To measure the height of the cloud cover at an airport, a worker shines a spotlight upward at an angle 75° from the horizontal. an observer at a distance d = 590 m away measures the angle of elevation to the spot of light to be 45°. find the height h of the cloud cover,

The angle from the Spotlight is 75º, the angle of the elevation of the spotlight is 45º, so the 3rd angle is 60º(180-75-45 =60).
Use the law of sines to find the distance "s" from the observer to the cloud:

515/sin(60) = s/sin(75)
s = 515*sin(75)/sin(60)
s = 574.41 meters
Then, the height is s*sin(45)
s = 406.17 meters