TRIGONOMETRY - functions & angles in standard position

Question

Find the value of cos (angle) in (angle) is in standard position and the point (5,-12) is on the terminal side of (angle). 

b) -5/13

c) 5/13

The correct answer is C, but I chose B. Why is the answer negative? How am I supposed to know that the hypotenuse is negative if I am not given the Quadrant in which it is located?

Answer 1

The coordinate lies in the 4th quadrant.  Therefore, if we were to draw a diagonal in which the endpoints are the point (0, 0) and (5, -12), and draw a vertical line so that we get a right triangle, the diagonal will be located in the 4th quadrant.  cosine is positive in this quadrant.  However, terminal angles are angles between the diagonal and the positive x-axis.  So the terminal angle is between 270 degree and 360 degrees.

 

The diagonal is below the x-axis, so the terminal side is negative.

 

 

 

Also, the hypotenuse is never negative  Think of the diagonal as a vector.  This vector distance is always positive no matter its orientation.