document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1181851>Question 1181851</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Evaluate cos(𝛼 + 𝛽) given sin(𝛼) = 5/6, 𝛼 𝑖𝑛 𝑄𝐼𝐼 and tan(𝛽) = 1/2, 𝛽 𝑖𝑛 𝑄𝐼𝐼𝐼 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #811780 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=811780\">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 use A and B in place of 𝛼 and 𝛽\r<BR>\n" );
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document.write( "If sin(A) = 5/6, then cos(A) = -sqrt(11)/6. You can use the pythagorean trig identity cos^2+sin^2 = 1 to see why this works. Or you can draw out a right triangle to find the missing side (using the pythagorean theorem). So it's not a coincidence that the term 'pythagorean' comes up with this trig identity.\r<BR>\n" );
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document.write( "Note: cosine is negative in Q2\r<BR>\n" );
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document.write( "If tan(B) = 1/2, then we have an opposite side of 1 and adjacent side of 2. <BR>\n" );
document.write( "This leads to a hypotenuse sqrt(1^2+2^2) = sqrt(5), which is again where the pythagorean theorem comes up.\r<BR>\n" );
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document.write( "Through a bit of algebra, you should find these two facts:<BR>\n" );
document.write( "sin(B) = -sqrt(5)/5<BR>\n" );
document.write( "cos(B) = -2sqrt(5)/5\r<BR>\n" );
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document.write( "Both sine and cosine are negative in Q3\r<BR>\n" );
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document.write( "In the previous two sections, we found the following four facts<BR>\n" );
document.write( "cos(A) = -sqrt(11)/6<BR>\n" );
document.write( "cos(B) = -2sqrt(5)/5<BR>\n" );
document.write( "sin(A) = 5/6<BR>\n" );
document.write( "sin(B) = -sqrt(5)/5\r<BR>\n" );
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document.write( "Use those four items in the identity below<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%28A%2BB%29+=+cos%28A%29%2Acos%28B%29+-+sin%28A%29%2Asin%28B%29 ALIGN=MIDDLE  ALT=\"cos%28A%2BB%29+=+cos%28A%29%2Acos%28B%29+-+sin%28A%29%2Asin%28B%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=cos%28A%2BB%29+=+%28-sqrt%2811%29%2F6%29%2A%28-2%2Asqrt%285%29%2F5%29+-+%285%2F6%29%2A%28-sqrt%285%29%2F5%29 ALIGN=MIDDLE   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=cos%28A%2BB%29+=+2%2Asqrt%2811%2A5%29%2F%286%2A5%29+%2B+5%2Asqrt%285%29%2F%286%2A5%29 ALIGN=MIDDLE  ALT=\"cos%28A%2BB%29+=+2%2Asqrt%2811%2A5%29%2F%286%2A5%29+%2B+5%2Asqrt%285%29%2F%286%2A5%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=cos%28A%2BB%29+=+2%2Asqrt%2855%29%2F30+%2B+5%2Asqrt%285%29%2F30 ALIGN=MIDDLE  ALT=\"cos%28A%2BB%29+=+2%2Asqrt%2855%29%2F30+%2B+5%2Asqrt%285%29%2F30\"   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=cos%28A%2BB%29+=+%282%2Asqrt%2855%29+%2B+5%2Asqrt%285%29%29%2F30 ALIGN=MIDDLE  ALT=\"cos%28A%2BB%29+=+%282%2Asqrt%2855%29+%2B+5%2Asqrt%285%29%29%2F30\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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