MA1310 Week 4 Other Trigonometric Functions Lab

MA1310: Week 4 Other Trigonometric Functions This lab requires You To

This assignment involves understanding and graphing various trigonometric functions, including the tangent, cotangent, secant, and cosecant functions. Additionally, it requires the use of inverse trigonometric functions, evaluation using calculators, and application of these concepts to solve problems. The tasks include analyzing the properties of graphs, deriving equations of functions, computing exact and approximate values, and solving real-world problems involving angles and distances.

Paper For Above instruction

Trigonometric functions are fundamental in understanding periodic phenomena, oscillations, and wave behaviors in mathematics and physics. Functions such as tangent, cotangent, secant, and cosecant extend the basic sine and cosine functions, exhibiting unique characteristics that are essential for advanced mathematical analysis and real-world applications.

Graphing Variations of y = tan x

The tangent function, y = tan x, exhibits a periodic behavior with asymptotes where the function is undefined. The general form for a shifted and scaled tangent function is y = A tan(Bx – C). Understanding how to derive and interpret A, B, and C in this context is crucial.

Given the graph with points of y = -1 and y = 1 between asymptotes, the value of A is determined from the amplitude of the tangent's maximum and minimum values. For tangent functions, these extrema are not bounded like sine and cosine, but at points between asymptotes, the tangent takes values approaching ± infinity. Thus, the given y-values suggest the amplitude A essentially scales the tangent graph.

Part a: Find A

The points on the graph indicate that between successive asymptotes, the tangent function reaches y = 1 and y = -1. Since the standard tangent function has no amplitude (it naturally takes all real values), scaling by A modifies the amplitude, leading to A being the coefficient such that the points y = ±1 correspond to the tangent function's scaled maximums.

Part b: Find the period of the function

The period of y = tan(x) is π, but if scaled horizontally by B, the period becomes π/|B|. The period can be found from the distance between two consecutive asymptotes, which are vertical lines where the function is undefined. Measuring this distance on the graph gives the period.

Part c: Find the coefficient B

The period corresponds to the interval between asymptotes, so B can be calculated via B = π/period length.

Part d: Find the phase shift C

The phase shift indicates horizontal translation of the graph. It is calculated based on the location of the asymptotes and the corresponding phase of the standard tangent function.

Part e: Find the value of C

Once the asymptote positions are known, C is derived to shift the standard tangent graph accordingly.

Part f: Write the equation of the tangent graph

The general form, incorporating A, B, and C, follows the structure y = A tan(Bx – C). The reasoning involves analyzing the asymptote positions, the maximum and minimum y-values, and the period to determine the precise coefficients.

Evaluating Inverse Trigonometric Functions

These functions include sin^–1 x, cos^–1 x, and tan^–1 x. They are used to determine angles based on ratio inputs and are critical in applications involving angles of elevation, depression, and positional calculations.

Calculation Example:

Using a calculator, sin^–1 0.47 yields approximately 0.487 radians, rounded to two decimal places as 0.49 radians, which corresponds to about 27.89 degrees (since 1 radian ≈ 57.2958 degrees). This evaluation demonstrates how inverse sine functions convert ratio inputs into angles, facilitating geometric and physical interpretations.

Application Problem: Angle of Elevation

In the given scenario, a point on the ground 90 feet from the base of a building measures the angle of elevation to the top of the 200-foot tall building. Using trigonometric ratios, specifically tangent, the angle θ can be found via:

tan θ = opposite / adjacent = 200 / 90 ≈ 2.222

Evaluating θ = tan^–1(2.222) yields an angle of approximately 65.4 degrees. This calculation demonstrates the utility of inverse tangent functions in real-world applications, such as determining heights or angles in surveying and navigation.

Conclusion

This exploration of trigonometric functions encompasses graph analysis, derivation of function equations, evaluation of inverse functions, and practical applications. Mastery of these concepts is essential for advancing in mathematical understanding and for applying mathematics to solve real-world problems involving angles and periodic phenomena.

References

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