This mathematical work is a new theory, which complements the existing theories of the Theodorus, Archimedean, Logarithmic and Hyperbolic spirals. Starting from assumptions that a radius, an apothem, an arrow, a chord, a circular sector, are growing or decreasing, in the space of a plane of two or three dimensions in a spiral or exponential or other form. Just as a circle is having partitions to generate its wave function and wavelets. Applying the graphical and mathematical methodology, to draw triangles where all the basic trigonometric identities can be defined. With this new methodology, new trigonometric identities were found for the mean angles as well as two new mathematical equations for the circular sector. Also, laws governing trigonometric spirals were discovered. These laws define trigonometric spirals, thus distinguishing them from the current spirals.
It is also shown how the radius is inversely proportional to the sine or cosine of its angle. These equations indicate which of the two ways can be taken, to analyze, or to make the graphs of the trigonometric spirals. This theory leads us to analyze at infinity each of the properties of a circle, in order to observe how its calculations and graphs are.