document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=826685>Question 826685</A>: <I><SPAN STYLE=\"word-break: break-all;\"> How can I algebraically find all zeros of y= (x)= 3sin(2x) + cos(x) over the interval -(pi)less than or equal to x which is less than or equal to (pi)? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #498287 by </B><A HREF=/tutors/aboutme.mpl?userid=jsmallt9><B>jsmallt9(3758)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jsmallt9><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=498287\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Finding the zeros of<BR>\n" );
document.write( "y = 3sin(2x) + cos(x)<BR>\n" );
document.write( "means finding the solutions to:<BR>\n" );
document.write( "0 = 3sin(2x) + cos(x)<br><BR>\n" );
document.write( "Solving Trig equations like this usually starts with using algebra and/or Trig properties to transform the equation into one or more equations of the general form:<BR>\n" );
document.write( "TrigFunction(expression) = number<br><BR>\n" );
document.write( "To solve this equation we will start by using the sin(2x) = 2sin(x)cos(x) identity. Substituting for sin(2x) we get:<BR>\n" );
document.write( "0 = 3(2sin(x)cos(x)) + cos(x)<BR>\n" );
document.write( "which simplifies to:<BR>\n" );
document.write( "0 = 6sin(x)cos(x) + cos(x)<BR>\n" );
document.write( "Now we can factor out cos(x):<BR>\n" );
document.write( "0 = cos(x)(6sin(x)+1)<BR>\n" );
document.write( "From the Zero Product Property:<BR>\n" );
document.write( "cos(x) = 0 or 6sin(x)+1 = 0<BR>\n" );
document.write( "(Note how we now have two equations. This is one way to get more than one equation from a single equation.) Solving these we get:<BR>\n" );
document.write( "cos(x) = 0 or sin(x) = -1/6<BR>\n" );
document.write( "We now have the equations in the desired form.<br><BR>\n" );
document.write( "Next we find the general solution.<BR>\n" );
document.write( "For cos(x) = 0 we should recognize that 0 is a special angle value for cos. We should know that cos is 0 at <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=pi%2F2 ALIGN=MIDDLE  ALT=\"pi%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3pi%2F2 ALIGN=MIDDLE  ALT=\"3pi%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > and any other angle which is co-terminal with these. So the general solution for cos(x) = 0 is:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+pi%2F2+%2B+2pi%2An ALIGN=MIDDLE  ALT=\"x+=+pi%2F2+%2B+2pi%2An\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+3pi%2F2+%2B+2pi%2An ALIGN=MIDDLE  ALT=\"x+=+3pi%2F2+%2B+2pi%2An\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "For sin(x) = -1/6 we should recognize that 1/6 is not a special angle value for sin. So we will need our calculators. We will use <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%5E%28-1%29%281%2F6%29 ALIGN=MIDDLE  ALT=\"sin%5E%28-1%29%281%2F6%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > to find the reference angle. (Note: Do not use negative values when finding reference angles!) With my calculator set to radian mode, I get a reference angle, rounded to 4 places, of 0.1674. Since the 1/6 was negative (<i>Here</i> is where we use the fact that the sin is negative!) and since sin is negative in the 3rd and 4th quadrants we get the following general solution equations for sin(x) = -1/6:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+pi+%2B+0.1674+%2B+2pi%2An ALIGN=MIDDLE  ALT=\"x+=+pi+%2B+0.1674+%2B+2pi%2An\"   DISABLED_style= \"max-width:90%; height:auto;\" > for the 3rd quadrant<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+-0.1674+%2B+2pi%2An ALIGN=MIDDLE  ALT=\"x+=+-0.1674+%2B+2pi%2An\"   DISABLED_style= \"max-width:90%; height:auto;\" > for the 4th quadrant<br><BR>\n" );
document.write( "Together, the general solution equations for 0 = 3sin(2x) + cos(x) are:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+pi%2F2+%2B+2pi%2An ALIGN=MIDDLE  ALT=\"x+=+pi%2F2+%2B+2pi%2An\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+3pi%2F2+%2B+2pi%2An ALIGN=MIDDLE  ALT=\"x+=+3pi%2F2+%2B+2pi%2An\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+pi+%2B+0.1674+%2B+2pi%2An ALIGN=MIDDLE  ALT=\"x+=+pi+%2B+0.1674+%2B+2pi%2An\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+-0.1674+%2B+2pi%2An ALIGN=MIDDLE  ALT=\"x+=+-0.1674+%2B+2pi%2An\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "The general solution equations express the infinite set of solutions to the equation. Many problems, like this one, ask for solutions which are in a given interval. This time the interval is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-pi+%3C=+x+%3C=+pi ALIGN=MIDDLE  ALT=\"-pi+%3C=+x+%3C=+pi\"   DISABLED_style= \"max-width:90%; height:auto;\" > To find these answers we replace the n's in the equations with various integers until we have found all the x's which are in the interval.<br><BR>\n" );
document.write( "From <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+pi%2F2+%2B+2pi%2An ALIGN=MIDDLE  ALT=\"x+=+pi%2F2+%2B+2pi%2An\"   DISABLED_style= \"max-width:90%; height:auto;\" >...<BR>\n" );
document.write( "If n = 0 then <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+pi%2F2 ALIGN=MIDDLE  ALT=\"x+=+pi%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "If n = 1 (or greater) then x is too large for the interval.<BR>\n" );
document.write( "If n = -1 (or smaller) then x is too small for the interval.<br><BR>\n" );
document.write( "From <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+3pi%2F2+%2B+2pi%2An ALIGN=MIDDLE  ALT=\"x+=+3pi%2F2+%2B+2pi%2An\"   DISABLED_style= \"max-width:90%; height:auto;\" >...<BR>\n" );
document.write( "If n = 0 (or larger) then x is too large for the interval.<BR>\n" );
document.write( "If n = -1 then <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+-pi%2F2 ALIGN=MIDDLE  ALT=\"x+=+-pi%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "If n = -2 or smaller then x is too small for the interval.<br><BR>\n" );
document.write( "From <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+pi+%2B+0.1674+%2B+2pi%2An ALIGN=MIDDLE  ALT=\"x+=+pi+%2B+0.1674+%2B+2pi%2An\"   DISABLED_style= \"max-width:90%; height:auto;\" >...<BR>\n" );
document.write( "If n = 0 (or larger) then x is too large for the interval.<BR>\n" );
document.write( "If n = -1 then <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+-pi%2B0.16674 ALIGN=MIDDLE  ALT=\"x+=+-pi%2B0.16674\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "If n = -2 or smaller then x is too small for the interval.<br><BR>\n" );
document.write( "From <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+-0.1674+%2B+2pi%2An ALIGN=MIDDLE  ALT=\"x+=+-0.1674+%2B+2pi%2An\"   DISABLED_style= \"max-width:90%; height:auto;\" >...<BR>\n" );
document.write( "If n = 0 then <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+-0.1674 ALIGN=MIDDLE  ALT=\"x+=+-0.1674\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "If n = 1 (or greater) then x is too large for the interval.<BR>\n" );
document.write( "If n = -1 (or smaller) then x is too small for the interval.<br><BR>\n" );
document.write( "Altogether, the solutions to 0 = 3sin(2x) + cos(x) (and therefore the zeros of f(x) = 3sin(2x) + cos(x)) within the specified interval are:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+pi%2F2 ALIGN=MIDDLE  ALT=\"x+=+pi%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+-pi%2F2 ALIGN=MIDDLE  ALT=\"x+=+-pi%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+-pi%2B0.1674 ALIGN=MIDDLE  ALT=\"x+=+-pi%2B0.1674\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+-0.1674 ALIGN=MIDDLE  ALT=\"x+=+-0.1674\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "P.S. Feel free to replace the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=pi ALIGN=MIDDLE  ALT=\"pi\"   DISABLED_style= \"max-width:90%; height:auto;\" > in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-pi%2B0.1674 ALIGN=MIDDLE  ALT=\"-pi%2B0.1674\"   DISABLED_style= \"max-width:90%; height:auto;\" > with a decimal approximation and then add the two decimals. </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );