document.write( "Question 766846: In ABC , a=22cam,b=19cm and ABC =55 degrees. Sketch the triangle and solve the triangle. Round answers to the nearest tenth. \n" ); document.write( "
Algebra.Com's Answer #467326 by Edwin McCravy(20056)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "This is a problem of the AMBIGUOUS case SSA, (side-side-angle) \r\n" );
document.write( "where you are asked to solve a triangle given two sides and a \r\n" );
document.write( "non-included angle.  If I have drawn this triangle accurately\r\n" );
document.write( "to scale, then it appears that there are two solutions to\r\n" );
document.write( "this problem.  If we open a compass to 19 units, put the sharp\r\n" );
document.write( "point of the compass at C and swing an arc, like this,\r\n" );
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document.write( "So it appears that there are two solutions, these two triangles:\r\n" );
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document.write( "We will use the law of sines to determine angle A\r\n" );
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document.write( "\"a%2Fsin%28A%29\"\"%22%22=%22%22\"\"b%2Fsin%28B%29\"\r\n" );
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document.write( "b·sin(A) = a·sin(B)\r\n" );
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document.write( "sin(A) = \"a%2Asin%28B%29%2Fb\"\r\n" );
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document.write( "sin(A) = \"%2822%2Asin%28%2255%B0%22%29%29%2F19\"\r\n" );
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document.write( "sin(A) = 0.948918408  <--- If this had been greater than 1, there\r\n" );
document.write( "                           would have been no solution.  If this had \r\n" );
document.write( "                           been exactly 1, there would have been 1 right\r\n" );
document.write( "                           triangle angle solution.  But since it is less \r\n" );
document.write( "                           than 1, we can tell that there is either 1 or 2\r\n" );
document.write( "                           solutions.  \r\n" );
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document.write( "If you use your calculator with the inverse sine,\r\n" );
document.write( "you get \r\n" );
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document.write( "A = 71.5303934°.\r\n" );
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document.write( "That is a correct value for angle A.  However there is another \r\n" );
document.write( "possible angle with that same sine, which is a second quadrant \r\n" );
document.write( "angle, and it is found by subtracting A = 71.5303934° from 180°.\r\n" );
document.write( "We will use primes on the letters to indicate the possible\r\n" );
document.write( "second solution: \r\n" );
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document.write( "A' = 180° - 71.5303934° = 108.4696066°\r\n" );
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document.write( "So we put those two values in:\r\n" );
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document.write( "First solution               Second solution (maybe)\r\n" );
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document.write( "A = 71.5303934°              A' = 108.4696066°\r\n" );
document.write( "B = 55°                      B' = 55°  \r\n" );
document.write( "C =                          C' = \r\n" );
document.write( "a = 22                       a' = 22  \r\n" );
document.write( "b = 19                       b' = 19  \r\n" );
document.write( "c =                          c' =  \r\n" );
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document.write( "To find out whether there are 2 solutions or only 1,\r\n" );
document.write( "we calculate the angles C and C', and see if both\r\n" );
document.write( "are possibilities:\r\n" );
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document.write( "We calculate C by using 180°-A-B = 180°-71.5303934°-55° = 53.4696066°\r\n" );
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document.write( "We calculate C' by using 180°-A'-B' = 180°-108.4696066°-55° = 16.5303934°\r\n" );
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document.write( "Since C' came out positive, we know that there are two solutions,\r\n" );
document.write( "[If C' had come out negative, there would have been only 1 solution].\r\n" );
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document.write( "First solution               Second solution\r\n" );
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document.write( "A = 71.5303934°              A' = 108.4696066°\r\n" );
document.write( "B = 55°                      B' = 55°  \r\n" );
document.write( "C = 53.4696066°              C' = 16.5303934° \r\n" );
document.write( "a = 22                       a' = 22  \r\n" );
document.write( "b = 19                       b' = 19  \r\n" );
document.write( "c =                          c' = \r\n" );
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document.write( "Now we just have to calculate sides c and c'\r\n" );
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document.write( "To calculate side c:\r\n" );
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document.write( "\"b%2Fsin%28B%29\" = \"c%2Fsin%28C%29\"\r\n" );
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document.write( "\"19%2Fsin%28%2255%B0%22%29\" = \"c%2Fsin%28%2253.4696066%B0%22%29\" \r\n" );
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document.write( "c·sin(55°) = 19·sin(53.4696066°)  \r\n" );
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document.write( "c = \"%2819%2Asin%28%2253.4696066%B0%22%29%29%2Fsin%28%2255%B0%22%29\"\r\n" );
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document.write( "c = 18.63791123\r\n" );
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document.write( "---\r\n" );
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document.write( "To calculate side c':\r\n" );
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document.write( "\"%22b%27%22%2Fsin%28%22B%27%22%29\" = \"%22c%27%22%2Fsin%28%22C%27%22%29\"\r\n" );
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document.write( "\"19%2Fsin%28%2255%B0%22%29\" = \"%22c%27%22%2Fsin%28%2216.5303934%B0%22%29\" \r\n" );
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document.write( "c'·sin(55°) = 19·sin(16.5303934°)  \r\n" );
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document.write( "c' = \"%2819%2Asin%28%2216.5303934%B0%B0%22%29%29%2Fsin%28%2255%B0%22%29\"\r\n" );
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document.write( "c' = 6.599451972\r\n" );
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document.write( "So we end up with:\r\n" );
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document.write( "First solution               Second solution \r\n" );
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document.write( "A = 71.5303934°              A' = 108.4696066°\r\n" );
document.write( "B = 55°                      B' = 55°  \r\n" );
document.write( "C = 53.4696066°              C' = 16.5303934° \r\n" );
document.write( "a = 22                       a' = 22  \r\n" );
document.write( "b = 19                       b' = 19  \r\n" );
document.write( "c = 18.63791123              c' = 6.599451972\r\n" );
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document.write( "Incidentally the drawings above are to scale,\r\n" );
document.write( "because I calculated them before drawing them.\r\n" );
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document.write( "You can round the solutions to tenths. \r\n" );
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document.write( "A = 71.5°                    A' = 108.5°\r\n" );
document.write( "B = 55°                      B' = 55°  \r\n" );
document.write( "C = 53.5°                    C' = 16.5° \r\n" );
document.write( "a = 22                       a' = 22  \r\n" );
document.write( "b = 19                       b' = 19  \r\n" );
document.write( "c = 18.6                     c' = 6.6\r\n" );
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document.write( "However, you should wait until the end to round off\r\n" );
document.write( "because if you round earlier, the answers will likely\r\n" );
document.write( "be too inaccurate. \r\n" );
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document.write( "Edwin
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