7 Trigonometric Identities, Complex Exponentials and Periodic Sequences 26 7.1 Basic Trigonometric Identities . . . . . . . . . . . . . . . . . . 26 7.2 Basic Trigonometric Integrals . . . . . . . . . . . . . . . . . . 28 7.3 Basic Properties of Complex Numbers . . . . . . . . . . . . . 29 7.4 Complex Numbers and Trigonometrical Identities . . . . . . . 31 7.5 The Exponential of a Complex Numbe...

this paper presents a modified 32-bit rom-based direct digital frequency synthesizer (ddfs). maximum output frequency of the ddfs is limited by the structure of the accumulator used in the ddfs architecture. the hierarchical pipeline accumulator (hpa) presented in this paper has less propagation delay time rather than the conventional structures. therefore, it results in both higher maximum ope...

Cosines and sines around the unit circle In mathematics, trigonometric identities are equalities that involve trigonometric functions and are true for every single value of the occurring variables. Geometrically, these are identities involving certain functions of one or more angles. They are distinct from triangle identities, which are identities involving both angles and side lengths of a tri...

On page 335 in his lost notebook, Ramanujan recorded without proofs two identities involving finite trigonometric sums and doubly infinite series of Bessel functions. The two identities are intimately connected with the classical circle and divisor problems, respectively. For each of Ramanujan’s identities, there are three possible interpretations for the double series. In two earlier papers, t...

In this work, we use the generalized exponential function in the fractional-order domain to define generalized cosine and sine functions. We then re-visit some important trigonometric identities and generalize them from the narrow integer-order subset to the more general fractional-order domain. It is clearly shown that trigonometric functions and trigonometric identities in the transient-time ...

We calculate the values of the trigonometric functions for angles: π3 and π 6 , by [16]. After defining some trigonometric identities, we demonstrate conventional trigonometric formulas in the triangle, and the geometric property, by [14], of the triangle inscribed in a semicircle, by the proposition 3.31 in [15]. Then we define the diameter of the circumscribed circle of a triangle using the d...