A series of new formulas to approximate the Sine and Cosine functions

In our analysis, we approximate the sine and cosine trigonometric functions within the interval (cid:2) 0, π 2 (cid:3) . This examination yields two distinct formulas for approximating sine and cosine. Initially, we endeavor to derive the formula that involves a square root, and then we develop an alternative formula that does not require any use of a square root. However, even after establishing the square root-free procedure, we seek to enhance its accuracy and subsequently derive yet another formula for more precise approximations of trigonometric functions within the interval (cid:2) 0, π 2 (cid:3) . Thus, our analysis presents two primary procedures. One is utilizing square roots, and the other abstaining from them. Our focus extends to ensuring the accuracy of these trigonometric functions specifically within the interval (cid:2) 0, π 2 (cid:3) . This precision assessment is visually represented through graphs, illustrating the disparity between the values generated by the formulated functions and the exact values of these functions. Consequently, these graphs serve as indicators of the error within the interval (cid:2) 0, π 2 (cid:3) . Finally, we conduct a comparative analysis between our approximation and the 7th-century Indian mathematician Bhaskara I’s approximation formula.