document.write( "Question 149726: In triangle ABC A=35 degrees, a=43 and c=20. Determine whether triangle ABC has no solution, one solution, or two solutions. Then solve the triangle. Round to the nearest tenth. \n" ); document.write( "
Algebra.Com's Answer #109957 by Edwin McCravy(20056)\"\" \"About 
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In triangle ABC A=35 degrees, a=43 and c=20. Determine whether triangle ABC has no solution, one solution, or two solutions. Then solve the triangle. Round to the nearest tenth.
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document.write( "In solving the ambiguous case, \"Angle-Side-Side\", \"ASS\", \r\n" );
document.write( "we use the law of sines in this fashion:\r\n" );
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document.write( "Solve for \"sin%28ANGLE_OPPOSITE_OTHER_GIVEN_SIDE%29\"\r\n" );
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document.write( "1. If this is greater than 1 there are no solutions.\r\n" );
document.write( "2. If this is less than 1, we have either one or two solutions \r\n" );
document.write( "3. In the rare case this equals exactly 1, the\r\n" );
document.write( "   \"ANGLE_OPPOSITE_THE_OTHER_GIVEN_SIDE\" is 90°, and you \r\n" );
document.write( "   solve it as a right triangle.\r\n" );
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document.write( "In case 2, \r\n" );
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document.write( "I. use the inverse sine button on your calculator to find one \r\n" );
document.write( "certain possibility for the \"ANGLE_OPPOSITE_THE_OTHER_GIVEN_SIDE\".\r\n" );
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document.write( "II. Find the supplement of the angle found in step I, by subtracting\r\n" );
document.write( "    from 180°.\r\n" );
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document.write( "III. Add the result of step II to the given angle and subtract from\r\n" );
document.write( "     180°.\r\n" );
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document.write( "IV.  If the result of III is negative there is only one solution. If it\r\n" );
document.write( "     is positive then there are two solutions, one with the result of II\r\n" );
document.write( "     and one with the result of III for the \"ANGLE_OPPOSITE_THE_OTHER_GIVEN_SIDE\".\r\n" );
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document.write( "In your problem we use the law of sines in this fashion:\r\n" );
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document.write( "\"%28GIVEN_OPPOSITE_SIDE_OF_GIVEN_ANGLE%29=a=43\"\r\n" );
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document.write( "\"%28GIVEN_ANGLE%29=35\"°\r\n" );
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document.write( "\"%28OTHER_GIVEN_SIDE%29=c=20\"\r\n" );
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document.write( "\"%28ANGLE_OPPOSITE_OTHER_GIVEN_SIDE%29\" is the unknown angle \"C\"\r\n" );
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document.write( "\"a%2Fsin%28A%29%2Fsin%28C%29=c%2Fsin%28C%29\"\r\n" );
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document.write( "\"43%2Fsin%2835%29%2Fsin%28C%29=20%2Fsin%28C%29\"\r\n" );
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document.write( "Solve for \"sin%28C%29\"\r\n" );
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document.write( "\"sin%28C%29+=+20sin%2835%29%2F43+=+0.2667797378\"\r\n" );
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document.write( "This is less than 1, so we have case 2 above, and\r\n" );
document.write( "there is either one or two solutions. So\r\n" );
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document.write( "I. We use the inverse sine button on the calculator to find one \r\n" );
document.write( "certain solution for the angle C.\r\n" );
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document.write( "angle C = 15.47273197°\r\n" );
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document.write( "II. Find the supplement of that angle, by subtracting from 180°:\r\n" );
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document.write( "180° - 15.47273197° = 164.527268°\r\n" );
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document.write( "III. Add the result of step II to the given angle and subtract from\r\n" );
document.write( "     180°.\r\n" );
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document.write( "164.527268° + 35° = 199.527268°\r\n" );
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document.write( "180° - 199.527268° = -19.52726803°.\r\n" );
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document.write( "IV.  This result is negative, so there is only one solution. \r\n" );
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document.write( "Now to find the other parts.  We can find angle B by adding\r\n" );
document.write( "the given angle A = 35° to Angle C = 15.47273197°, and then\r\n" );
document.write( "subtracting from 180° to find Angle B:\r\n" );
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document.write( "35° + 15.47273197° = 50.47273197°\r\n" );
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document.write( "180° - 50.47273197° = 129.527268° = Angle B\r\n" );
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document.write( "Using the law of sines again to find side \"b\"\r\n" );
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document.write( "\"b%2Fsin%28B%29=a%2Fsin%28A%29\"\r\n" );
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document.write( "\"b=%28a%2Asin%28B%29%29%2Fsin%28A%29+=+%2843sin%28129.527268%29%29%2Fsin%2835%29=57.82461456\"\r\n" );
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document.write( "So we round everything to the nearest tenth:\r\n" );
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document.write( "angle C = 15.5°, angle B = 129.5°, side b = 57.8\r\n" );
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document.write( "Edwin

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