find the angle between two vectors and check if the difference is between 45 degrees

I have an object in the middle of screen that have a field of view of 90 degrees. It keeps rotating every frame about its origin.

There is a target object with random (x,y) vector. I want this target object to change to red color when it is inside the field of view of the object at the center of screen.

If you see the image above, the target is inside of field of view, then it must be changed to red color. But unfortunately I could not make this happen.

I have tried two formulas to achieve this:

1. Finding the arccos of the angle
θ = cos-1 [ (a. b) / (|a| |b|) ]

2. Finding the atang of the angle:
θ = atan2(a.y - b.y, a.x - b.x)

Here is the code of number 1:

let dotProduct = p5.Vector.dot(target, centerView)
let productMag = target.mag() * centerView.mag();
let result = dotProduct / productMag;
let angleInRad = acos(result);
if (angleInRad > HALF_FOV) {
  // Change target color to red
  fill(color(255, 0, 0))
}

Here is the code of number 2:

let diff = p5.Vector.sub(target, centerView);
let theta = atan2(diff.y, diff.x)
let result = theta
if (result < HALF_FOV && result > HALF_FOV * -1) {
  // Change target color to red
  fill(color(255, 0, 0))
}

When I run these codes, the target turns to red color, but it never changes when it is inside the field of view.

You can check and run my current code (p5.js) here.

Any ideas? Thank you

------UPDATE-------

Thanks to @MBo that show me how to fix this.
Here is the working example if someone want to check

OK, found error - you erroneously mixed points and vectors. Working variant :

 let targetvec = createVector(width - width / 4 - origin.x, height / 2 - origin.y)
  let centerViewvec = createVector(centerView.x - origin.x, centerView.y - origin.y)
  
   let dotProduct = p5.Vector.dot(targetvec, centerViewvec)
   let productMag = targetvec.mag() * centerViewvec.mag();
   let result = dotProduct / productMag;
   let angleInRad = acos(result);
  //  console.log(angleInRad)
   if (angleInRad < HALF_FOV) {
       fill(color(255, 0, 0))
   }

Note that it would better to store unit direction vectors if you work with directions a lot.