The angular velocity of the Earth around its axis is 0.000417 degrees per second. The angular velocity of the Earth around the sun is 0.986 degrees per day.
The Earth can be thought of as simultaneously rotating, or turning, on its axis throughout the cycle of a 24-hour day and rotating around the sun once each year, creating the seasons. Therefore, this question has two answers depending upon which rotation of the Earth is considered. These values can be calculated with a high degree of accuracy because the Earth's velocity around its axis and around the sun are both known constants.