Question

This piece of code has been taken from a game built with XNA framework. I’d like some explanation of how it works in terms of trig and physics.

ball.velocity = new
Vector2((float)Math.Cos(cannon.rotation),
(float)Math.Sin(cannon.rotation));

ball.rotation is the rotation of a sprite in what i should think, radians.

Why is it that they can use the angle in radians only to find the x position then the same thing to find the y position of a direction of where the hypotenuse is pointing.

Reason why I asked this. I would like to get a feel of how this frameworks does calculations for trig. I am trying to get a sprite to turn in the direction of where the mouse is, that is: x and y is known, i just need the angle.

So there are 2 questions here. explaining that code above and pointing a sprite in the direction of a known point.

Update:

I found out that the point a which the object is at is not (0,0) because xna uses inverse coordinate system. So now the variables I have are these:

point of object.
point of mouse.

Answer 1

Every angle corresponds to a point on the unit circle (the unit circle is the unique circle centered at the origin with radius one; that is, the unit circle is the set of points satisfying x^2 + y^2 = 1). The correspondence is the following: given an angle theta, theta corresponds to the point (cos theta, sin theta). Why does (cos theta, sin theta) live on the unit circle? Because of everyone’s favorite identity

cos^2 theta + sin^2 theta = 1.

That is with x = cos theta and y = sin theta, the point (x, y) satisfies x^2 + y^2 = 1 so that (x, y) is on the unit circle.

To reverse this, given a point on the unit circle you can find the angle by using the inverse tangent (perhaps known to you as arctan or atan and sometimes tan-1). Precisely, given (x, y) on the unit circle you can find the angle corresponding to (x, y) by computing theta = arctan(y / x).

Of course, there are some messy details here. The function arctan can’t tell the difference between the inputs (x, y) and (-x, -y) because y / x and (-y / -x) have the same sign. Further, arctan can’t handle inputs where x = 0. So we typically handle these by defining the function atan2 that will handle these messy details for us

atan2(y, x) = arctan(y / x)       if x > 0
            = pi + arctan(y / x)  if y >= 0, x < 0
            = -pi + arctan(y / x) if y < 0, x < 0
            = pi / 2              if y > 0, x = 0
            = -pi / 2             if y < 0, x = 0
            = NaN                 if y = 0, x = 0

In C#, Math.Atan is the function arctan that I have referred to above, and Math.Atan2 is the function atan2 that I have referred to above.