document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=282667>Question 282667</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Solve the equation 5cos(2a)+3=0 for all values of a on the interval 0<a<360. Round all answers to the nearest tenth of a degree. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #205285 by </B><A HREF=/tutors/aboutme.mpl?userid=Theo><B>Theo(13342)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Theo><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=Theo><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=205285\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Equation is:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "5*cos(2a)+3=0\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Subtract 3 from both sides of this equation to get:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "5*cos(2a) = -3\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Divide both sides of this equation by 5 to get:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "cos(2a) = -3/5 = -.6 \r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "arccos(-3/5) = 126.8698976 degrees.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "This means that 2a = 126.8698976 degrees.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "This means that a = 63.43494882 degrees.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "I thought this might be the only one, but it turns out I was wrong.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "There is another angle between 0 and 360 whose cosine is equal to -.6.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "I found that out through graphing, and then confirmed through the use of the calculator.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "It turns out the other angle is 233.1301024 degrees.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "I should have known based on the following facts.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "The cosine is negative in quadrants 2 and 3 only.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Within each quadrant, the value of the cosine either goes from 1 to 0 or 0 to -1.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "This means that the angle I was looking for had to be in quadrants 2 and 3 which means that there had to be 2 angles I was looking for, and not 1.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "The graph of the cosine of x confirms that.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "That graph is shown below\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph+%28400%2C200%2C-7%2C7%2C-1%2C1%2Ccos%28x%29%2C-.6%29 ALIGN=MIDDLE  ALT=\"graph+%28400%2C200%2C-7%2C7%2C-1%2C1%2Ccos%28x%29%2C-.6%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "The graph is in radian format.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "0 to 360 degrees goes from 0 radians to 6.283 radians.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "This corresponds to x = 0 to x = 6.283.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "The resolution isn't that great, so I would use 6.3 as a rough measure.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "You can see that the graph of cosine (x) goes from 1 at 0 radians to 0 at 1.6 radians to -1 at 3.1 radians to 0 at 4.7 radians to 1 at 6.3 radians.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "The value of x is the value of the radians.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "The line at y = -.6 intersects the graph of the equation of cosine(x) at 2 points between 0 and 6.3 radians.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Those intersections are at:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "x = 2.2 radians and at:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "x = 4.1 radians.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "This corresponds to the angles at:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "126.8698976 degrees, and at:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "233.1301024 degrees.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Those angles, however, are double the angles you are looking for.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Those correspond to 2 times the angle you are looking for.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "cos(2a) = -.6 is the equation.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Once you find 2a, you have to cut it in half to find a.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "The angles you are looking for are therefore:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "a = 63.43494882 and a = 116.5650512\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Rounded to the nearest 10th of a degree, those angles becomes:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "a = 63.4 and a = 116.6 degrees.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "To confirm these angles were good, I substituted in the original equation to get:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "5*cos(2a) = -3\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "This equation became:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "5*cos(126.8698976) = -3 which became 5 * -.6 = -3 which became -3 = -3.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "5*cos(233.1301024) = -3 which became 5 * -.6 = -3 which became -3 = -3.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( " \r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );