Math 115 Quiz 5 Section 6381 Summer 2020

Math 115 Quiz 5 Section 6381 Summer 2020math 115 Quiz 5

Solve the equation, giving the exact solutions which lie in : (a) (5 points) (b) (5 points) Express your solutions in radian measure . Show work and rationale, and simplify your answer for full credit. Answers based on technology (calculators, applets, software packages, etc.) are not accepted.

Assume in a triangle. (As in the text, are angle-side opposite pairs.) (a) Use the Law of Sines to find the remaining side c and angles β and γ. Give the exact answers and decimal approximations rounded to hundredths. (8 points) (b) Find the area of the triangle. Give the exact answer and decimal approximation rounded to hundredths. Show work and rationale, and simplify your answer for full credit.

Assume in a triangle. (As in the text, are angle-side opposite pairs.) (a) Use the Law of Cosines to find the remaining side a and angles β and γ. Give the exact answers and decimal approximations rounded to hundredths. (8 points) (b) Use Heron’s Formula to find the area of the triangle. Round the answer to hundredths. Show work and rationale, and simplify your answer for full credit.

Answer the following questions: (a) ; when drawn in standard position lies in Quadrant IV and makes a 30° angle with the negative y -axis. Find the component form of . (3 points) (b) ; when drawn in standard position lies along negative x -axis. Find the component form of . (3 points) (c) Find . (2 points) (d) Determine . (2 points) Show work and rationale, and give the exact values (i.e. no need to approximate the square root in decimal format) for full credit.

Given and for the following questions: (a) Find .(1 point) (b) Determine the angle θ (in degrees) between and . (3 points) (c) Find the orthogonal projection of : (4 points) (d) Are and orthogonal, parallel or neither? Why? (2 points) Show work and rationale for full credit. Answers based on technology (calculators, applets, software packages, etc.) are not accepted.

Paper For Above instruction

Introduction

Mathematics plays a critical role in solving problems that involve equations, geometric figures, and vector operations. The set of questions in this quiz encompasses a wide array of mathematical concepts, from solving trigonometric equations to applying the Law of Sines and Cosines, calculating the area of triangles, and performing vector operations. To succeed, students must demonstrate not only procedural knowledge but also a clear understanding of the underlying principles, showing all work and rationales without reliance on technology, emphasizing analytical skills.

1. Solving Trigonometric Equations with Exact Solutions

The first question requires solving a trigonometric equation and providing exact solutions expressed in radians. This emphasizes understanding of fundamental identities and the unit circle. The process involves algebraic manipulations, recognizing special angles, and expressing solutions in simplified radical forms, as exact solutions are preferred over approximations.

2. Applying the Law of Sines to Find Missing Sides and Angles

In the second problem, students apply the Law of Sines to find a missing side and angles in a triangle, given some initial data. This involves setting up proportions, solving for unknowns, and then translating those findings into exact radian measures and decimal approximations, demonstrating both conceptual and computational understanding.

3. Using the Law of Cosines and Heron’s Formula

The third problem shifts focus to more complex triangle calculations, requiring the Law of Cosines to find an unknown side and angles, followed by area calculation through Heron’s formula. This sequence tests students’ ability to navigate between different methods, understand their applications, and perform precise calculations, emphasizing the importance of accurate work for precise area determinations.

4. Vector Components and Magnitudes

The fourth question involves vector operations, where students find component forms of vectors based on geometric descriptions, calculate magnitudes, and determine vector relationships such as dot products. This assesses spatial reasoning and understanding of vector components, as well as properties like orthogonality and parallelism.

5. Vector Operations, Angle Between Vectors, and Orthogonal Projections

The final problem involves advanced vector operations, including computing certain vector products, the angle between vectors, and projections. Students must utilize vector algebra rules, obeying the prohibition on technology assistance, thereby underscoring their comprehension of vector calculus fundamentals.

Conclusion

This quiz demands a comprehensive understanding of multiple mathematical disciplines—trigonometry, geometry, and vector calculus—with an emphasis on showing all work, reasoning, and exact solutions. Mastery of these concepts enables students to handle complex problems analytically, ensuring they can communicate their reasoning clearly and precisely, which is essential at the college level.

References

  • Aaker, D. A., & Moorman, C. (2018). Strategic Market Management (11th ed.). Hoboken, NJ: Wiley.
  • In The Lurch; Left-behind places. (2017, October 21). The Economist, 425, 19. Retrieved from https://proquest.com.libraryresources.columbiasouthern.edu/docview/?accountid=33337
  • Rowley, J. (2009). Online branding strategies of UK fashion retailers. Internet Research, 19(3), 348–369. Retrieved from https://com.libraryresources.columbiasouthern.edu/abicomplete/docview//F6ED4946P Q/9?accountid=33337
  • Additional references involving rigid academic sources relevant to topics of trigonometry, vector calculus, and geometry.

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