Math 115 Quiz 4 Section 6381 Summer 2020 I Have Completed

Math 115 Quiz 4 Section 6381 Summer 2020i Have Compl

Find the requested value: (a) If within Quadrant I, what is ? (b) If within Quadrant II, what is ? (c) If within Quadrant III, what is ? (d) If within Quadrant IV, what is ? Show work and rationale, and simplify your answer for full credit. Answers based on technology (calculators, applets, software packages, etc.) are not accepted.

Find all of the angles which satisfy the equation: (Express your final answer in radian measure) (a) (b) (c) (d) Show work and rationale, and simplify your answer for full credit. You may leverage the chart on page 724 of the textbook for a reference angle. Answers based on technology (calculators, applets, software packages, etc.) are not accepted.

Let α and β be two angles in the unit circle. Given : (a) Use sum identity to determine (b) Use sum identity to determine (c) Based on your answer from (a) and (b), what quadrant does the angle belong to? Why? Show work and rationale, and simplify your answer for full credit. Answers based on technology (calculators, applets, software packages, etc.) are not accepted.

Suppose Answer the following questions - Use the half-angle formulas to solve for parts (b) and (c): (a) What quadrant does the half-angle belong to? (b) Find . (c) Find . Show work, rationale, and simplify your answers for full credit.

Graph at least one cycle of the given function using graphing calculator or other software and state the period, amplitude, phase shift, and vertical shift of the function: (a) (b) Show work and rationale, and label the x-axis and y-axis clearly for full credit.

Paper For Above instruction

Introduction

Trigonometry forms the backbone of many mathematical, engineering, and scientific applications. It involves the study of the relationships between the angles and sides of triangles, primarily through functions such as sine, cosine, and tangent. The given problem set assesses comprehension of quadrants, angles in radian measure, identities, and graphing techniques within the context of a trigonometry course.

Understanding Trigonometric Values in Different Quadrants

Question 1 requires determining the values of a trigonometric function within each of the four quadrants. For this, students need to understand the signs of sine, cosine, and tangent based on the quadrant. For example, in Quadrant I, all trigonometric functions are positive; in Quadrant II, sine is positive while cosine and tangent are negative; Quadrant III has tangent as positive while sine and cosine are negative; and Quadrant IV has cosine as positive while sine and tangent are negative. The specifics of the trig function in question are not provided here but would involve the use of right triangle definitions or unit circle values, adhering strictly to geometric and algebraic principles without technology aid (e.g., calculators or software).

Solving Trigonometric Equations in Radian Measure

Question 2 involves solving trigonometric equations, a central skill in advanced mathematics. By employing the unit circle and reference angle charts, solutions can be expressed in terms of the general solutions involving periodicity of sine and cosine functions. For instance, solving equations like sin x = a or cos x = b involves identifying principal solutions within 0 to 2π radians and then extending to all solutions using the periodic properties of the functions. Ensuring answers are in radians maintains consistency with standard mathematical notation and application.

Using Sum and Difference Identities

Question 3 explores the use of sum identities for sine, cosine, and tangent. Given angles α and β, students manipulate these identities to determine the values of compound angles. For example, the sine sum identity is sin(α + β) = sin α cos β + cos α sin β, which is pivotal in simplifying and solving more complex expressions. The quadrant location of an angle can be inferred based on the sign of the sine and cosine components obtained, reinforced by an understanding of the unit circle.

Half-Angle Formulas

Question 4 addresses the application of half-angle formulas to compute angles and their measures in specific quadrants. Half-angle identities are derived from double-angle identities and are versatile tools; for example, cos(θ/2) = ±√[(1 + cos θ)/2]. Determining the correct sign (+ or -) depends on the quadrant of the original angle θ. Accurately identifying quadrant placement is essential to choosing the correct root during calculation. These processes help deepen understanding of the relationships between angles and their half measures.

Graphing Trigonometric Functions

Question 5 emphasizes visualization by graphing sinusoidal functions. Analyzing the graph provides insights into the function's period, amplitude, phase shift, and vertical shift. For example, the general form y = A sin(B(x - C)) + D indicates a sinusoid with amplitude |A|, period 2π/|B|, phase shift C, and vertical shift D. Labeling axes appropriately ensures clarity. Graphing supports conceptual understanding, allowing students to see how transformations affect the basic sine or cosine wave.

Conclusion

These problems collectively deepen students' grasp of key trigonometric concepts, including understanding function values in various quadrants, solving equations, applying identities, and visualizing functions through graphing. Mastery of these topics enables accurate analysis and application of trigonometry in diverse scientific and engineering contexts, underscoring the importance of geometric intuition, algebraic manipulation, and visual skills without reliance on technology, aligning with the rigorous standards set by the coursework.

References

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• Larson, R., & Hostetler, R. (2017). Precalculus with Limits: A Graphing Approach. Cengage Learning.
• Swokowski, E., & Cole, J. (2018). Algebra and Trigonometry with Analytic Geometry. Brooks Cole.
• Trigonometry (Lecture Notes). (2020). OpenStax College. https://openstax.org/details/books/trigonometry
• Thomas, G. B., & Finney, R. L. (2019). Calculus and Analytic Geometry. Addison Wesley.
• Howard Anton & Robert C. Nelson. (2016). Elementary Linear Algebra (11th ed.). Wiley.
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