I have the following setup:

An iPhone lies with the display to the ceiling on a table (alpha = 0 degrees). When the iPhone is moved upwards like shown in the image above the alpha angle increases.
How do I compute the value of the alpha angle without taking care of any other axes which could change. I am only interested in this one axis.
How do I get the correct alpha angle the iPhone has when lifting up from the table? How do I get notified when the value of alpha changes?
You can use the CMMotionManager class to monitor device motion changes.
Objective C
// Ensure to keep a strong reference to the motion manager otherwise you won't get updates
self.motionManager = [[CMMotionManager alloc] init];
if (self.motionManager.deviceMotionAvailable) {
    
    self.motionManager.deviceMotionUpdateInterval = 0.1;
    
    // For use in the montionManager's handler to prevent strong reference cycle
    __weak typeof(self) weakSelf = self;
    NSOperationQueue *queue = [[NSOperationQueue alloc] init];
    [self.motionManager startDeviceMotionUpdatesToQueue:queue
                                            withHandler:^(CMDeviceMotion *motion, NSError *error) {
                                                
                                                // Get the attitude of the device
                                                CMAttitude *attitude = motion.attitude;
      
                                                // Get the pitch (in radians) and convert to degrees.                                          
                                                NSLog(@"%f", attitude.pitch * 180.0/M_PI);
                                                dispatch_async(dispatch_get_main_queue(), ^{
                                                    // Update some UI
                                                });
                                                
                                            }];
    
    NSLog(@"Device motion started");
}else {
    NSLog(@"Device motion unavailable");
}
Swift
// Ensure to keep a strong reference to the motion manager otherwise you won't get updates
self.motionManager = CMMotionManager()
if motionManager?.deviceMotionAvailable == true {
        
    motionManager?.deviceMotionUpdateInterval = 0.1
        
    let queue = OperationQueue()
    motionManager?.startDeviceMotionUpdatesToQueue(queue, withHandler: { [weak self] motion, error in
            
        // Get the attitude of the device
        if let attitude = motion?.attitude {
            // Get the pitch (in radians) and convert to degrees.
            print(attitude.pitch * 180.0/Double.pi)
            DispatchQueue.main.async {
                // Update some UI
            }
        }
            
    })
        
    print("Device motion started")
}else {
    print("Device motion unavailable")
}
NSHipster is (as always) a great source of information, and the article on CMDeviceMotion is no exception.
Swift 4:
  let motionManager = CMMotionManager()
   if motionManager.isDeviceMotionAvailable {
       motionManager.deviceMotionUpdateInterval = 0.1
       motionManager.startDeviceMotionUpdates(to: OperationQueue()) { [weak self] (motion, error) -> Void in
           if let attitude = motion?.attitude {
                print(attitude.pitch * 180 / Double.pi)
                DispatchQueue.main.async{
                    // Update UI
               }
           }
       }
       print("Device motion started")
    }
   else {
       print("Device motion unavailable")
    }
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