The coordinate angle formula originated from the need to describe and measure angles in a systematic way. It is a fundamental concept in trigonometry, which is a branch of mathematics that deals with the relationships between the lengths of the sides and the angles of triangles. The formula itself is based on the unit circle, a circle with a radius of one, where the coordinates of any point on the circle can be represented by (cos θ, sin θ), where θ is the angle between the positive x-axis and the line connecting the origin to the point.

The origin of the coordinate angle formula can be traced back to the ancient Greeks, who were the first to develop trigonometry. The Pythagoreans, in particular, were interested in the properties of right triangles and how to calculate their sides and angles. They used a unit circle to define the trigonometric functions, which are the sine, cosine, and tangent of an angle. These functions relate the angle to the coordinates of a point on the unit circle.

The coordinate angle formula was further developed by mathematicians such as Ptolemy and Archimedes. Ptolemy, in his Almagest, provided a comprehensive treatise on trigonometry that included the use of the unit circle and the coordinate angle formula. Archimedes, on the other hand, used geometric methods to approximate the values of trigonometric functions, which laid the groundwork for the development of calculus.

The coordinate angle formula became more refined and precise with the advent of calculus in the 17th century. Mathematicians like Isaac Newton and Gottfried Wilhelm Leibniz used calculus to define the trigonometric functions in terms of integrals and derivatives, which provided a more rigorous foundation for the formula. This development was crucial for the advancement of science and engineering, as it allowed for the precise calculation of angles and distances in various applications.

Data from historical texts and mathematical treatises confirm the evolution of the coordinate angle formula. For instance, the works of Ptolemy and Archimedes provide evidence of the early use of the unit circle and the coordinate angle formula. The development of calculus in the 17th century is well-documented in the works of Newton and Leibniz, who applied their new mathematical tools to refine the formula.

In conclusion, the coordinate angle formula originated from the need to describe and measure angles in a systematic way, with roots in ancient Greek trigonometry and further developed through the works of Ptolemy, Archimedes, and the advent of calculus. The formula's evolution has been instrumental in the advancement of mathematics and its applications in various fields.