document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=325704>Question 325704</A>: <I><SPAN STYLE=\"word-break: break-all;\"> the book asks to find all solutions in the interval(0.2Pi).<BR>\n" );
document.write( " the problem is:<BR>\n" );
document.write( " 1-sin t=sqroot3 cos t <BR>\n" );
document.write( "What I did was bring the right side over to the left by subtracting it from both sides. (1-sin t)-(sqroot3 cos t)=0 using zero factor theory, then I would have: sin t= 1,the solution is Pi/2, and I'm not quite sure how to solve the other term , I thought I could divide the sqrt3 from both sides leaving cos t=0 which the answer for that would also be Pi/2. The answers in the back of the book are: 11Pi/6, and Pi/2. so I know I have one right but I can't figure out how to get the second answer. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #233173 by </B><A HREF=/tutors/aboutme.mpl?userid=solver91311><B>solver91311(24713)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=solver91311><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=solver91311><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=233173\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font face=\"Garamond\" size=\"+2\">\r<BR>\n" );
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document.write( "Let's see, you started with:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ 1\ -\ \sin(t)\ =\ \sqrt{3}\,\cos(t)\">\r<BR>\n" );
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document.write( "And then you wrote:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ 1\ -\ \sin(t)\ -\ \sqrt{3}\,\cos(t)\ =\ 0\">\r<BR>\n" );
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document.write( "Which is perfectly ok, in and of itself, but as we will soon see, not very helpful.\r<BR>\n" );
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document.write( "Then you really went astray.  You tried to apply the Zero Factor Property.\r<BR>\n" );
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document.write( "But <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large 1\ -\ \sin(t)\"> and <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \sqrt{3}\,\cos(t)\"> are NOT factors that when multiplied together equal zero.  The Zero Factor Property applies to a situation like <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large ab\ =\ 0\"> from which we can tell that either <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large a\ =\ 0\"> or <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large b\ =\ 0\">.  But your situation is <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large a\ -\ b\ =\ 0\">.  All you can tell from that is that <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large a\ =\ b\"> -- but you already knew that from the original equation.\r<BR>\n" );
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document.write( "Start with the original equation and square both sides, applying FOIL to the LHS.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ 1\ -\ 2\,\sin(t)\ +\ \sin^2(t)\ =\ 3\,\cos^2(t)\">\r<BR>\n" );
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document.write( "Ewww! That's ugly!  Have faith, Grasshopper...\r<BR>\n" );
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document.write( "Recall the Pythagorean Identity, <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \cos^2{\varphi}\ +\ \sin^2{\varphi}\ =\ 1\">, which can also be written: <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \cos^2{\varphi}\ =\ 1\ -\ \sin^2{\varphi}\">.\r<BR>\n" );
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document.write( "Then make the substitution into your equation:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ 1\ -\ 2\,\sin(t)\ +\ \sin^2(t)\ =\ 3\left(1\ -\ \sin^2{t}\right)\">\r<BR>\n" );
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document.write( "Distribute the 3 in the RHS, and then collect like terms with everything in the LHS:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ 2\,sin^2(t)\ -\ \sin(t)\ -\ 1\ =\ 0\">\r<BR>\n" );
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document.write( "Let <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large u\ =\ \sin(t)\">, then:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ 2u^2\ -\ u\ -\ 1\ =\ 0\">\r<BR>\n" );
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document.write( "Ah Ha!  Now it is starting to look like something familiar, no?\r<BR>\n" );
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document.write( "Factor:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ (u\ -\ 1)(2u\ +\ 1)\ =\ 0\">\r<BR>\n" );
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document.write( "NOW you can use the Zero Factor Property.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ u\ -\ 1\ =\ 0\ \Rightarrow\ u\ =\ 1\">\r<BR>\n" );
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document.write( "or\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ 2u\ +\ 1\ =\ 0\ \Rightarrow\ u\ =\ -\frac{1}{2}\">\r<BR>\n" );
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document.write( "Substitute back:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \sin(t)\ =\ 1\ \Rightarrow\ t\ =\ \sin^{-1}\left(1\right)\">\r<BR>\n" );
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document.write( "or\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \sin(t)\ =\ -\frac{1}{2}\ \Rightarrow\ t\ =\ \sin^{-1}\left(-\frac{1}{2}\right)\">\r<BR>\n" );
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document.write( "Look at the Unit Circle:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" src=\"http://www.math.ucsd.edu/~jarmel/math4c/Unit_Circle_Angles.png\">\r<BR>\n" );
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document.write( "And recall that on the unit circle, <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \sin{\theta}\"> is represented by the <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large y\">-coordinate.\r<BR>\n" );
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document.write( "In the interval [<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large 0\">,<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \pi\">], there is one angle that has a <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large y\">-coordinate of 1, namely <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \frac{\pi}{2}\">.  There are 2 angles that have a <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large y\">-coordinate of <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large -\frac{1}{2}\">, namely <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \frac{7\pi}{6}\"> and <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \frac{11\pi}{6}\">.\r<BR>\n" );
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document.write( "However, you need to check your answers, which is left as an exercise for the student.  Submit each of the angle values to the original equation.  2 of them check, but one of them does not.\r<BR>\n" );
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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( "</font> </SPAN>\n" );
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