Given the point (3,-3) on the terminal side of an angle, what is the exact value of CSC Θ

Answer 1

sqr2

cscx = 1/sinx = 1/(-3/3sqr2) = sqr2

Answer 2

Draw the ray form the origin to the point (3,-3) to create the angle between the x-axis and the ray, theta. Construct a line segment from the point (3,-3) which is perpendicular to the x-axis (intersecting at (3,0). The hypotenuse of the resulting triangle is 3 * root 2 (by Pythagoras), so sin [opposite over hypotenuse] of theta is 3/(3 * root 2). Csc is the inverse of sin, so the answer is root 2. Theta is negative pi/4 (or more generally, 2kpi+7 pi/4) where k is an integer.