285%29%2B2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The approximate value of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%285%29%2B2 ALIGN=MIDDLE  ALT=\"sqrt%285%29%2B2\"   DISABLED_style= \"max-width:90%; height:auto;\" > is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4.236 ALIGN=MIDDLE  ALT=\"4.236\"   DISABLED_style= \"max-width:90%; height:auto;\" > . </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );