A New Approach to Learn Trigonometry

TL;DR

The paper introduces the Primary Gasing Triangle framework, defining $\sin$ and $\cos$ as projections of a unit vector at angle $\theta$ and deriving the remaining trig functions via similarity-based scaling of two derived triangles. It provides trig identities, a trig-based proof of the Pythagorean theorem, and geometric derivations of the sum/difference formulas and the sine and cosine rules, all within a coherent, physically meaningful construction. The approach extends definitions to obtuse angles with a clear Cartesian-projection interpretation, addressing the limitations of unit-circle reasoning and aiming to deepen conceptual understanding for waves, vibrations, and other periodic phenomena. Overall, the method emphasizes visual, scalable reasoning over memorization, with practical problem-solving guidance and appendices containing multiple proofs.

Abstract

We introduce the Primary Gasing Triangle, a right triangle with a hypotenuse of 1 unit, to define the primary trigonometric functions: sine and cosine. This triangle serves as the foundational element in a new approach to learning trigonometry, enabling us to derive the Derived Gasing Triangle, where the other four trigonometric functions (tangent, secant, cotangent, and cosecant) are defined. Using the Primary Gasing Triangle, we derive key trigonometric formulas, provide several proofs of the Pythagorean theorem using trigonometry, and solve various trigonometry problems. This approach makes learning trigonometry simpler, easier, and more intuitive.

Figures (50)

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• Figure 5: The six trigonometric functions
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