document.write( "Question 524255: solve cos(56°-3x)=-0.79 in the range 0 < x < 360° \n" ); document.write( "
Algebra.Com's Answer #347695 by Edwin McCravy(20056)\"\" \"About 
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document.write( "cos(56°-3x) = -0.79 in the range 0 < x < 360°\r\n" );
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document.write( "We use the inverse cosine on a calculator with |-0.79| or +0.79\r\n" );
document.write( "to find the reference angle.  \r\n" );
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document.write( "(Caution: don't use the given negative value, as it will not give \r\n" );
document.write( "you the reference angle, which you need to get values between 0° \r\n" );
document.write( "and 360°.  When you study principle values of inverse trig functions\r\n" );
document.write( "you will understand why you should not use negative values.) \r\n" );
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document.write( "The reference angle for 56°-3x is 37.81448851°\r\n" );
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document.write( "There are two angles between 0° and 360° which have this reference angle.\r\n" );
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document.write( "One is in the 2nd quadrant and one is in the 3rd quadrant, since those\r\n" );
document.write( "are the two quadrants in which the cosine is negative. \r\n" );
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document.write( "To get the angle in the 2nd quadrant, subtract 37.81448851° from 180°,\r\n" );
document.write( "getting 142.1855115°.  We can add any positive or negative or zero\r\n" );
document.write( "multiple of 360° to this value and we will still have an angle whose\r\n" );
document.write( "cosine is -0.79.  So all such angles are 37.81448851°+360°k, where\r\n" );
document.write( "k is any integer. \r\n" );
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document.write( "To get the angle in the 3rd quadrant, add 37.81448851° to 180°,\r\n" );
document.write( "getting 217.8144885°.  Also we can add any positive or negative or zero\r\n" );
document.write( "multiple of 360° to this value and we will still have an angle whose\r\n" );
document.write( "cosine is -0.79.  So all such angles are 217.8144885°+360°k, where\r\n" );
document.write( "k is any integer. \r\n" );
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document.write( "So we have two equations:\r\n" );
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document.write( "56°-3x = 142.1855115°+360°k     and   56°-3x = 217.8144885°+360°k\r\n" );
document.write( "   -3x = 86.18551149°+360°k              -3x = 161.8144885°+360°k  \r\n" );
document.write( "     x = -28.72850383°-120°k               x = -53.93816284-120°k\r\n" );
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document.write( "To get the answers between 0° and 360°, we have these inequalities:\r\n" );
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document.write( "0° < -28.72850383°-120°k < 360°       0° < -53.93816284-120°k < 360°\r\n" );
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document.write( "Taking the first one. We first\r\n" );
document.write( "solve this inequality:\r\n" );
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document.write( "28.72850383° < -120°k < 388.7285038°\r\n" );
document.write( "-.2394041986 > k > -3.239404198\r\n" );
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document.write( "So we take k = -3,-2, and -1 as the only\r\n" );
document.write( "integers between those decimal values:\r\n" );
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document.write( "Therefore\r\n" );
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document.write( "using k=-3\r\n" );
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document.write( "     x = -28.72850383°-120°(-3)\r\n" );
document.write( "     x = 331.2714962°\r\n" );
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document.write( "using k=-2\r\n" );
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document.write( "     x = -28.72850383°-120°(-2)\r\n" );
document.write( "     x = 211.2714962°\r\n" );
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document.write( "using k=-1\r\n" );
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document.write( "     x = -28.72850383°-120°(-1)\r\n" );
document.write( "     x = 91.27149617°\r\n" );
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document.write( "Now we take the other equation:\r\n" );
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document.write( "x = -53.93816284-120°k\r\n" );
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document.write( "First solve the inequality for k:\r\n" );
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document.write( "0° < -53.93816284-120°k < 360°\r\n" );
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document.write( "53.93816284° < -120°k < 413.9381628°\r\n" );
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document.write( "-.4494846903 > k > -3.44948469 \r\n" );
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document.write( "So we take k = -3,-2, and -1 as the only\r\n" );
document.write( "integers between those decimal values:\r\n" );
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document.write( "x = -53.93816284°-120°k\r\n" );
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document.write( "Using k=-3,\r\n" );
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document.write( "x = -53.93816284°-120°(-3)\r\n" );
document.write( "x = 306.0618372°\r\n" );
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document.write( "Using k=-2,\r\n" );
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document.write( "x = -53.93816284°-120°(-2)\r\n" );
document.write( "x = 186.0618372°\r\n" );
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document.write( "Using k=-1,\r\n" );
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document.write( "x = -53.93816284°-120°(-1)\r\n" );
document.write( "x = 66.0618372°\r\n" );
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document.write( "So there are six solutions between 0° and 360°:\r\n" );
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document.write( "66.06183716°\r\n" );
document.write( "91.27149617°\r\n" );
document.write( "186.0618372°\r\n" );
document.write( "211.2714962°\r\n" );
document.write( "306.0618372°\r\n" );
document.write( "331.2714962°   \r\n" );
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document.write( "Edwin
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