
Trigonometry Ratios for Angles based on 0^{o},30^{o},45^{o},60^{o},90^{o}  Game Tips:  The four quadrants of a Math graph resemble the 4 regions of a map. Starting from the centre, the NorthEast region is quadrant 1. The NorthWest region is quadrant 2. The SouthWest region is quadrant 3, and the SouthEast region is quadrant 4.  Positive angles in standard position at the origin (0,0) are formed using an initial arm always pointing East and rotating the second arm of the angle counterclockwise to a terminal position. Angles that rotate clockwise are negative angles in Mathematics.  When the terminal arm goes from (0,0) to (0,1) then the angle is 90^{o}.  When the terminal arm goes from (0,0) to (1,0) then the angle is 180^{o}.  When the terminal arm goes from (0,0) to (0,1) then the angle is 270^{o}.  When the terminal arm goes from (0,0) to (1,0) then the angle is 0^{o} or 360^{o}.  The special angles based on 60^{o}, are 120^{o}, 240^{o}, and 300^{o}.  When the terminal arm goes from (0,0) to (0.5,0.866) then the angle is 60^{o}.  When the terminal arm goes from (0,0) to (0.5,0.866) then the angle is 120^{o}.  When the terminal arm goes from (0,0) to (0.5,0.866) then the angle is 240^{o}.  When the terminal arm goes from (0,0) to (0.5,0.866) then the angle is 300^{o}.  The special angles based on 30^{o}, are 150^{o}, 210^{o}, and 330^{o}.  When the terminal arm goes from (0,0) to (0.866,0.5) then the angle is 30^{o}.  When the terminal arm goes from (0,0) to (0.866,0.5) then the angle is 150^{o}.  When the terminal arm goes from (0,0) to (0.866,0.5) then the angle is 210^{o}.  When the terminal arm goes from (0,0) to (0.866,0.5) then the angle is 330^{o}.  The special angles based on 45^{o}, are 135^{o}, 225^{o}, and 315^{o}.  When the terminal arm goes from (0,0) to (0.707,0.707) then the angle is 45^{o}.  When the terminal arm goes from (0,0) to (0.707,0.707) then the angle is 135^{o}.  When the terminal arm goes from (0,0) to (0.707,0.707) then the angle is 225^{o}.  When the terminal arm goes from (0,0) to (0.707,0.707) then the angle is 315^{o}.  Notice 30,60,90 right triangles have sides in the special ratio 0.5 / 0.866 / 1.0 approx.  Notice 45,45,90 right triangles have sides in the special ratio 0.707 / 0.707 / 1.0 approx.  When angle EAD 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).  The CAST Rule is a memory aid to help remember that the Cosine ratio is positive in the 4th quadrant, All the primary Trig ratios are positive in the first quadrant, the Sine ratio is positive in the 2nd quadrant, and the Tangent ratio is positive in the 3rd quadrant.  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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