document.write( "Question 130107: on the same set of axes, sketch and label the graphs of the equations y= 2 cos 2x and y = -2 sin x in the interval 0 < x < \"pi\"
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Algebra.Com's Answer #95066 by Edwin McCravy(20056)\"\" \"About 
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document.write( "On the same set of axes, sketch and label the 
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document.write( "graphs of the equations\r
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document.write( "y = 2cos(2x) \r
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document.write( "and \r
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document.write( "y = -2sin(x) \r
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document.write( "in the interval 0 < x < \"pi\".\r
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document.write( "find the values which 2 cos 2x = -2 sin x\r
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document.write( "The rule for the graphs of:\r
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document.write( "y = Asin(Bx) and y = Acos(Bx)\r
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document.write( "both have these important points on the x-axis:\r
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document.write( "0, ±\"%28n%2Api%29%2F%282B%29\" where n is any integer.\r
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document.write( "The peaks all have y-coordinate |A|.
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document.write( "The valleys all have y-coordinates -|A|\r
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document.write( "If A is positive,
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document.write( "y = Asin(Bx) circulates from x-intercept at the origin, peak, x-intercept, valley, x-intercept, peak, etc. \r
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document.write( "y = Acos(Bx) circulates from a peak at the origin, x-intercept, valley, x-intercept, peak, etc.\r
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document.write( "Peaks and valleys in the rules above reverse if A is negative.\r
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document.write( "So for y = 2cos(2x)\r
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document.write( "A = 2, B = 2.  This graph has important points at \r
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document.write( "x = 0, \"pi%2F4\", \"%282pi%29%2F4\", \"%283pi%29%2F4\", \"4pi%2F4\" \r
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document.write( "which simplify to \r
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document.write( "x = 0, \"pi%2F4\", \"%28pi%29%2F42\", \"%283pi%29%2F4\", \"pi\"   \r
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document.write( "It has 
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document.write( "a peak at (0,2), 
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document.write( "an x-intercept at (\"pi%2F4\",0),
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document.write( "a valley at (\"%28pi%29%2F2\",-2),
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document.write( "an x-intercept at (\"%283pi%29%2F4\",0),
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document.write( "and a peak at (\"pi\",2)\r
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document.write( "This is the graph:\r
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document.write( "Now for y = -2sin(x)\r
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document.write( "A = -2, B = 1.  This graph has important points at \r
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document.write( "x = \"%280pi%29%2F2\", \"%281pi%29%2F2\", \"%282pi%29%2F2\", \"%283pi%29%2F2\", \"4pi%2F2\" \r
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document.write( "which simplify to \r
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document.write( "x = \"0\", \"%28pi%29%2F2\", \"pi\", \"%283pi%29%2F2\", \"2pi\"\r
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document.write( "but only the first three are within the interval 0 < x < \"pi\"  \r
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document.write( "Since A is negative, the peaks and valleys switch from the 
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document.write( "rules above and there are only these on the interval 0 < x < \"pi\"\r
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document.write( "an x-intercept at (0,0),
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document.write( "a valley at (\"%28pi%29%2F2\",-2),
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document.write( "an x-intercept at (\"pi\",0)\r
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document.write( "This is the graph (it only has one arch on this
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document.write( "interval 0 < x < \"pi\".\r
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document.write( "Now let's put them both on the same set of axes:\r
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document.write( "To find the solution of\r
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document.write( "2 cos 2x = -2 sin x\r
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document.write( "we just observe that the solution will be the x-coordinate of
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document.write( "the point where the two curves intersect.\r
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document.write( "Well it looks as though they intersect only at one point,
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document.write( "which is (\"pi%2F2\",-2), their common valley point.\r
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document.write( "So the solution in the interval 0 < x < \"2pi\" is x = \"pi%2F2\"\r
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document.write( "Edwin
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