\r\n" );
document.write( "There are three cases for the law of cosines and 2 cases for\r\n" );
document.write( "the law of sines:\r\n" );
document.write( "\r\n" );
document.write( "Case 1:\r\n" );
document.write( "a² = b²+c²-2·b·c·cos(A) when ∠A and ∠B are both acute angles.\r\n" );
document.write( "
\r\n" );
document.write( "\r\n" );
document.write( "Draw altitude CD ⊥ AB. Label CD h for the height of ߡABC.\r\n" );
document.write( "Label the left part of c, AD as x and the right part of c DB as c-x.\r\n" );
document.write( "\r\n" );
document.write( "
\r\n" );
document.write( " \r\n" );
document.write( "ߡADC and ߡBDC are right triangles, so by the Pythagorean\r\n" );
document.write( "theorem,\r\n" );
document.write( "\r\n" );
document.write( "h² = b²-x² and also h² = a²-(c-x)²\r\n" );
document.write( "\r\n" );
document.write( "Therefore equate their right sides:\r\n" );
document.write( "\r\n" );
document.write( "b²-x² = a²-(c-x)²\r\n" );
document.write( "b²-x² = a²-(c²-2cx+x²)\r\n" );
document.write( "b²-x² = a²-c²+2cx-x²\r\n" );
document.write( "Add x² to both sides:\r\n" );
document.write( "b² = a²-c²+2cx\r\n" );
document.write( "\r\n" );
document.write( "From ߡADC, 
= cos(A) or x = b·cos(A)\r\n" );
document.write( "\r\n" );
document.write( "Substituting that for x:\r\n" );
document.write( "\r\n" );
document.write( "b² = a²-c²+2c·b·cos(A)\r\n" );
document.write( "\r\n" );
document.write( "Isolate a² on the right side:\r\n" );
document.write( "\r\n" );
document.write( "b² + c² - 2c·b·cos(A) = a²\r\n" );
document.write( "\r\n" );
document.write( "That is equivalent to\r\n" );
document.write( "\r\n" );
document.write( "a² = b²+c²-2·c·b·cos(A)\r\n" );
document.write( "\r\n" );
document.write( "or since c·b = b·c,\r\n" );
document.write( "\r\n" );
document.write( "a² = b²+c²-2·b·c·cos(A) \r\n" );
document.write( "\r\n" );
document.write( "-----------------------\r\n" );
document.write( "The law of sines for this case\r\n" );
document.write( "\r\n" );
document.write( "
= sin(A) and 
= sin(B)\r\n" );
document.write( "\r\n" );
document.write( "h = b·sin(A) and h =a·sin(B)\r\n" );
document.write( "\r\n" );
document.write( "So b·sin(A) = a·sin(B)\r\n" );
document.write( "\r\n" );
document.write( "Divide both sides by sin(A)sin(B)\r\n" );
document.write( "\r\n" );
document.write( " 
= 
\r\n" );
document.write( "\r\n" );
document.write( "That's not complete yet, for we haven't shown that those\r\n" );
document.write( "equal to 
but it will be when we finish \r\n" );
document.write( "the next case. \r\n" );
document.write( "\r\n" );
document.write( "--------------------------------------------\r\n" );
document.write( "\r\n" );
document.write( "a² = b²+c²-2·b·c·cos(A) when ∠A is acute and ∠B is obtuse.\r\n" );
document.write( "\r\n" );
document.write( "
\r\n" );
document.write( "\r\n" );
document.write( "Extend AB and draw CD ⊥ AB. Label CD h for the height of ߡABD.\r\n" );
document.write( "Label the left part of c, AD as x and the right part of c DB as c-x.\r\n" );
document.write( "\r\n" );
document.write( "
\r\n" );
document.write( " \r\n" );
document.write( "ߡADC and ߡBDC are right triangles, so by the Pythagorean\r\n" );
document.write( "theorem,\r\n" );
document.write( "\r\n" );
document.write( "h² = b²-(c+x)² and also h² = a²-x²\r\n" );
document.write( "\r\n" );
document.write( "Therefore equate their right sides:\r\n" );
document.write( "\r\n" );
document.write( "b²-(c+x)² = a²-x²\r\n" );
document.write( "b²-(c²+2cx+x²) = a²-x²\r\n" );
document.write( "b²-c²-2cx-x² = a²-x²\r\n" );
document.write( "Add x² to both sides:\r\n" );
document.write( "b²-c²-2cx = a²\r\n" );
document.write( "\r\n" );
document.write( "From ߡADC, 
= cos(A)\r\n" );
document.write( " c+x = b·cos(A)\r\n" );
document.write( " x = b·cos(A)-c\r\n" );
document.write( "\r\n" );
document.write( "Substituting in\r\n" );
document.write( "b²-c²-2cx = a²\r\n" );
document.write( "\r\n" );
document.write( " b²-c²-2c(b·cos(A)-c) = a²\r\n" );
document.write( "b²-c²-2·c·b·cos(A)+2c² = a²\r\n" );
document.write( " b²+c²-2·c·b·cos(A) = a²\r\n" );
document.write( "\r\n" );
document.write( "That is equivalent to\r\n" );
document.write( "\r\n" );
document.write( "a² = b²+c²-2·c·b·cos(A)\r\n" );
document.write( "\r\n" );
document.write( "or since c·b = b·c,\r\n" );
document.write( "\r\n" );
document.write( "a² = b²+c²-2·b·c·cos(A)\r\n" );
document.write( " \r\n" );
document.write( "---------------------------\r\n" );
document.write( "The law of sines for this case\r\n" );
document.write( "\r\n" );
document.write( " = sin(A) and = sin(∠CBD)\r\n" );
document.write( "\r\n" );
document.write( "h = b·sin(A) and h =a·sin(∠CBD)\r\n" );
document.write( "\r\n" );
document.write( "So b·sin(A) = a·sin(∠CBD)\r\n" );
document.write( "\r\n" );
document.write( "Divide both sides by sin(A)sin(∠CBD)\r\n" );
document.write( "\r\n" );
document.write( " 
= \r\n" );
document.write( "\r\n" );
document.write( "Now we use the fact that the sine of an angle \r\n" );
document.write( "is equal to the sine of its supplement. Therefore\r\n" );
document.write( "if we erase the extended part we can label ∠CBD as ∠B.\r\n" );
document.write( "\r\n" );
document.write( " = \r\n" );
document.write( "\r\n" );
document.write( "The law of sines is now complete for the first case \r\n" );
document.write( "was for two angles being acute, and this case is for \r\n" );
document.write( "when one angle is obtuse.\r\n" );
document.write( "Since the other two angles are necessarily acute, \r\n" );
document.write( "the first case takes care of them and we have the\r\n" );
document.write( "complete law of sines:\r\n" );
document.write( "\r\n" );
document.write( " = = \r\n" );
document.write( " \r\n" );
document.write( "------------------------------------\r\n" );
document.write( "\r\n" );
document.write( "To prove the law of cosines when A is an obtuse angle,\r\n" );
document.write( "we have to accept the definition from xy-plane \r\n" );
document.write( "trigonometry that cos(180°-A) = -cos(A).\r\n" );
document.write( "\r\n" );
document.write( "We'll just take the above triangle and swap angles \r\n" );
document.write( "A and B and sides a and b:\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "
\r\n" );
document.write( "\r\n" );
document.write( "Extend BA and draw CD ⊥ BD. Label CD h for the height of ߡABC.\r\n" );
document.write( "Label the extended segment, AD, as x.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( " \r\n" );
document.write( "ߡADC and ߡBDC are right triangles, so by the Pythagorean\r\n" );
document.write( "theorem,\r\n" );
document.write( "\r\n" );
document.write( "h² = b²-x² and h² = a²-(c+x)²\r\n" );
document.write( "\r\n" );
document.write( "Therefore equate their right sides:\r\n" );
document.write( "\r\n" );
document.write( "b²-x² = a²-(c+x)²\r\n" );
document.write( "b²-x² = a²-(c²+2cx+x²)\r\n" );
document.write( "b²-x² = a²-c²-2cx-x²\r\n" );
document.write( "Add x² to both sides:\r\n" );
document.write( " b² = a²-c²-2cx\r\n" );
document.write( "\r\n" );
document.write( "From ߡADC, = cos(CAD),\r\n" );
document.write( " x = b·cos(CAD)\r\n" );
document.write( "\r\n" );
document.write( "and since ∠CAD and ∠BAC are supplementary,\r\n" );
document.write( "\r\n" );
document.write( "cos(∠CAD) = -cos(∠BAC)\r\n" );
document.write( "and\r\n" );
document.write( " x = -b·cos(∠BAC)\r\n" );
document.write( "\r\n" );
document.write( " b² = a²-c²-2cx\r\n" );
document.write( "\r\n" );
document.write( "becomes:\r\n" );
document.write( "\r\n" );
document.write( " b² = a²-c²-2c[-b·cos(∠BAC)]\r\n" );
document.write( "\r\n" );
document.write( " b² = a²-c²+2c·b·cos(∠BAC)\r\n" );
document.write( "\r\n" );
document.write( "Isolate a² on the right\r\n" );
document.write( "\r\n" );
document.write( " b²+c²-2c·b·cos(∠BAC) = a²\r\n" );
document.write( "\r\n" );
document.write( "which is equivalent to\r\n" );
document.write( " \r\n" );
document.write( "a² = b²+c²-2c·b·cos(∠BAC)\r\n" );
document.write( "\r\n" );
document.write( "or since c·b = b·c,\r\n" );
document.write( "\r\n" );
document.write( "a² = b²+c²-2·b·c·cos(∠BAC)\r\n" );
document.write( "\r\n" );
document.write( "and if we erase the extended \r\n" );
document.write( "segment x, we can write ∠BAC as\r\n" );
document.write( "∠A and have\r\n" );
document.write( "\r\n" );
document.write( "a² = b²+c²-2·b·c·cos(A)\r\n" );
document.write( "\r\n" );
document.write( "---------------------------\r\n" );
document.write( "Edwin
\n" );
document.write( " \n" );
document.write( "![\"\"](\"/images/stars/stars-12.gif\")
