Ma1310 Week 4: Other Trigonometric Functions This Lab Requir

Ma1310 Week 4 Other Trigonometric Functionsthis Lab Requires You To

This assignment involves understanding and graphing variations of basic trigonometric functions, specifically y = tan x, y = cot x, y = csc x, and y = sec x. Additionally, the task includes understanding and applying inverse sine, cosine, and tangent functions using a calculator to evaluate inverse functions and find exact values of composite functions. The final goal is to solve applied problems involving these functions.

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Trigonometric functions play a fundamental role in various branches of mathematics and applied sciences. These functions—sine, cosine, tangent, cosecant, secant, and cotangent—are essential for modeling periodic phenomena, analyzing angles, and solving geometric problems. Their variations and transformations, especially shifts and stretches, enable handling complex real-world situations such as wave motions, signal processing, and structural analysis.

In the context of this lab, the primary focus is on understanding the behavior of the tangent function and its transformations. The tangent function, y = tan x, exhibits periodic behavior with vertical asymptotes occurring at points where cosine equals zero. Variations in y = tan(Bx - C) involve amplitude scaling, phase shifts, and period modifications. Similarly, understanding y = cot x, y = csc x, and y = sec x supports comprehensive analysis of all basic trigonometric functions.

Graphing of y = tan x and its variations involves recognizing the period, amplitude (though tangent has no maximum amplitude but is unbounded), and shifts. For example, the general form y = A tan(Bx - C) encompasses transformations such as vertical stretch/shrink (A), horizontal compression/expansion (B), and phase shifts (C). The period of the tangent function is given by π/|B|, which depends on the value of B, while the phase shift is determined by C.

Understanding inverse trigonometric functions requires familiarity with the inverse sine (arcsin), cosine (arccos), and tangent (arctan). These functions are instrumental in solving equations where the angle must be determined from a ratio—such as in physics for projectile motion or in engineering for signal phase shifts. Using a calculator, precise evaluations of these functions facilitate solving complex problems, especially when exact algebraic solutions are difficult or impossible.

An example problem involves calculating the value of sin-1 0.47. This requires inputting the value into a calculator set to the correct mode (radians or degrees) depending on the context. The result provides an angle measure representative of the ratio used in the inverse sine calculation, crucial for solving real-world triangle problems, such as finding the angle of elevation or depression.

Application example: angle of elevation involves right triangle trigonometry where, given a distance (ground level) and height (building), the tangent function relates these quantities to the angle of elevation. Solving such problems involves setting up equations, using inverse functions, and applying known trigonometric identities.

Overall, mastery of these functions and their transformations enhances problem-solving skills in mathematics and beyond. This assignment requires a combination of graphing, algebraic manipulation, calculator use, and application of concepts to real-world scenarios to develop a comprehensive understanding of trigonometric methods.

References

  • Anton, H., Bivens, I., & Davis, S. (2016). Calculus: Early Transcendentals (10th ed.). Wiley.
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  • Thomas, G. B., Weir, M. D., & Hass, J. (2014). Thomas' Calculus (13th ed.). Pearson.
  • Stewart, J. (2015). Calculus: Early Transcendentals (8th ed.). Cengage Learning.
  • Kievit, R., & Clegg, M. (2017). Trigonometry: A Unit Circle Approach. Pearson.
  • Koening, A., & Spackman, L. K. (2020). Trigonometry and Its Applications. Wiley.
  • Redden, J., & Bellini, M. (2018). Applied Trigonometry for Engineers. Springer.
  • Hewitt, P. G. (2014). Conceptual Physics (12th ed.). Pearson.
  • Myer, D. C. (2019). Engineering Mathematics with Applications. Oxford University Press.
  • Brown, R. F., & Hooper, P. (2015). Using Trigonometric Functions in Modeling and Applications. Academic Press.

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