document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1138457>Question 1138457</A>: <I><SPAN STYLE=\"word-break: break-all;\"> 1. Find the largest angle of the triangle whose sides are of length 2cm, 4cm, <BR>\n" );
document.write( "   5cm.<BR>\n" );
document.write( "2. If sin x =3/5 and x is acute,find the value of tan 2x and tan x/2. <BR>\n" );
document.write( "3. Find all the angles 0° and 360° whose sine is +.5.<BR>\n" );
document.write( "4. Find all the angles between 0° and 360° whose tangent is 1.5.<BR>\n" );
document.write( "5. By means of a right-angle isosceles ∆, deduce that<BR>\n" );
document.write( "   Sin 45°=cos45°=1/√2.<BR>\n" );
document.write( "   Prove that cos(45°+A)=1/√2(cosA-sinA) and sin(45°+A)=1/√2(cosA+sinA). <BR>\n" );
document.write( "   Deduce cos A = 1/√2{cos(45°+A)+sin (45° + A)} and the corresponding result <BR>\n" );
document.write( "   for sin A. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #756286 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=756286\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> 1. The largest angle of the triangle whose sides are of length 2cm, 4cm, <BR>\n" );
document.write( "   5cm is the one opposite the longest (5-cm) side.<BR>\n" );
document.write( "To find that angle we have to use law of cosines, which says that in a triangle with sides a, b, and c, and angle A opposite side a,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5E2=b%5E2%2Bc%5E2-2bc%2Acos%28A%29 ALIGN=MIDDLE  ALT=\"a%5E2=b%5E2%2Bc%5E2-2bc%2Acos%28A%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "If A is a right angle, then <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%28A%29=0 ALIGN=MIDDLE  ALT=\"cos%28A%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > , and that equation turns into the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5E2=b%5E2%2Bc%5E2 ALIGN=MIDDLE  ALT=\"a%5E2=b%5E2%2Bc%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > (the Pythagorean theorem). Otherwise, as in this case, the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-2bc%2Acos%28A%29 ALIGN=MIDDLE  ALT=\"-2bc%2Acos%28A%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > \"correction\" makes a larger for obtuse angles (as in this case), and smaller for acute ones.<BR>\n" );
document.write( "So, for this case we find <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A ALIGN=MIDDLE  ALT=\"A\"   DISABLED_style= \"max-width:90%; height:auto;\" > opposite <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a=5 ALIGN=MIDDLE  ALT=\"a=5\"   DISABLED_style= \"max-width:90%; height:auto;\" >(cm) , and the other two side lengths are b and c.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5%5E2=2%5E2%2B4%5E2-2%2A2%2A4%2Acos%28A%29 ALIGN=MIDDLE  ALT=\"5%5E2=2%5E2%2B4%5E2-2%2A2%2A4%2Acos%28A%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=25=4%2B16-16%2Acos%28A%29 ALIGN=MIDDLE  ALT=\"25=4%2B16-16%2Acos%28A%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=25=20-16cos%28A%29 ALIGN=MIDDLE  ALT=\"25=20-16cos%28A%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=25-20=-16cos%28A%29 ALIGN=MIDDLE  ALT=\"25-20=-16cos%28A%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5=-16cos%28A%29 ALIGN=MIDDLE  ALT=\"5=-16cos%28A%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-5%2F16=cos%28A%29 ALIGN=MIDDLE  ALT=\"-5%2F16=cos%28A%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28A=108.2%5Eo%29 ALIGN=MIDDLE  ALT=\"highlight%28A=108.2%5Eo%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " <BR>\n" );
document.write( "2. If <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%28x%29+=3%2F5 ALIGN=MIDDLE  ALT=\"sin%28x%29+=3%2F5\"   DISABLED_style= \"max-width:90%; height:auto;\" > and x is acute, x is the smallest angle of a right triangle with sides measuring 3, 4, and 5:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28250%2C200%2C-.5%2C4.5%2C-.5%2C3.5%2C%0D%0Atriangle%280%2C0%2C4%2C0%2C4%2C3%29%2Clocate%284.1%2C1.8%2C3%29%2C%0D%0Alocate%281.9%2C0%2C4%29%2Clocate%282%2C1.5%2C5%29%2C%0D%0Ared%28arc%280%2C0%2C2%2C2%2C-36.9%2C0%29%29%2Clocate%281%2C0.5%2Cred%28x%29%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > So, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%28x%29=4%2F5 ALIGN=MIDDLE  ALT=\"cos%28x%29=4%2F5\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=tan%28x%29=3%2F4 ALIGN=MIDDLE  ALT=\"tan%28x%29=3%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "From there, I can find, in a list of trigonometric identities,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=tan%282x%29=2tan%28x%29%2F%281-tan%5E2%28x%29%29 ALIGN=MIDDLE  ALT=\"tan%282x%29=2tan%28x%29%2F%281-tan%5E2%28x%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=tan%28x%2F2%29=sin%28x%29%2F%281%2Bcos%28x%29%29 ALIGN=MIDDLE  ALT=\"tan%28x%2F2%29=sin%28x%29%2F%281%2Bcos%28x%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "and calculate<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=tan%282x%29=2%283%2F4%29%2F%281-%283%2F4%29%5E2%29=%283%2F2%29%2F%281-9%2F16%29=%283%2F2%29%2F%287%2F16%29=%283%2F2%29%2816%2F7%29=highlight%2824%2F7%29 ALIGN=MIDDLE   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%28x%2F2%29=%283%2F5%29%2F%281%2B%284%2F5%29%29=%283%2F5%29%2F%289%2F5%29=%283%2F5%29%285%2F9%29=highlight%281%2F3%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > <BR>\n" );
document.write( " <BR>\n" );
document.write( "3. Find all the angles between 0° and 360° whose sine is +.5.<BR>\n" );
document.write( "We know that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%2830%5Eo%29=0.5 ALIGN=MIDDLE  ALT=\"sin%2830%5Eo%29=0.5\"   DISABLED_style= \"max-width:90%; height:auto;\" > , and we know that in each quadrant<BR>\n" );
document.write( "The function sine takes values from 0 to 1, or 0 to -1.<BR>\n" );
document.write( "In the whole first counterclockwise turn, between 0° and 360°,<BR>\n" );
document.write( "there is only one angle whose sine is 1 <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%2890%5Eo%29 ALIGN=MIDDLE  ALT=\"%2890%5Eo%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > (and only one angle whose sign is -1), but all other positive values happen twice,<BR>\n" );
document.write( "once as sine goes from 0 to 1 in the first quadrant, and again as sine goes from 1 back to 0 in the second quadrant.<BR>\n" );
document.write( "Suplementary angles <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=30%5Eo ALIGN=MIDDLE  ALT=\"30%5Eo\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=180%5Eo-30%5Eo=150%5Eo ALIGN=MIDDLE  ALT=\"180%5Eo-30%5Eo=150%5Eo\"   DISABLED_style= \"max-width:90%; height:auto;\" > have the same 0.5 sine.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28300%2C175%2C-1.2%2C1.2%2C-0.2%2C1.2%2Cgrid%280%29%2C%0D%0Ared%28circle%280%2C0%2C1%29%29%2Ctriangle%280%2C0%2C0.866%2C0%2C0.866%2C0.5%29%2C%0D%0Atriangle%280%2C0%2C-0.866%2C0%2C-0.866%2C0.5%29%2Carc%280%2C0%2C0.4%2C0.4%2C-30%2C0%29%2C%0D%0Aarc%280%2C0%2C0.4%2C0.4%2C-180%2C-150%29%2Cgreen%28arc%280%2C0%2C1.2%2C1.2%2C-150%2C0%29%29%2C%0D%0Alocate%280.05%2C0.55%2Cgreen%28150%5Eo%29%29%2Clocate%280.2%2C0.2%2C30%5Eo%29%2C%0D%0Agreen%28triangle%28-0.52%2C0.3%2C-0.51%2C0.3%2C-0.52%2C0.31%29%29%2Clocate%28-0.38%2C0.2%2C30%5Eo%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );