document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1118695>Question 1118695</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Solve the following equations for 0 &#8804; &#952; &#8804; 180 degrees.<BR>\n" );
document.write( "6.a. sin^2 &#952; = 1<BR>\n" );
document.write( "b. 3sin &#952; = 2cos &#952;<BR>\n" );
document.write( "c. sin &#952; - 4cos &#952; = 0 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #734096 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=734096\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> you can solve these graphically.\r<BR>\n" );
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document.write( "the graph of 6a is shown below:\r<BR>\n" );
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document.write( "the graph of 6b is shown below:\r<BR>\n" );
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document.write( "the graph of 6c is shown below:\r<BR>\n" );
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document.write( "to make graphing easier, i used x instead of theta.\r<BR>\n" );
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document.write( "basically i created separate equations from the expression on the left side of the equal sign and the expression on the right side of the equal sign.\r<BR>\n" );
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document.write( "the intersection of those 2 equations is the common solution to both.\r<BR>\n" );
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document.write( "i only looked for a solution between 0 and 180 degrees.\r<BR>\n" );
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document.write( "that's the unshaded region of the graphs.\r<BR>\n" );
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document.write( "to solve algebraically, i did the following/\r<BR>\n" );
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document.write( "6a equation is sin^2(x) = 1.\r<BR>\n" );
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document.write( "remember, i'm using x in place of theta.<BR>\n" );
document.write( "this makes it easier to type and also easier to graph.\r<BR>\n" );
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document.write( "take the square root of both sides of the equation to get:\r<BR>\n" );
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document.write( "sin(x) = plus or minus sqrt(1) = plus or minus 1.\r<BR>\n" );
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document.write( "my calculator tells me:\r<BR>\n" );
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document.write( "arcsin(1) = 90 degrees.<BR>\n" );
document.write( "arcsin(-1) = -90 degrees.\r<BR>\n" );
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document.write( "to convert to all positive angles between 0 and 360 degrees, i do the following:\r<BR>\n" );
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document.write( "90 degrees = 90 degrees<BR>\n" );
document.write( "-90 degrees = 360 - 90 = 270 degrees.\r<BR>\n" );
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document.write( "i therefore get:\r<BR>\n" );
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document.write( "arcsin(1) = 90 degrees.<BR>\n" );
document.write( "arcsin(-1) = 270 degrees.\r<BR>\n" );
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document.write( "the only solution between 0 and 180 degrees is arcsin1) = 90 degrees.\r<BR>\n" );
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document.write( "therefore 90 degrees is my only solution.\r<BR>\n" );
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document.write( "6b equation is 3sin(x) = 2cos(x).\r<BR>\n" );
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document.write( "i'm using x instead of theta for purposes already described above.\r<BR>\n" );
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document.write( "divide both sides of this equaation by 3 and divide both sides of this equation by cos(x) to get:\r<BR>\n" );
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document.write( "sin(x)/cos(x) = 2/3.\r<BR>\n" );
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document.write( "since sin(x)/cos(x) is equivalent to tan(x), the equation becomes:\r<BR>\n" );
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document.write( "tan(x) = 2/3.\r<BR>\n" );
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document.write( "arctan(2/3) = 33.69006753 degrees which can be rounded to 33.69 degrees as shown in the graph for 6b.\r<BR>\n" );
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document.write( "that would be in the first quadrant.\r<BR>\n" );
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document.write( "tangent is positive in the first and third quadrant.\r<BR>\n" );
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document.write( "third quadrant is not between 0 and 180 degrees, therefore the only solution is:\r<BR>\n" );
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document.write( "x = 33.69 degrees, as shown in the graph for 6b.\r<BR>\n" );
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document.write( "6c equation is sin(x) - 4cos(x) = 0\r<BR>\n" );
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document.write( "again, i use x in place of theta.\r<BR>\n" );
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document.write( "add 4cos(x) to both sides of the equation to get:\r<BR>\n" );
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document.write( "sin(x) = 4cos(x)\r<BR>\n" );
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document.write( "divide both sides of the equation by cos(x) to get:\r<BR>\n" );
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document.write( "sin(x) / cos(x) = 4\r<BR>\n" );
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document.write( "this is equivalent to tan(x) = 4\r<BR>\n" );
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document.write( "arctan(4) = 75.96375653 degrees.\r<BR>\n" );
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document.write( "this can be rounded to 75.964 degrees as shown in the graph for 6c.\r<BR>\n" );
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document.write( "tan is positive in the first and third quadrants only.\r<BR>\n" );
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document.write( "third quadrant is not between 0 and 180 degrees.\r<BR>\n" );
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document.write( "therefore the only solution is 75.964 degrees. </SPAN>\n" );
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