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Using Excel, one can plot these functions and observe how the approximations differ from actual values as the angle increases. The graphical representation shows that at angles exceeding 11 degrees, the approximations can vary significantly from the true values.
Conclusion
Understanding linear approximations of trigonometric functions is a crucial aspect of engineering and physics. This knowledge allows for effective simplification of calculations in various applications, particularly when angles are small. By recognizing the applicability and limitations of these approximations—in particular, the environments in which sine, cosine, and tangent can be simplified to their linear forms—engineers can solve complex problems with greater efficiency.