06.02b Applying the Laws of Sines and Cosines

• Respond to the following prompt in a word processing document. Describe, in detail, when to use the law of cosines, the law of sines, and the law of sines with the ambiguous case. Provide general guidelines, in your own words, for each law that can be applied to any triangle situation with which you are presented. To aid in your explanation, you may refer to specific problems from the text.
• Your response must include:
  • A discussion of
    • The law of cosines
    • The law of sines
    • The ambiguous case (law of sines)
  • General guidelines in your own words that can be applied to any triangle.
  • At least 100 words in complete sentences with appropriate grammar and spelling.
  • Provide the source used for information (i.e. the website, the book, etc.). Not providing your source may result in an Academic Integrity violation.

Part 1 - Application An airplane is flying on a bearing of 170 at 495 mph. Find the component form of the velocity of the airplane. Be sure to show and explain your work.

Part 2 - Find the error(s) and solve the problem correctly. Be sure to show and explain your work. An airplane is flying on a compass heading (bearing) of 170 at 460 mph. A wind is blowing with the bearing 200 at 80 mph. a. Find the component form of the velocity of the airplane. b. Find the actual ground speed and direction of the airplane Answer: a. v = 460 = < -453.01, 79.88 > b. Find the wind vector w = 80 = <-75.18, -27.36> Velocity vector = v + w = < -528.19, 52.52> Actual speed | |v + w| | ( 528. 19) 530.79 mph

2 + (52. 52) 2 Actual direction: = 180 + tan )= 95.68

1 ( 528.19 52.52

Part 3 - Discussion Question What determines when you can use the Law of Sines or the Law of Cosines? Sketch an example of your explanation.

Part 4 - Reflection This question required: 1. What new information have you learned from this module? What surprised you about what you learned? What knowledge from your past courses did you use in this module?

Choose 2 of the following questions: 2. Over the course of this module, what concept did you find the most challenging? What did you do to clarify the concepts in this module to further your learning?How did I move through roadblocks or challenges? 3. Describe a specific LSS resource in addition to the course content that you used in this module and how it helped you learn the concepts. 4. Why do students need to understand vectors? What job do you think would use vectors other than a pilot?

Use mathematical induction to prove the statement is true for all positive integers n, or show why it is false. (4 points each.)

1. 46 + 57 + 68 + ... + 4n( 4n + 2) =
2. 1²+ 4²+ 7²+ ... + (3n - 2)²=

For the given statement P_n, write the statements P₁, P_k, and P_k+1. (2 points)

1. 2 + 4 + 6 + . . . + 2n = n(n+1)

Be sure to show and explain all work. 10 points each Application The table below shows the February balance of a simple interest savings account each year from 2015 to 2021 Year 2015 2016 2017 2018 2019 2020 2021 Balance

$12,000 14018 16036 18054 20072 22090 24108

Do the balances form an arithmetic or geometric sequence? What is the d or the r? Write a formula for the balance in the account n years after February 2015. Find the sum of the February balances from 2015 to 2032, inclusive.

Find the error(s) and solve the problem correctly. Determine whether the sequence converges or diverges. How do you know if it converges or diverges? If it converges, give the limit. 5n1 { n+1 } Answer: Diverges because of the degree of the numerator and denominator.

Discussion Question Describe the difference between a recursive rule and an explicit rule when working with sequences.

Reflection This question required: 1. What new information have you learned from this module? What surprised you about what you learned? What knowledge from your past courses did you use in this module?

Choose 2 of the following questions: 2. Does your work in this chapter truly reflect your effort? If so, how? If not, how could you improve? 3. Describe a specific LSS resource in addition to the course content that you used in this module and how it helped you learn the concepts. 4. Write a short story describing your performance and understanding throughout this module. 5. The course resources and your teacher says that note taking is really helpful in mathematics. As you went through this module would you agree or disagree with that? What additional advice would you give to a student that is beginning the course today?"

Determine the conic section given by each of the following equations. Be sure to show all work to find the standard form of the equation of each conic section. 10 points each. Correct conic section (2 points) All work shown (8 points) 1. 3x 2 + 5y 2 12x + 30y = 42 2. x 2 + 3y 2 + 4x = 5 3. x 2 + 2x + 4y = 7 4. 9x 2 + 25y 2 54x 50y 281 = 0 5. 12x 2 4y 2 72x 16y + 44 = 0

Graph each pair of parametric equations. (2 points each)

1. x = 3 sin³t y = 3 cos³t

2. x = 7 sin t + sin 7t y = 7 cos t + cos 7t

2. x = 2t y = t + 5, -2 t 3

2. x = 2t - 1 y = t²+ 5, -4 t 4

2. x = 6 sin t y = 6 cos t, 0 t 2

Part 1 - Application Write an equation for the orbit of Saturn in the form of x 2 a 2 + y 2 b 2 = 1

Part 2 - Find the error(s) and solve the problem correctly. Convert the polar equation to rectangular form and identify the graph. Support your answer by sketching the graph. Show and explain your work. r = 4 cos Answer: Cosine graph reflected over x-axis with amplitude 4

Part 3 - Discussion Question In your own words, explain the discriminant test on page 600 in your ebook. Use the discriminant test to decide whether the equation represents a parabola, ellipse or a hyperbola and explain why you know this is true. x 2 4xy + 3x + 25y 6 = 0

Part 4 - Reflection This question required: 1. Briefly explain a concept in this module and give at least one real world example that demonstrates the concept that you have selected. Share an example not given in the course content or in the application question above.

Choose 2 of the following questions: 2. Over the course of this module, what concept did you find the most challenging? What did you do to clarify the concepts in this module to further your learning?How did I move through roadblocks or challenges? 3. Does your work in this chapter truly reflect your effort? If so, how? If not, how could you improve? 4. The course resources and your teacher say that note taking is really helpful in mathematics. As you went through this module would you agree or disagree with that? What additional advice would you give to a student that is beginning the course today?" 5. What new information have you learned from this module? What surprised you about what you learned? What knowledge from your past courses did you use in this module?

Part 1 - Exploration Let f(x) = 30 1 + 2 3x

a. Use tables and graphs to find and x lim f(x) x lim f(x)

b. Identify any horizontal asymptotes c. What is the y-intercept? d. How is the numerator of the fraction for f related to part b? e. Sketch the graph.

Part 2 - Find the error(s) and solve the problem correctly. Find the derivative, if it exists, of the function at the specified point. f(x) = x at 2 4x + 3 x = 1

Answer: f(1) = 1 = 0 2 4(1) + 3

Part 3 - Discussion Question Explain in your own words how the derivative of a function can be viewed geometrically.

Part 4 - Reflection This question required: 1. Think back over this entire semester. What concept do you still need clarification on? Have you reached out to your teacher about this module? What are 3 ways that you can be successful on your final exam? What assignments do you need to review and study before you take the final exam? Choose 2 of the following questions: 2. Over the course of this module, what concept did you find the most challenging? What did you do to clarify the concepts in this module to further your learning?How did I move through roadblocks or challenges? 3. Briefly explain a concept in this module and give at least one real world example that demonstrates the concept that you have selected. Share an example not given in the course content or in the application question above. 4. What new information have you learned from this module? What surprised you about what you 5. Describe the notebook that you used in this course. Did you stay organized the entire year? What changes would you make as you prepare for your next math course? What advice would you give students that are starting their first day of PreCalculus?