document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1199044>Question 1199044</A>: <I><SPAN STYLE=\"word-break: break-all;\"> If tan 𝛼 = − 12/35 and cot 𝛽 = 3/4<BR>\n" );
document.write( " for a second-quadrant angle 𝛼 and a third-quadrant angle 𝛽, find the following.<BR>\n" );
document.write( "(a)    sin(𝛼 + 𝛽) <BR>\n" );
document.write( "(c)    tan(𝛼 + 𝛽) </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #833005 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3838)</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=833005\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "I'll do part (a) to get you started.\r<BR>\n" );
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document.write( "𝛼 = alpha<BR>\n" );
document.write( "𝛽 = beta\r<BR>\n" );
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document.write( "Adj = adjacent side<BR>\n" );
document.write( "Opp = opposite side<BR>\n" );
document.write( "Hyp = hypotenuse\r<BR>\n" );
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document.write( "alpha is in quadrant Q2, which is in the northwest.<BR>\n" );
document.write( "Any (x,y) point located here has <BR>\n" );
document.write( "x < 0<BR>\n" );
document.write( "y > 0<BR>\n" );
document.write( "This will mean the adjacent side will be labeled negative, and opposite side positive\r<BR>\n" );
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document.write( "Informally we can think of it like this<BR>\n" );
document.write( "x = adjacent<BR>\n" );
document.write( "y = opposite\r<BR>\n" );
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document.write( "tan = opp/adj<BR>\n" );
document.write( "tan(alpha) = -12/35 = 12/(-35)<BR>\n" );
document.write( "opp = 12<BR>\n" );
document.write( "adj = -35\r<BR>\n" );
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document.write( "The hypotenuse is 37 found by using 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;\" ><BR>\n" );
document.write( "Plug in a = 35, b = 12 and solve for c to get c = 37.<BR>\n" );
document.write( "I'll let you do these steps.\r<BR>\n" );
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document.write( "Diagram:<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%2C0.75%2Calpha%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%2212%22%29%2C%0D%0Alocate%28-2%2C-0.6%2C%22-35%22%29%2C%0D%0Alocate%28-1%2C1.9%2C%2237%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( "From this diagram we can determine the following:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%28alpha%29+=+opp%2Fhyp+=+12%2F37 ALIGN=MIDDLE  ALT=\"sin%28alpha%29+=+opp%2Fhyp+=+12%2F37\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%28alpha%29+=+adj%2Fhyp+=+-35%2F37 ALIGN=MIDDLE  ALT=\"cos%28alpha%29+=+adj%2Fhyp+=+-35%2F37\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Continuation of part (a)\r<BR>\n" );
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document.write( "cot(beta) = 3/4<BR>\n" );
document.write( "beta is in Q3, which is in the southwest.\r<BR>\n" );
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document.write( "This is when x < 0 and y < 0<BR>\n" );
document.write( "So I'll have both the opposite and adjacent sides to be labeled negative.\r<BR>\n" );
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document.write( "cot = adj/opp = 3/4 = -3/(-4)<BR>\n" );
document.write( "adj = -3<BR>\n" );
document.write( "opp = -4<BR>\n" );
document.write( "Diagram:<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%2C-4%29%29%2C%0D%0Ared%28line%28-3%2C-4%2C-3%2C0%29%29%2C%0D%0Ared%28line%28-3%2C0%2C0%2C0%29%29%2C%0D%0Alocate%28-1%2C-0.5%2Cbeta%29%2C%0D%0A%0D%0Alocate%28-4%2C-2%2C%22Opp%22%29%2C%0D%0Alocate%28-2%2C-0.1%2C%22Adj%22%29%2C%0D%0Alocate%28-1.5%2C-2.4%2C%22Hyp%22%29%2C%0D%0A%0D%0Alocate%28-4%2C-1.5%2C%22-4%22%29%2C%0D%0Alocate%28-2%2C0.6%2C%22-3%22%29%2C%0D%0Alocate%28-1%2C-1.9%2C%225%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;\" >\r<BR>\n" );
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document.write( "The hypotenuse of 5 units is found using the pythagorean theorem (use a = 3, b = 4 to solve for c). I'll let the student perform these steps.\r<BR>\n" );
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document.write( "From that second diagram, we can determine:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%28beta%29+=+opp%2Fhyp+=+-4%2F5 ALIGN=MIDDLE  ALT=\"sin%28beta%29+=+opp%2Fhyp+=+-4%2F5\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%28beta%29+=+adj%2Fhyp+=+-3%2F5 ALIGN=MIDDLE  ALT=\"cos%28beta%29+=+adj%2Fhyp+=+-3%2F5\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "--------------------------------------------------<BR>\n" );
document.write( "Continuation of part (a)\r<BR>\n" );
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document.write( "To summarize what each diagram gives us:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%28alpha%29+=+12%2F37 ALIGN=MIDDLE  ALT=\"sin%28alpha%29+=+12%2F37\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%28alpha%29+=+-35%2F37 ALIGN=MIDDLE  ALT=\"cos%28alpha%29+=+-35%2F37\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%28beta%29+=+-4%2F5 ALIGN=MIDDLE  ALT=\"sin%28beta%29+=+-4%2F5\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%28beta%29=+-3%2F5 ALIGN=MIDDLE  ALT=\"cos%28beta%29=+-3%2F5\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Then we can use the trig identity shown below <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%28alpha+%2B+beta%29+=+sin%28alpha%29cos%28beta%29+%2B+cos%28alpha%29sin%28beta%29 ALIGN=MIDDLE  ALT=\"sin%28alpha+%2B+beta%29+=+sin%28alpha%29cos%28beta%29+%2B+cos%28alpha%29sin%28beta%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%28alpha+%2B+beta%29+=+%2812%2F37%29%2A%28-3%2F5%29+%2B+%28-35%2F37%29%2A%28-4%2F5%29 ALIGN=MIDDLE  ALT=\"sin%28alpha+%2B+beta%29+=+%2812%2F37%29%2A%28-3%2F5%29+%2B+%28-35%2F37%29%2A%28-4%2F5%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%28alpha+%2B+beta%29+=+-36%2F185+%2B+140%2F185 ALIGN=MIDDLE  ALT=\"sin%28alpha+%2B+beta%29+=+-36%2F185+%2B+140%2F185\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%28alpha+%2B+beta%29+=+%28-36+%2B+140%29%2F185 ALIGN=MIDDLE  ALT=\"sin%28alpha+%2B+beta%29+=+%28-36+%2B+140%29%2F185\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%28alpha+%2B+beta%29+=+104%2F185 ALIGN=MIDDLE  ALT=\"sin%28alpha+%2B+beta%29+=+104%2F185\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Answer to part (a) is the fraction <font color=red>104/185</font>\r<BR>\n" );
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document.write( "--------------------------------------------------\r<BR>\n" );
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document.write( "I'll let you handle part (c).\r<BR>\n" );
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document.write( "There are at least two approaches you could take here.\r<BR>\n" );
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document.write( "Method 1 is to compute <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%28alpha%2Bbeta%29 ALIGN=MIDDLE  ALT=\"cos%28alpha%2Bbeta%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > using the trig identity <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%28alpha%2Bbeta%29+=+cos%28alpha%29cos%28beta%29-sin%28alpha%29sin%28beta%29 ALIGN=MIDDLE  ALT=\"cos%28alpha%2Bbeta%29+=+cos%28alpha%29cos%28beta%29-sin%28alpha%29sin%28beta%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Then compute <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=tan%28alpha%2Bbeta%29+=+%28sin%28alpha%2Bbeta%29%29%2F%28cos%28alpha%2Bbeta%29%29 ALIGN=MIDDLE  ALT=\"tan%28alpha%2Bbeta%29+=+%28sin%28alpha%2Bbeta%29%29%2F%28cos%28alpha%2Bbeta%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and simplify.\r<BR>\n" );
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document.write( "Method 2 involves this trig identity<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=tan%28alpha%2Bbeta%29+=+%28tan%28alpha%29%2Btan%28beta%29%29%2F%281-tan%28alpha%29%2Atan%28beta%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > \r<BR>\n" );
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document.write( "Other approaches could be possible.\r<BR>\n" );
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document.write( "A list of identities 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( "--------------------------------------------------\r<BR>\n" );
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document.write( "Edit:\r<BR>\n" );
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document.write( "This in response to what @MathLover1 posted.<BR>\n" );
document.write( "She made an error when going from something like <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%28alpha%29+=+12%2F37 ALIGN=MIDDLE  ALT=\"sin%28alpha%29+=+12%2F37\"   DISABLED_style= \"max-width:90%; height:auto;\" > (which is correct) to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%2812%2F37%29 ALIGN=MIDDLE  ALT=\"sin%2812%2F37%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > which is not correct. <BR>\n" );
document.write( "The two ideas are different. <BR>\n" );
document.write( "Similar errors are made elsewhere.\r<BR>\n" );
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document.write( "This means both <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%28alpha%2Bbeta%29+=+-0.2721971 ALIGN=MIDDLE  ALT=\"sin%28alpha%2Bbeta%29+=+-0.2721971\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=tan%28alpha+%2B+beta%29=-0.2828782 ALIGN=MIDDLE  ALT=\"tan%28alpha+%2B+beta%29=-0.2828782\"   DISABLED_style= \"max-width:90%; height:auto;\" > are incorrect as well.<BR>\n" );
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