document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1201490>Question 1201490</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Given sin B=5/13 in QII,and(6,-8) is on the terminal side of a, find the exact value of sin (A+B)<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 #835878 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=835878\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "Answer: <font color=red size=4>63/65</font>\r<BR>\n" );
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document.write( "Explanation:\r<BR>\n" );
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document.write( "sine = opposite/hypotenuse<BR>\n" );
document.write( "cosine = adjacent/hypotenuse<BR>\n" );
document.write( "tangent = opposite/adjacent\r<BR>\n" );
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document.write( "we can rewrite those formulas into this shorter notation<BR>\n" );
document.write( "sin(theta) = y/r<BR>\n" );
document.write( "cos(theta) = x/r<BR>\n" );
document.write( "tan(theta) = y/x<BR>\n" );
document.write( "where (x,y) is the terminal point's location and x^2+y^2 = r^2<BR>\n" );
document.write( "x = adjacent<BR>\n" );
document.write( "y = opposite<BR>\n" );
document.write( "r = hypotenuse = distance from origin to terminal point<BR>\n" );
document.write( "Using this notation will avoid having to memorize things like \"cosine is negative in Q2, cosine is positive in Q4, etc\".<BR>\n" );
document.write( "All you'll need to worry about is the terminal point's coordinates.\r<BR>\n" );
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document.write( "sin(B) = 5/13<BR>\n" );
document.write( "opposite = y = 5<BR>\n" );
document.write( "hypotenuse = r = 13<BR>\n" );
document.write( "angle B is in quadrant QII, aka Q2<BR>\n" );
document.write( "This is the northwest quadrant<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28400%2C400%2C-5%2C5%2C-5%2C5%2C%0D%0Aline%28-5-3%2C0%2C5%2B3%2C0%29%2C%0D%0Aline%280%2C-5-3%2C0%2C5%2B3%29%2C%0D%0Ared%28line%280%2C0%2C-3%2C4%29%29%2C%0D%0Ared%28line%28-3%2C4%2C-3%2C0%29%29%2C%0D%0Ared%28line%28-3%2C0%2C0%2C0%29%29%2C%0D%0Alocate%28-1%2C0.5%2CB%29%2C%0D%0A%0D%0Alocate%28-4%2C2%2C%22Opp%22%29%2C%0D%0Alocate%28-2%2C-0.1%2C%22Adj%22%29%2C%0D%0Alocate%28-1.5%2C2.4%2C%22Hyp%22%29%2C%0D%0A%0D%0Alocate%28-4%2C1.5%2C%225%22%29%2C%0D%0Alocate%28-2%2C-0.6%2C%22-12%22%29%2C%0D%0Alocate%28-1%2C1.9%2C%2213%22%29%2C%0D%0A%0D%0Acircle%28-3%2C4%2C0.04%29%2Ccircle%28-3%2C4%2C0.05%29%2Ccircle%28-3%2C4%2C0.06%29%2Ccircle%28-3%2C4%2C0.07%29%2Ccircle%28-3%2C4%2C0.08%29%2Ccircle%28-3%2C4%2C0.09%29%2Ccircle%28-3%2C4%2C0.1%29%2C%0D%0Alocate%28-2.8%2C4.2%2C%22%28-12%2C5%29%22%29%2C%0D%0A%0D%0Alocate%285-0.4%2C5-0.4%2C%22Q1%22%29%2C%0D%0Alocate%28-4.6%2C5-0.4%2C%22Q2%22%29%2C%0D%0Alocate%28-4.6%2C-4.6%2C%22Q3%22%29%2C%0D%0Alocate%285-0.4%2C-4.6%2C%22Q4%22%29%2C%0D%0Alocate%280.3%2C-4%2Cmatrix%281%2C2%2C%22Diagram%22%2C%22not%22%29%29%2C%0D%0Alocate%280.3%2C-4.4%2Cmatrix%281%2C2%2C%22to%22%2C%22scale%22%29%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The adjacent side of 12 is found through the pythagorean theorem <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5E2%2Bb%5E2+=+c%5E2 ALIGN=MIDDLE  ALT=\"a%5E2%2Bb%5E2+=+c%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > (translating to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2%2By%5E2+=+r%5E2 ALIGN=MIDDLE  ALT=\"x%5E2%2By%5E2+=+r%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >)<BR>\n" );
document.write( "Plug in y = 5 and r = 13 to find x = 12 or x = -12<BR>\n" );
document.write( "We will go for x = -12 since we're in Q2.\r<BR>\n" );
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document.write( "From that drawing, we can determine:<BR>\n" );
document.write( "cos(B) = adjacent/hypotenuse<BR>\n" );
document.write( "cos(B) = x/r<BR>\n" );
document.write( "cos(B) = -12/13\r<BR>\n" );
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document.write( "The terminal side of angle A is at (6,-8)<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%0D%0Adrawing%28400%2C400%2C-5%2C5%2C-5%2C5%2C%0D%0Aline%28-5-3%2C0%2C5%2B3%2C0%29%2C%0D%0Aline%280%2C-5-3%2C0%2C5%2B3%29%2C%0D%0Ared%28line%280%2C0%2C3%2C-4%29%29%2C%0D%0Ared%28line%283%2C-4%2C3%2C0%29%29%2C%0D%0Ared%28line%283%2C0%2C0%2C0%29%29%2C%0D%0Alocate%281%2C-0.5%2CA%29%2C%0D%0A%0D%0Alocate%283.5%2C-2%2C%22Opp%22%29%2C%0D%0Alocate%282%2C1.2%2C%22Adj%22%29%2C%0D%0Alocate%281.5-1%2C-2.4%2B1%2C%22Hyp%22%29%2C%0D%0A%0D%0Alocate%283.5%2C-1.5%2C%22-8%22%29%2C%0D%0Alocate%282%2C0.6%2C%226%22%29%2C%0D%0Alocate%281%2C-1.9%2C%2210%22%29%2C%0D%0A%0D%0Acircle%283%2C-4%2C0.04%29%2Ccircle%283%2C-4%2C0.05%29%2Ccircle%283%2C-4%2C0.06%29%2Ccircle%283%2C-4%2C0.07%29%2Ccircle%283%2C-4%2C0.08%29%2Ccircle%283%2C-4%2C0.09%29%2Ccircle%283%2C-4%2C0.1%29%2C%0D%0Alocate%283%2C-4%2C%22%286%2C-8%29%22%29%2C%0D%0A%0D%0Alocate%285-0.4%2C5-0.4%2C%22Q1%22%29%2C%0D%0Alocate%28-4.6%2C5-0.4%2C%22Q2%22%29%2C%0D%0Alocate%28-4.6%2C-4.6%2C%22Q3%22%29%2C%0D%0Alocate%285-0.4%2C-4.6%2C%22Q4%22%29%2C%0D%0Alocate%28-3%2C-4%2Cmatrix%281%2C2%2C%22Diagram%22%2C%22not%22%29%29%2C%0D%0Alocate%28-3%2C-4.4%2Cmatrix%281%2C2%2C%22to%22%2C%22scale%22%29%29%0D%0A%29%0D%0A ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Angle A is in quadrant Q4, in the southeast.<BR>\n" );
document.write( "The coordinates of the point (6,-8) lead directly to the adjacent and opposite sides in that order.<BR>\n" );
document.write( "The hypotenuse is determined through the pythagorean theorem.<BR>\n" );
document.write( "Plug in x = 6 and y = -8 to find r = 10.\r<BR>\n" );
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document.write( "sin(A) = opp/hyp = y/r = -8/10 = -4/5<BR>\n" );
document.write( "cos(A) = adj/hyp = x/r = 6/10 = 3/5\r<BR>\n" );
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document.write( "------------------\r<BR>\n" );
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document.write( "Let's recap what we have so far\r<BR>\n" );
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document.write( "sin(A) = -4/5<BR>\n" );
document.write( "cos(A) = 3/5<BR>\n" );
document.write( "and<BR>\n" );
document.write( "sin(B) = 5/13<BR>\n" );
document.write( "cos(B) = -12/13\r<BR>\n" );
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document.write( "We'll then use a trig identity to finish up the problem.\r<BR>\n" );
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document.write( "A reference sheet can be found here<BR>\n" );
document.write( "<a  rel=nofollow HREF= \"https://tutorial.math.lamar.edu/pdf/Trig_Cheat_Sheet.pdf\">https://tutorial.math.lamar.edu/pdf/Trig_Cheat_Sheet.pdf</a>\r<BR>\n" );
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document.write( "sin(A+B) = sin(A)cos(B) + cos(A)sin(B)<BR>\n" );
document.write( "sin(A+B) = (-4/5)*(-12/13) + (3/5)*(5/13)<BR>\n" );
document.write( "sin(A+B) = 48/65 + 15/65<BR>\n" );
document.write( "sin(A+B) = (48 + 15)/65<BR>\n" );
document.write( "sin(A+B) = <font color=red>63/65</font><BR>\n" );
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