Quadrant determines the functions sign?

[Keyser Söze]

Hello! In my book we have defined trigonometric functions for all angles. Everything is pretty much the same as when we were using acute angles (same ratio of sides), except that in a quadrant one trigonometric function might be positive, another might be negative, so on. The book has a table for the six functions (sine, cosine, and tangent and their reciprocals) and what their signs are in each quadrant. What makes the signs change? It's much easier for me to remember something if I understand it. Thanks.

 

[Defennder]

Have you learned the ASTC quadrants mnemonic rule? That makes it a lot easier to remember the signs.

 

[nicksauce]

Ok, suppose I draw a vector (ie a line with some length L, making an angle A with the positive x-axis). The x-value lining up with the tip of the vector must be L*cos(A) and the y-value lining up with the tip of the vector must be L*sin(A). Try to draw a picture if you can't see it in your head. (Someone correct me if I'm wrong here) These can be taken as the definitions of the sine and cosine. Therefore for consistency, we require that the sine be positive in the first and second quadrants (where y is positive) and negative in the third and fourth (where y is negative), and that the cosine be positive in the first and fourth quadrants (where x is positive) and negative in the second and third (where x is negative). The signs of the other functions follow from this as they are all defined in terms of the sine and cosine.Cheers,
Nick