The clear difference between positive and negative rake angles is that for the positive rake angle,the leading edge of the cutting tool is behind the perpendicular of the material to be cut.While the tool leading edge is ahead or beyond the perpendicular for the negative rake angle.
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The cosine of an angle is the adjacent side of the angle of a triangle divided the hypotenuse. If you plot the adjacent side as x on an x -y graph, for negative angles less than 90 degrees the adjacent side is positive and the hypotenuse is always positive, so you get a positive. The cosine is positive int e upper right and lower right quadrants
Tangent and cotangent positive; other 4 negative.
Certainly. A negative angle is simply clockwise instead of counterclockwise from the zero angle, but this is usually not indicated as such on a drawing.
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It is a reflex angle because it is greater than 180 degrees.