document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1176366>Question 1176366</A>: <I><SPAN STYLE=\"word-break: break-all;\"> For an angle a in the interval [0,π2) it is given that sin(a)=4/5<BR>\n" );
document.write( "Determine cos(a) exactly.\r<BR>\n" );
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document.write( "From my understanding, the angles is in the first quadrant (0 -> 90 deg), but I don't understand, it there a formula I can use?  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #803173 by </B><A HREF=/tutors/aboutme.mpl?userid=Solver92311><B>Solver92311(821)</B></A><img src=\"/images/stars/stars-11.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Solver92311><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=Solver92311><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=803173\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font face=\"Times New Roman\" size=\"+2\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \cos\varphi\ =\ \pm\sqrt{1\ -\ \sin^2\varphi}\">\r<BR>\n" );
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document.write( "The sine is positive in QI and QII which means that there are two possible arguments for the function that have a positive sine value, but the cosine is positive in QI and negative in QII.  So, as posed the question has two answers as indicated by the <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \pm\"> in the formula.\r<BR>\n" );
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document.write( "The formula is a consequence of the Pythagorean Identity: <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \cos^2\varphi\ +\ \sin^2\varphi\ =\ 1\">\r<BR>\n" );
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document.write( "Let's say that <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \sin\varphi\ =\ x\">.  We square both sides of this equation to obtain:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \sin^2\varphi\ =\ x^2\">\r<BR>\n" );
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document.write( "Applying the Pythagorean Identity:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ 1\ -\ cos^2\varphi\ =\ x^2\">\r<BR>\n" );
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document.write( "A little algebra music, Maestro:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \cos^2\varphi\ =\ 1\ -\ x^2\">\r<BR>\n" );
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document.write( "Taking the square root (remembering to account for positive and negative roots):\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \cos\varphi\ =\ \pm\sqrt{1\ -\ x^2}\">\r<BR>\n" );
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document.write( "But we know from earlier that <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE x^2\ =\ \sin^2\theta\"> we can substitute to get the desired formula.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \cos\varphi\ =\ \pm\sqrt{1\ -\ \sin^2\varphi}\">\r<BR>\n" );
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document.write( "Further, given the symmetrical nature of the Pythagorean Identity, it should be intuitively obvious that the following formula holds as well:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \sin\varphi\ =\ \pm\sqrt{1\ -\ \cos^2\varphi}\">\r<BR>\n" );
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document.write( "The following will help you choose which sign to affix based on your knowledge of the Quadrant location of the function argument:\r<BR>\n" );
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document.write( "<PRE STYLE=\"white-space: pre-wrap; white-space: -moz-pre-wrap; white-space: -pre-wrap; white-space: -o-pre-wrap; word-wrap: break-word;\">\r\n" );
document.write( "Quadrant  Sine Cosine Tangent\r\n" );
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document.write( "   I        +     +      +\r\n" );
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document.write( "   II       +     -      -\r\n" );
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document.write( "   III      -     -      +\r\n" );
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document.write( "   IV       -     +      -\r\n" );
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document.write( "</PRE>\r<BR>\n" );
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document.write( "Also, secant has the same sign as cosine and cosecant has the same sign as sin.\r<BR>\n" );
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document.write( "																<BR>\n" );
document.write( "John<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE e^{i\pi}\ +\ 1\ =\ 0\"><BR>\n" );
document.write( "My calculator said it, I believe it, that settles it<BR>\n" );
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document.write( "From <https://www.algebra.com/cgi-bin/upload-illustration.mpl> <BR>\n" );
document.write( "I > Ø<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \ \\r\"> </SPAN>\n" );
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