Exploring sine, cosine, and tangent without a calculator can be achieved through geometric diagrams and hand calculations. Techniques such as using Maclaurin series and consulting tables of values were common before calculators became widespread. It's important to note that these functions are transcendental and cannot be expressed as polynomials, which can lead to discrepancies in manual calculations. The small angle approximation is useful for angles less than 0.2 radians, where sine and tangent values are nearly equal to the angle itself. Understanding these functions involves recognizing their definitions on the unit circle, where sine and cosine correspond to the y and x coordinates, respectively.