document.write( "Question 117922: solve. Give all positive values of the angle between 0degrees and 360degrees. Give any approximate value to the nearest minute only.
\n" ); document.write( " a.) cos 4x= sin2x
\n" ); document.write( "
\n" ); document.write( " and if you can this is a problem I also had trouble with
\n" ); document.write( " b.) 3 sin(theta)-4 cos (theta)= 2 \n" ); document.write( "
Algebra.Com's Answer #85915 by Edwin McCravy(20056)\"\" \"About 
You can put this solution on YOUR website!
\r\n" );
document.write( "solve. Give all positive values of the angle between 0° and 360°. \r\n" );
document.write( "Give any approximate value to the nearest minute only.\r\n" );
document.write( "a.) cos4x = sin2x\r\n" );
document.write( "\r\n" );
document.write( "Use the identity: cos(2A) = 1 - 2sin²A and the fact that\r\n" );
document.write( "4x = 2(2x) to replace the left side of\r\n" );
document.write( "\r\n" );
document.write( "                 cos 4x = sin2x\r\n" );
document.write( "\r\n" );
document.write( "             cos[2(2x)] = sin2x\r\n" );
document.write( "\r\n" );
document.write( "            1 - 2sin²2x = sin2x\r\n" );
document.write( "\r\n" );
document.write( "   -2sin²2x - sin2x + 1 = 0\r\n" );
document.write( "\r\n" );
document.write( "    2sin²2x + sin2x - 1 = 0\r\n" );
document.write( "\r\n" );
document.write( "(2sin2x - 1)(sin2x + 1) = 0 \r\n" );
document.write( "\r\n" );
document.write( "2sin2x - 1 = 0                 sin2x + 1 = 0\r\n" );
document.write( "    2sin2x = 1                     sin2x = -1  \r\n" );
document.write( "     sin2x = 1/2           \r\n" );
document.write( "\r\n" );
document.write( "To get all values for x between 0° and 360°,\r\n" );
document.write( "we must take 2x between 0° and 720°:\r\n" );
document.write( "\r\n" );
document.write( "2x = 30°, 150°, 390°, 510°      2x = 270°, 630°\r\n" );
document.write( " x = 15°,  75°, 195°, 255°       x = 135°, 315°\r\n" );
document.write( "\r\n" );
document.write( "---------------------------------------------------\r\n" );
document.write( "\r\n" );
document.write( "and if you can this is a problem I also had trouble with\r\n" );
document.write( "b.) 3 sinq - 4 cosq = 2 \r\n" );
document.write( "\r\n" );
document.write( "The best way to handle a linear combination of sine and cosine,\r\n" );
document.write( "such as\r\n" );
document.write( "\r\n" );
document.write( "m·sinq ± n·cosq is \r\n" );
document.write( "\r\n" );
document.write( "1. To draw a right triangle with legs m and n,\r\n" );
document.write( "2. Find an acute angle of that triangle.\r\n" );
document.write( "3. Solve for m and n in terms of that acute angles\r\n" );
document.write( "of the triangle:\r\n" );
document.write( "\r\n" );
document.write( "In this case m=3 and n=4, so draw the triangle:\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "We calculate agle F using tan(F) = \"4%2F3\" or F = 53.13010235°\r\n" );
document.write( "\r\n" );
document.write( "We find the hypotenuse to be 5 by the Pythagorean theorem:\r\n" );
document.write( "\r\n" );
document.write( "c² = 3² + 4²\r\n" );
document.write( "c² = 9 + 16\r\n" );
document.write( "c² = 25\r\n" );
document.write( " c = 5\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Now 3 = m = 5cosF, and 4 = n = 5sinF\r\n" );
document.write( "\r\n" );
document.write( "so substitute for the 3 and the 4\r\n" );
document.write( "\r\n" );
document.write( " 3·sinq - 4·cosq = 2\r\n" );
document.write( "\r\n" );
document.write( "5cosF·sinq - 5sinF·cosq = 2\r\n" );
document.write( "\r\n" );
document.write( "Now divide through by 5\r\n" );
document.write( "\r\n" );
document.write( "  cosF·sinq - sinF·cosq = \"2%2F5\"\r\n" );
document.write( "\r\n" );
document.write( "Now use the identity sin(A-B) = sinA·cosB - cosA·sinB,\r\n" );
document.write( "reversing the factors in the two terms on the left side:\r\n" );
document.write( "\r\n" );
document.write( "  sinq·cosF - cosq·sinq = \"2%2F5\"\r\n" );
document.write( "\r\n" );
document.write( "               sin(q-F) = .4\r\n" );
document.write( "\r\n" );
document.write( "Now since        0° < q < 360°\r\n" );
document.write( "              0°-F < q-F < 360°-F \r\n" );
document.write( "\r\n" );
document.write( "and since F = 53.13010235°\r\n" );
document.write( "\r\n" );
document.write( "         0 - 53.13010235° < q-F < 360° - 53.13010235°\r\n" );
document.write( "\r\n" );
document.write( "            -53.13010235° < q-F < 306.8608976°\r\n" );
document.write( "\r\n" );
document.write( "So q-F = 23.57617646°, 156.4218215°\r\n" );
document.write( "\r\n" );
document.write( "     q = 23.57617646°+F, 156.4218215°+F\r\n" );
document.write( "\r\n" );
document.write( "     q = 23.57617646°+53.13010235°, 156.4218215°+53.13010235° \r\n" );
document.write( "\r\n" );
document.write( "     q = 76.70627881°, 209.5519239°\r\n" );
document.write( "\r\n" );
document.write( "To change to minutes, multiply the decimal parts by 60\r\n" );
document.write( "\r\n" );
document.write( "(.70627881)(60) = 42.3767286 minutes or 42 minutes to the nearest minute\r\n" );
document.write( "(.5519239)(60) = 33.115434 minutes or 33 minutes to the nearest minute\r\n" );
document.write( "\r\n" );
document.write( "So the answers are:\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "     q = 76°42', 209°33'\r\n" );
document.write( "\r\n" );
document.write( "Edwin
\n" ); document.write( "
\n" );