document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1196285>Question 1196285</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find the exact value of arcsin(1/8) + arccos(1/8). </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #829048 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=829048\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "A = arcsin(1/8)<BR>\n" );
document.write( "B = arccos(1/8)<BR>\n" );
document.write( "The goal is to find A+B<BR>\n" );
document.write( "As a slight detour, let's find out what sin(A+B) is\r<BR>\n" );
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document.write( "Recall from the list of trig identities that<BR>\n" );
document.write( "sin(A+B) = sin(A)cos(B)+cos(A)sin(B)\r<BR>\n" );
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document.write( "Let's compute each piece<BR>\n" );
document.write( "sin(A) = sin(arcsin(1/8))<BR>\n" );
document.write( "sin(A) = 1/8<BR>\n" );
document.write( "The sine and inverse sine functions cancel each other out.\r<BR>\n" );
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document.write( "cos(B) = cos(arccos(1/8))<BR>\n" );
document.write( "cos(B) = 1/8<BR>\n" );
document.write( "We have similar cancellation going on.\r<BR>\n" );
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document.write( "cos(A) = cos(arcsin(1/8))<BR>\n" );
document.write( "cos(A) = 3*sqrt(7)/8<BR>\n" );
document.write( "See the figure below<BR>\n" );
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document.write( "I used the pythagorean theorem to determine the 3*sqrt(7) portion\r<BR>\n" );
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document.write( "sin(B) = sin(arccos(1/8))<BR>\n" );
document.write( "sin(B) = 3*sqrt(7)/8<BR>\n" );
document.write( "The steps are similar to the previous section\r<BR>\n" );
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document.write( "Now we can put everything together<BR>\n" );
document.write( "sin(A+B) = sin(A)cos(B)+cos(A)sin(B)<BR>\n" );
document.write( "sin(A+B) = (1/8)*(1/8) + ( 3*sqrt(7)/8 )( 3*sqrt(7)/8 )<BR>\n" );
document.write( "sin(A+B) = 1/64 + 63/64<BR>\n" );
document.write( "sin(A+B) = (1 + 63)/64<BR>\n" );
document.write( "sin(A+B) = 64/64<BR>\n" );
document.write( "sin(A+B) = 1<BR>\n" );
document.write( "A+B = arcsin(1)<BR>\n" );
document.write( "A+B = pi/2<BR>\n" );
document.write( "arcsin(1/8) + arccos(1/8) = pi/2\r<BR>\n" );
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document.write( "Answer: pi/2 radians (which is equivalent to 90 degrees)\r<BR>\n" );
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document.write( "As pointed out by the other tutor, there's nothing particularly special about the 1/8. It turns out that arcsin(x) + arccos(x) = pi/2 for any x on the interval of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-1+%3C=+x+%3C=+1 ALIGN=MIDDLE  ALT=\"-1+%3C=+x+%3C=+1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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