document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1181850>Question 1181850</A>: <I><SPAN STYLE=\"word-break: break-all;\"> 4𝑐𝑜𝑠(2𝜃) + 1 = 2, over the interval 0 ≤ 𝜃 < 360° </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #811784 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=811784\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> i used x instead of theta.<BR>\n" );
document.write( "equation is:<BR>\n" );
document.write( "4 * cos(2x) + 1 = 2<BR>\n" );
document.write( "subtract 1 from both sides to get:<BR>\n" );
document.write( "4 * cos(2x) = 1<BR>\n" );
document.write( "divide both sides by 4 to get:<BR>\n" );
document.write( "cos(2x) = 1/4<BR>\n" );
document.write( "solve for 2x to get:<BR>\n" );
document.write( "2x = arccos(1/4) = 75.52248781<BR>\n" );
document.write( "solve for x to get:<BR>\n" );
document.write( "x = 37.76124391 degrees.\r<BR>\n" );
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document.write( "since the frequency is 2, the period is 360/2 = 180.<BR>\n" );
document.write( "that means two full cycles of the cosine function in 360 degrees.<BR>\n" );
document.write( "when the period is 180 degrees, the 4 quadrants are:<BR>\n" );
document.write( "0 to 45, 45 to 90, 90 to 135, 135 to 180.<BR>\n" );
document.write( "the cosine function is positive in the first and fourth quadrant.<BR>\n" );
document.write( "the angle is 37.76124391 in the first quadrant and 180 minus 37.76124391 in the fourth quadrant = 142.2387561.<BR>\n" );
document.write( "in the second 180 degree period, the angle in the first quadrant of that = 37.76124391 + 180 = 217.7612439 degrees and the angle in the fourth quadrant of that = 142.2387561 + 180 = 322.2387561.\r<BR>\n" );
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document.write( "the graph of the two equations of y = 4 * cos(2x) + 1 and y = 2 is shown below.<BR>\n" );
document.write( "since 4 * cos(2x) + 1 = 2, the intersection of thoose two equations is the solution.<BR>\n" );
document.write( "the graph repeats endlessly in either direction, but the focus is on the interval between 0 degrees and 360 degrees.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2021/061301.jpg\" >\r<BR>\n" );
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document.write( "the general form of the cosine equation is:\r<BR>\n" );
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document.write( "y = a * cos(b * (x-c)) + d<BR>\n" );
document.write( "a is the amplitude<BR>\n" );
document.write( "b is the frequency<BR>\n" );
document.write( "c is the horizontal shift<BR>\n" );
document.write( "d is the vertical shift\r<BR>\n" );
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document.write( "in the equation of y = 4 * cos(2x) + 1, .....<BR>\n" );
document.write( "a = 4<BR>\n" );
document.write( "b = 2<BR>\n" );
document.write( "c = 0<BR>\n" );
document.write( "d = 1\r<BR>\n" );
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document.write( "to graph the equation, i basically broke up the one equation of 4 * cos(2x) + 1 = 2 into 2 equations.<BR>\n" );
document.write( "they were y = 4 * cos(2x) + 1 and y = 2.<BR>\n" );
document.write( "the intersection of those 2 equations is where the solution was.\r<BR>\n" );
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document.write( " </SPAN>\n" );
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