
Special Trigonometry Ratios for 30^{o},60^{o},45^{o} Angles  Math Teaching Help  Game Tips:  A 60 degree angle is special since it is found in every equilateral triangle. Splitting an equilateral triangle in half, yields two right triangles, each having 30, 60 & 90 degrees.  For each 30,60,90 right triangle, the sides have special ratios. Begin with hypotenuse length = 1 unit. The side opposite the 30^{o} angle is half the length of the side opposite the 90^{o} angle and the side opposite the 60^{o} angle can be found from x^{2} + y^{2} = r^{2} , namely (1/2)^{2} + y^{2} = (1)^{2} . So each 30,60,90 right triangle has sides in the special ratio 0.5 / 0.866 / 1.0 approx.  Basic Trigonometry uses the ratios of the 'lengths of sides' of triangles as related to 'angle size'.  A 45 degree angle is also special since it is found twice in every right isosceles triangle.  For each 45,45,90 right triangle, the sides have special ratios. Use hypotenuse r=1. A side opposite one 45^{o} angle can be found from x^{2} + y^{2} = r^{2} , namely x^{2} + x^{2} = (1)^{2} . So each 45,45,90 right triangle has sides in the special ratio 0.707 / 0.707 / 1.0 approx.  When angle DAE is in standard position at a graph's origin (0,0) then sin A equals (y/r), cos A equals (x/r), and tan A = (y/x).  When r=1 then the primary Trig ratios simplify to sin A = y, cos A = x, and tan A = (y/x).  It may take a minute to load the game at dialup speeds.  The game can be played using the mouse by itself or using the keyboard by itself.  If the game doesn't respond to keyboard input, click inside the game area to reset the game's focus.  Adjust the game's speed by pressing the + or  key repeatedly. 
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