Video Summary:
To define the six trigonometric functions, start with an angle in standard position. Choose any point P having coordinates (x,y) on the terminal side as seen in the figure below. Notice that r > 0 since distance is never negative.
The six trigonometric functions are sine, cosine, tangent, cotangent, secant, and cosecant. In the definitions of the trigonometric functions, the distance r is never negative, so r > 0.
Choose a point (x,y) in quadrant I, then both x and y will be positive, so the values of the six trigonometric functions will be positive in quadrant I.
A point (x,y) in quadrant II has x < 0 and y > 0. This makes sine and cosecant positive for quadrant II angles, while the other four functions take on negative values.
Similar results can be obtained for the other quadrants.