document.write( "Question 255679: how do i do the 45 45 90 and the 30 60 90 triangles \n" ); document.write( "
Algebra.Com's Answer #187933 by Edwin McCravy(20056)\"\" \"About 
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how do i do the 45 45 90 and the 30 60 90 triangles
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document.write( "For the 45°-45°-90° right triangle\r\n" );
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document.write( "Start with a 1x1 square,\r\n" );
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document.write( "It has four right, or 90° angles.\r\n" );
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document.write( "Now cut it into two congruent right triangles by drawing\r\n" );
document.write( "a diagonal from the lower left to the upper right, like this:\r\n" );
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document.write( "That diagonal bisects the right angles on the lower left\r\n" );
document.write( "and upper right corners, each into two 45° angles. Now\r\n" );
document.write( "take away the top triangle, leaving only the bottom half:\r\n" );
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document.write( "Now you have a 45°-45°-90° right triangle.\r\n" );
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document.write( "Use the Pythagorean theorem to calculate its hypotenuse:\r\n" );
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document.write( "\"c%5E2=a%5E2%2Bb%5E2\"\r\n" );
document.write( "\"c%5E2=1%5E2%2B1%5E2\"\r\n" );
document.write( "\"c%5E2=1%2B1\"\r\n" );
document.write( "\"c%5E2=2\"\r\n" );
document.write( "\"c=sqrt%282%29\"\r\n" );
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document.write( "So the length of the hypotenuse is \"sqrt%282%29\"\r\n" );
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document.write( "Now memorize the way this right triangle looks and the \r\n" );
document.write( "lengths of the three sides.  For you will need it to find \r\n" );
document.write( "the exact values for the sine, cosine, and tangent of 45°:\r\n" );
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document.write( "\"sin%28%2245%B0%22%29=%28opposite%29%2F%28hypotenuse%29=1%2Fsqrt%282%29=sqrt%282%29%2F2\", (after rationalizing the denominator)\r\n" );
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document.write( "\"cos%28%2245%B0%22%29=%28adjacent%29%2F%28hypotenuse%29=1%2Fsqrt%282%29=sqrt%282%29%2F2\", (after rationalizing the denominator)\r\n" );
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document.write( "[Notice that the sine and cosine of 45° are the same!]\r\n" );
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document.write( "\"tan%28%2245%B0%22%29=%28opposite%29%2F%28adjacent%29=1%2F1=1\".\r\n" );
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document.write( "For the 30°-60°-90° right triangle\r\n" );
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document.write( "Start with an equilateral triangle, each side of\r\n" );
document.write( "which has length 2,\r\n" );
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document.write( "It has three 60° angles.\r\n" );
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document.write( "Now cut it into two congruent triangles by drawing a median, which\r\n" );
document.write( "is also an altitude as well as a bisector of the upper 60°-vertex angle:\r\n" );
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document.write( "That bisects the upper 60° angle into two 30° angles. It also\r\n" );
document.write( "bisects the base of the equilateral into two parts, each with\r\n" );
document.write( "length 1. Now take away the triangle on the right, leaving only \r\n" );
document.write( "the one on the left:\r\n" );
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document.write( "Now you have a 30°-60°-90° right triangle.\r\n" );
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document.write( "Use the Pythagorean theorem to calculate its altitude:\r\n" );
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document.write( "\"c%5E2=a%5E2%2Bb%5E2\"\r\n" );
document.write( "\"2%5E2=1%5E2%2Bb%5E2\"\r\n" );
document.write( "\"4=1%2Bb%5E2\"\r\n" );
document.write( "\"3=b%5E2\"\r\n" );
document.write( "\"sqrt%283%29=b\"\r\n" );
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document.write( "So the length of the altitude is \"sqrt%283%29\"\r\n" );
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document.write( "  \r\n" );
document.write( "Now memorize the way this right triangle looks and the lengths \r\n" );
document.write( "of the three sides.  For you will need it to find the exact\r\n" );
document.write( "values for the sine, cosine, and tangent of BOTH 60° AND 30°:\r\n" );
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document.write( "\"sin%28%2260%B0%22%29=%28opposite%29%2F%28hypotenuse%29=sqrt%283%29%2F2\",\r\n" );
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document.write( "\"cos%28%2260%B0%22%29=%28adjacent%29%2F%28hypotenuse%29=1%2F2\", \r\n" );
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document.write( "\"tan%28%2260%B0%22%29=%28opposite%29%2F%28adjacent%29=sqrt%283%29%2F1=sqrt%283%29\".\r\n" );
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document.write( "\"sin%28%2230%B0%22%29=%28opposite%29%2F%28hypotenuse%29=1%2F2\",\r\n" );
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document.write( "\"cos%28%2230%B0%22%29=%28adjacent%29%2F%28hypotenuse%29=sqrt%283%29%2F2\", \r\n" );
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document.write( "\"tan%28%2230%B0%22%29=%28opposite%29%2F%28adjacent%29=+1%2Fsqrt%283%29=sqrt%283%29%2F3\",\r\n" );
document.write( "after rationalizing\r\n" );
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document.write( "[Notice that the sine of 60° and the cosine of 30° are the same,\r\n" );
document.write( "and also that the sine of 30° and the cosine of 60° are the same!]\r\n" );
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document.write( "Edwin
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