Final Exam Project - Chapter 7: Law of Sine and Cosine As a review, it is recommended that you complete & submit this assignment before taking the exam. Due Date: The original post is due by 11:59pm three (3) days before the Final Exam is due. Replies are due by 11:59pm of the last day that the Final Exam is due. INTRODUCTION Project: This project will enable you to explore various oblique triangles including the familiar right triangle. It will give insight on the relationship between angles of a triangle and the corresponding opposite sides. ACTIVITY/PROCESS AND GRADING (TOTAL = 50 (40 + 5 + 5) POINTS) You will submit your work to Final Exam Project forum in Blackboard. Only submit your work in one of the following ways: • Take a picture of your written project. Make sure it is readable. Upload the image to the discussion forum. • While working through this project, please save your work in a Word document at each step in order to submit the completed file in its entirety. Make sure you answer all questions in complete sentences. • Upload the file to the discussion forum. • To snip/crop & copy an image, pull up your image/photo on the screen: o Mac: Use Command + Shift + 4, click and drag cursor across the part of the image that you want to use. It will take a screenshot of your selected area and automatically save it to your desktop. o Windows: Go to Start Menu>>All Programs>>Accessories>>Snipping Tool. Drag the cursor around the area that you want to capture. Name and save to your desktop. This assignment is REQUIRED and will only be graded if resources and conclusion are part of the project. The point values for each section are noted below with an additional 10 points for replies to classmates. You are required to review at least two classmates’ projects and post a substantive reply to each (5 points for each reply for up to a total of 10 points). “Good Job” or “I didn’t think of that” will not do. You must post a follow-up question, an observation, make a suggestion, or apply some additional insight to what your classmate has posted. It is NOT your place to point out or correct errors. If you find an error that needs correcting, email your instructor for verification and the instructor will contact the student if your observation is correct. EXPLORING RATIOS: 10 Points In the Word document, add a heading called Exploring Ratios. Write the answers for this section under the heading. Continue to update your document as you continue through each section and submit one document at the end of this project. The Law of Sines is a relationship among angles of a triangle and their corresponding opposite sides. This law states that the ratio of the sine of any of the interior angles of a triangle to the length of the side opposite the angle is the same for all three interior angles. In other words, sin 𝐴 sin 𝐵 sin 𝐶 = = 𝑎 𝑏 𝑐 Final Exam Project - Chapter 7: Law of Sine and Cosine As a review, it is recommended that you complete & submit this assignment before taking the exam. Due Date: The original post is due by 11:59pm three (3) days before the Final Exam is due. Replies are due by 11:59pm of the last day that the Final Exam is due. In the equation a, b, and c are the lengths of the sides that are opposite of their angles A, B, C. In order to understand a ratio, let’s consider a ratio in everyday life. Determine how much one donut costs if a dozen donuts costs $4.20. In order to answer this question we will use ratios. The numerators will be the same units as well as the denominators. Our problem becomes $4.20/12 = $x/1 and therefore, one donut costs $.35 cents. In the same way that this problem uses ratios, ratios are used in trigonometry in the Law of Sines (notice both equations have fractions set equal to each other). 1) Using ratios, determine how much 6 donuts cost. Notice that when the denominator increases, the value of the unknown in the numerator increases. Now assume that donuts double in cost. Change the ratio to show this. If the denominator of the known value is increased by a factor of 2, what happens to our unknown? 2) Setup and solve an everyday problem using ratios to solve it. Keeping the fraction you have created, change one value on the side that has the unknown and solve the problem again. 3) As we change that one value, what happens to the unknown? Explain the various possibilities. EXPLORING THE LAW OF SINES WITH KNOWLEDGE WE KNOW: 15 points • • • • • To see this relationship, use the Wolfram Demonstrations Project website (http://www.wolfram.com/). Click on the “Try the Interactive CDF examples” link under Professional & Enterprise column on the left of the page (http://www.wolfram.com/cdf/uses-examples/?fp=left). Note: You may need to download the CDF player first. Scroll to the middle of the page & click on the red “Interact Now: Get the free Wolfram CDF Player” button. On the CDF Player page (http://www.wolfram.com/cdf-player/ or http://www.wolfram.com/cdfplayer/plugin/success.html?platform=WIN), click on “Explore demonstrations now” link at the bottom left of the page. Under the heading Wolfram Demonstrations Project, search for The Law of Sines and choose The Law of Sines. In Word, add a heading called Exploring the Law of Sines. Write the answers below. Submit one document at the end of this project. 1) To begin with information we know, move the dot at B and C (to change the angles) so that A becomes 90 degrees. Recall the value of sin (90) in order to get the triangle to look right. Draw the resulting triangle and label the values for all other angles and sides. Use the Pythagorean Theorem to verify that this triangle is a right triangle using values rounded to the hundredth digit. Note: This may not be an exact match but it should be very close. Use the snipping tool and capture this picture to place in Word. Add your algebraic work. 2) Keeping A equal to 90 degrees, increase α. Again draw the resulting triangle and label the values for all other angles and sides. Use the Pythagorean Theorem to verify that this is a right triangle using values rounded to the hundredth digit. Note: This may not be an exact match but it should be very close. Use the snipping tool and capture this picture to place in Word. Add your algebraic work. 3) Now change the angle of A so that it is not a right triangle. Draw the resulting triangle and label the values for all other angles and sides. Use the Pythagorean Theorem to verify that is not a Final Exam Project - Chapter 7: Law of Sine and Cosine As a review, it is recommended that you complete & submit this assignment before taking the exam. Due Date: The original post is due by 11:59pm three (3) days before the Final Exam is due. Replies are due by 11:59pm of the last day that the Final Exam is due. right triangle using values rounded to the hundredth digit. Use the snipping tool and capture this picture to place in Word. Add your algebraic work. 4) Use the Law of Sines to write the three equivalent ratios of your oblique triangle. Explain how ratios are working in this problem as they did in the problems above. Use the snipping tool and capture this picture to place in Word. Add your algebraic work. EXPLORING THE LAW OF COSINES: 15 points In Word, add a heading called Exploring the Law of Cosines and answer the questions below. The Law of Cosines relates the three side lengths of a triangle with the cosine of one of its angles. If this angle, call it A, is formed by sides b and c and the opposite side a, then according to the Law of Cosines, 𝑎 + = 𝑏 + + 𝑐 + − 2𝑏𝑐 cos 𝐴. In the equation, a, b, and c are the lengths of the sides that are opposite of their angles A, B, C. • • To see this relationship, use the Wolfram Demonstrations Project website (http://www.wolfram.com/). Under the heading Wolfram Demonstrations Project, search for Law of Cosines and choose Law of Cosines. 1) Let’s again start with something we know. Make angle A equal to 90 degrees. Recall the value of cos (90) in order to get the triangle to look right. Draw the resulting triangle and label the values for all other angles and sides except a. Using the Law of Cosines complete the algebra to find the value for a. Since A = 90 degrees, in a complete sentence answer the question what familiar formula does the Law of Cosines represent? Use the snipping tool to capture the picture and place it in Word under the heading Exploring the Law of Cosines. 2) Make the angle A an obtuse or acute triangle. Draw the resulting triangle and label the values for all other angles and sides except a. Using the Law of Cosines, complete the algebra to solve for a. Does this value match the value of 𝛼 shown on the demonstration? Use the snipping tool to capture the picture and place it in Word under the heading Exploring the Law of Cosines. 3) Create a real life example for your values. Add drawings or pictures to the significant landmarks as well as the measurements. If needed, reference the internet or Chapter 7.2 of your textbook for ideas. CONCLUSION & RESOURCES Write a summary (minimum of 3 sentences) of what you learned doing this project. Remember to list any resources you used for this project including books and or internet sites.
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