Question

1)The x- and y-components of a velocity vector of 100 N at 240 degrees would be?

2) A man pushes a lawn mower across the lawn exerting 5 N of force along the handle at an angle of 25 degrees with the ground. What is the x-component of the force he exerts?

Answer 1

1) The answer depends upon what reference direction the 240 degrees is measured from, and whether + means couterclockwise or counterclowise. In mathematics and physics, polar direction angles are measured counterclockwise from +x. I assume you are dealing with a Force (F) vector and that is how the angle is measured.
Then the vector is in the third quadrant and

Fx = 100N cos 240 = -100N cos 60 = -50N
Fy = 100N sin 240 = -100N sin 60 = ?

2) 5N cos 25